Structure real_topologyTheory
signature real_topologyTheory =
sig
type thm = Thm.thm
(* Definitions *)
val CLOSED_interval : thm
val OPEN_interval : thm
val at_def : thm
val at_infinity : thm
val at_neginfinity : thm
val at_posinfinity : thm
val ball_def : thm
val between : thm
val bilinear : thm
val bounded_def : thm
val cauchy_def : thm
val cball : thm
val closed_segment : thm
val closest_point : thm
val closure : thm
val collinear : thm
val compact : thm
val complete : thm
val components : thm
val condensation_point_of : thm
val connected : thm
val connected_component : thm
val content : thm
val continuous : thm
val continuous_on : thm
val dependent : thm
val dim : thm
val dist_def : thm
val euclidean_closed_def : thm
val euclidean_def : thm
val euclidean_open_def : thm
val eventually : thm
val frontier : thm
val hausdist : thm
val homeomorphic : thm
val homeomorphism : thm
val in_direction : thm
val independent : thm
val interior : thm
val interval_lowerbound : thm
val interval_upperbound : thm
val is_interval : thm
val isnet : thm
val limit_point_of_def : thm
val linear : thm
val locally : thm
val midpoint : thm
val net_TY_DEF : thm
val netlimit : thm
val open_segment : thm
val reallim : thm
val sequentially : thm
val set_diameter_def : thm
val set_dist_def : thm
val span : thm
val sphere : thm
val subspace : thm
val suminf_def : thm
val summable_def : thm
val sums_def : thm
val tendsto_real : thm
val trivial_limit : thm
val uniformly_continuous_on : thm
val within : thm
(* Theorems *)
val ABS_CAUCHY_SCHWARZ_ABS_EQ : thm
val ABS_CAUCHY_SCHWARZ_EQ : thm
val ABS_CAUCHY_SCHWARZ_EQUAL : thm
val ABS_SUM_TRIVIAL_LEMMA : thm
val ABS_TRIANGLE_EQ : thm
val ABS_TRIANGLE_LE : thm
val AFFINITY_INVERSES : thm
val ALWAYS_EVENTUALLY : thm
val APPROACHABLE_LT_LE : thm
val AT : thm
val AT_INFINITY : thm
val AT_NEGINFINITY : thm
val AT_POSINFINITY : thm
val BAIRE : thm
val BAIRE_ALT : thm
val BALL : thm
val BALL_EMPTY : thm
val BALL_EQ_EMPTY : thm
val BALL_INTERVAL : thm
val BALL_INTERVAL_0 : thm
val BALL_LINEAR_IMAGE : thm
val BALL_MAX_UNION : thm
val BALL_MIN_INTER : thm
val BALL_SCALING : thm
val BALL_SUBSET_CBALL : thm
val BALL_TRANSLATION : thm
val BALL_TRIVIAL : thm
val BALL_UNION_SPHERE : thm
val BANACH_FIX : thm
val BASIS_CARD_EQ_DIM : thm
val BASIS_EXISTS : thm
val BASIS_HAS_SIZE_DIM : thm
val BETWEEN_ABS : thm
val BETWEEN_ANTISYM : thm
val BETWEEN_IMP_COLLINEAR : thm
val BETWEEN_IN_SEGMENT : thm
val BETWEEN_MIDPOINT : thm
val BETWEEN_REFL : thm
val BETWEEN_REFL_EQ : thm
val BETWEEN_SYM : thm
val BETWEEN_TRANS : thm
val BETWEEN_TRANS_2 : thm
val BIGUNION_COMPONENTS : thm
val BIGUNION_CONNECTED_COMPONENT : thm
val BIGUNION_DIFF : thm
val BIGUNION_MONO : thm
val BIGUNION_MONO_IMAGE : thm
val BILINEAR_BOUNDED : thm
val BILINEAR_BOUNDED_POS : thm
val BILINEAR_CONTINUOUS_COMPOSE : thm
val BILINEAR_CONTINUOUS_ON_COMPOSE : thm
val BILINEAR_DOT : thm
val BILINEAR_LADD : thm
val BILINEAR_LMUL : thm
val BILINEAR_LNEG : thm
val BILINEAR_LSUB : thm
val BILINEAR_LZERO : thm
val BILINEAR_RADD : thm
val BILINEAR_RMUL : thm
val BILINEAR_RNEG : thm
val BILINEAR_RSUB : thm
val BILINEAR_RZERO : thm
val BILINEAR_SUM : thm
val BILINEAR_SUM_PARTIAL_PRE : thm
val BILINEAR_SUM_PARTIAL_SUC : thm
val BILINEAR_SWAP : thm
val BILINEAR_UNIFORMLY_CONTINUOUS_ON_COMPOSE : thm
val BOLZANO_WEIERSTRASS : thm
val BOLZANO_WEIERSTRASS_CONTRAPOS : thm
val BOLZANO_WEIERSTRASS_IMP_BOUNDED : thm
val BOLZANO_WEIERSTRASS_IMP_CLOSED : thm
val BOUNDED_BALL : thm
val BOUNDED_BIGINTER : thm
val BOUNDED_BIGUNION : thm
val BOUNDED_CBALL : thm
val BOUNDED_CLOSED_CHAIN : thm
val BOUNDED_CLOSED_IMP_COMPACT : thm
val BOUNDED_CLOSED_INTERVAL : thm
val BOUNDED_CLOSED_NEST : thm
val BOUNDED_CLOSURE : thm
val BOUNDED_CLOSURE_EQ : thm
val BOUNDED_COMPONENTWISE : thm
val BOUNDED_DECREASING_CONVERGENT : thm
val BOUNDED_DIFF : thm
val BOUNDED_DIFFS : thm
val BOUNDED_EMPTY : thm
val BOUNDED_EQ_BOLZANO_WEIERSTRASS : thm
val BOUNDED_FRONTIER : thm
val BOUNDED_HAS_INF : thm
val BOUNDED_HAS_SUP : thm
val BOUNDED_INCREASING_CONVERGENT : thm
val BOUNDED_INSERT : thm
val BOUNDED_INTER : thm
val BOUNDED_INTERIOR : thm
val BOUNDED_INTERVAL : thm
val BOUNDED_LINEAR_IMAGE : thm
val BOUNDED_NEGATIONS : thm
val BOUNDED_PARTIAL_SUMS : thm
val BOUNDED_POS : thm
val BOUNDED_POS_LT : thm
val BOUNDED_SCALING : thm
val BOUNDED_SING : thm
val BOUNDED_SPHERE : thm
val BOUNDED_SUBSET : thm
val BOUNDED_SUBSET_BALL : thm
val BOUNDED_SUBSET_CBALL : thm
val BOUNDED_SUBSET_CLOSED_INTERVAL : thm
val BOUNDED_SUBSET_CLOSED_INTERVAL_SYMMETRIC : thm
val BOUNDED_SUBSET_OPEN_INTERVAL : thm
val BOUNDED_SUBSET_OPEN_INTERVAL_SYMMETRIC : thm
val BOUNDED_SUMS : thm
val BOUNDED_SUMS_IMAGE : thm
val BOUNDED_SUMS_IMAGES : thm
val BOUNDED_TRANSLATION : thm
val BOUNDED_TRANSLATION_EQ : thm
val BOUNDED_UNIFORMLY_CONTINUOUS_IMAGE : thm
val BOUNDED_UNION : thm
val CARD_EQ_BALL : thm
val CARD_EQ_CBALL : thm
val CARD_EQ_EUCLIDEAN : thm
val CARD_EQ_INTERVAL : thm
val CARD_EQ_OPEN : thm
val CARD_EQ_REAL : thm
val CARD_EQ_REAL_IMP_UNCOUNTABLE : thm
val CARD_FRONTIER_INTERVAL : thm
val CARD_GE_DIM_INDEPENDENT : thm
val CARD_STDBASIS : thm
val CAUCHY : thm
val CAUCHY_CONTINUOUS_EXTENDS_TO_CLOSURE : thm
val CAUCHY_CONTINUOUS_IMP_CONTINUOUS : thm
val CAUCHY_CONTINUOUS_UNIQUENESS_LEMMA : thm
val CAUCHY_IMP_BOUNDED : thm
val CAUCHY_ISOMETRIC : thm
val CBALL_DIFF_BALL : thm
val CBALL_DIFF_SPHERE : thm
val CBALL_EMPTY : thm
val CBALL_EQ_EMPTY : thm
val CBALL_EQ_SING : thm
val CBALL_INTERVAL : thm
val CBALL_INTERVAL_0 : thm
val CBALL_LINEAR_IMAGE : thm
val CBALL_MAX_UNION : thm
val CBALL_MIN_INTER : thm
val CBALL_SCALING : thm
val CBALL_SING : thm
val CBALL_TRANSLATION : thm
val CBALL_TRIVIAL : thm
val CENTRE_IN_BALL : thm
val CENTRE_IN_CBALL : thm
val CLOPEN : thm
val CLOPEN_BIGUNION_COMPONENTS : thm
val CLOPEN_IN_COMPONENTS : thm
val CLOSED : thm
val CLOSED_APPROACHABLE : thm
val CLOSED_AS_GDELTA : thm
val CLOSED_BIGINTER : thm
val CLOSED_BIGINTER_COMPACT : thm
val CLOSED_BIGUNION : thm
val CLOSED_CBALL : thm
val CLOSED_CLOSURE : thm
val CLOSED_COMPACT_DIFFERENCES : thm
val CLOSED_COMPACT_SUMS : thm
val CLOSED_COMPONENTS : thm
val CLOSED_CONNECTED_COMPONENT : thm
val CLOSED_CONTAINS_SEQUENTIAL_LIMIT : thm
val CLOSED_DIFF : thm
val CLOSED_DIFF_OPEN_INTERVAL : thm
val CLOSED_EMPTY : thm
val CLOSED_FIP : thm
val CLOSED_FORALL : thm
val CLOSED_FORALL_IN : thm
val CLOSED_HALFSPACE_COMPONENT_GE : thm
val CLOSED_HALFSPACE_COMPONENT_LE : thm
val CLOSED_HALFSPACE_GE : thm
val CLOSED_HALFSPACE_LE : thm
val CLOSED_HYPERPLANE : thm
val CLOSED_IMP_FIP : thm
val CLOSED_IMP_FIP_COMPACT : thm
val CLOSED_IMP_LOCALLY_COMPACT : thm
val CLOSED_IN : thm
val CLOSED_INJECTIVE_IMAGE_SUBSPACE : thm
val CLOSED_INJECTIVE_LINEAR_IMAGE : thm
val CLOSED_INJECTIVE_LINEAR_IMAGE_EQ : thm
val CLOSED_INSERT : thm
val CLOSED_INTER : thm
val CLOSED_INTERVAL : thm
val CLOSED_INTERVAL_EQ : thm
val CLOSED_INTERVAL_IMAGE_UNIT_INTERVAL : thm
val CLOSED_INTERVAL_LEFT : thm
val CLOSED_INTERVAL_RIGHT : thm
val CLOSED_INTER_COMPACT : thm
val CLOSED_IN_CLOSED : thm
val CLOSED_IN_CLOSED_EQ : thm
val CLOSED_IN_CLOSED_INTER : thm
val CLOSED_IN_CLOSED_TRANS : thm
val CLOSED_IN_COMPACT : thm
val CLOSED_IN_COMPACT_EQ : thm
val CLOSED_IN_COMPONENT : thm
val CLOSED_IN_CONNECTED_COMPONENT : thm
val CLOSED_IN_INTER_CLOSED : thm
val CLOSED_IN_INTER_CLOSURE : thm
val CLOSED_IN_LIMPT : thm
val CLOSED_IN_REFL : thm
val CLOSED_IN_SING : thm
val CLOSED_IN_SUBSET_TRANS : thm
val CLOSED_IN_TRANS : thm
val CLOSED_IN_TRANS_EQ : thm
val CLOSED_LIMPT : thm
val CLOSED_LIMPTS : thm
val CLOSED_MAP_CLOSURES : thm
val CLOSED_MAP_FROM_COMPOSITION_INJECTIVE : thm
val CLOSED_MAP_FROM_COMPOSITION_SURJECTIVE : thm
val CLOSED_MAP_IFF_UPPER_HEMICONTINUOUS_PREIMAGE : thm
val CLOSED_MAP_IMP_OPEN_MAP : thm
val CLOSED_MAP_IMP_QUOTIENT_MAP : thm
val CLOSED_MAP_OPEN_SUPERSET_PREIMAGE : thm
val CLOSED_MAP_OPEN_SUPERSET_PREIMAGE_EQ : thm
val CLOSED_MAP_OPEN_SUPERSET_PREIMAGE_POINT : thm
val CLOSED_MAP_RESTRICT : thm
val CLOSED_NEGATIONS : thm
val CLOSED_OPEN_INTERVAL : thm
val CLOSED_POSITIVE_ORTHANT : thm
val CLOSED_SCALING : thm
val CLOSED_SEGMENT_LINEAR_IMAGE : thm
val CLOSED_SEQUENTIAL_LIMITS : thm
val CLOSED_SING : thm
val CLOSED_SPHERE : thm
val CLOSED_STANDARD_HYPERPLANE : thm
val CLOSED_SUBSET : thm
val CLOSED_SUBSET_EQ : thm
val CLOSED_SUBSTANDARD : thm
val CLOSED_UNION : thm
val CLOSED_UNION_COMPACT_SUBSETS : thm
val CLOSED_UNIV : thm
val CLOSEST_POINT_EXISTS : thm
val CLOSEST_POINT_IN_FRONTIER : thm
val CLOSEST_POINT_IN_INTERIOR : thm
val CLOSEST_POINT_IN_SET : thm
val CLOSEST_POINT_LE : thm
val CLOSEST_POINT_REFL : thm
val CLOSEST_POINT_SELF : thm
val CLOSURE_APPROACHABLE : thm
val CLOSURE_BALL : thm
val CLOSURE_BIGINTER_SUBSET : thm
val CLOSURE_BIGUNION : thm
val CLOSURE_BOUNDED_LINEAR_IMAGE : thm
val CLOSURE_CLOSED : thm
val CLOSURE_CLOSURE : thm
val CLOSURE_COMPLEMENT : thm
val CLOSURE_EMPTY : thm
val CLOSURE_EQ : thm
val CLOSURE_EQ_EMPTY : thm
val CLOSURE_HALFSPACE_COMPONENT_GT : thm
val CLOSURE_HALFSPACE_COMPONENT_LT : thm
val CLOSURE_HALFSPACE_GT : thm
val CLOSURE_HALFSPACE_LT : thm
val CLOSURE_HULL : thm
val CLOSURE_HYPERPLANE : thm
val CLOSURE_IMAGE_BOUNDED : thm
val CLOSURE_IMAGE_CLOSURE : thm
val CLOSURE_INJECTIVE_LINEAR_IMAGE : thm
val CLOSURE_INTERIOR : thm
val CLOSURE_INTERIOR_IDEMP : thm
val CLOSURE_INTERIOR_UNION_CLOSED : thm
val CLOSURE_INTERVAL : thm
val CLOSURE_INTER_SUBSET : thm
val CLOSURE_LINEAR_IMAGE_SUBSET : thm
val CLOSURE_MINIMAL : thm
val CLOSURE_MINIMAL_EQ : thm
val CLOSURE_NEGATIONS : thm
val CLOSURE_NONEMPTY_OPEN_INTER : thm
val CLOSURE_OPEN_INTERVAL : thm
val CLOSURE_OPEN_INTER_CLOSURE : thm
val CLOSURE_OPEN_INTER_SUPERSET : thm
val CLOSURE_OPEN_IN_INTER_CLOSURE : thm
val CLOSURE_SEQUENTIAL : thm
val CLOSURE_SING : thm
val CLOSURE_SUBSET : thm
val CLOSURE_SUBSET_EQ : thm
val CLOSURE_SUMS : thm
val CLOSURE_UNION : thm
val CLOSURE_UNION_FRONTIER : thm
val CLOSURE_UNIQUE : thm
val CLOSURE_UNIV : thm
val COBOUNDED_IMP_UNBOUNDED : thm
val COBOUNDED_INTER_UNBOUNDED : thm
val COLLINEAR_1 : thm
val COLLINEAR_2 : thm
val COLLINEAR_3 : thm
val COLLINEAR_3_EXPAND : thm
val COLLINEAR_3_TRANS : thm
val COLLINEAR_4_3 : thm
val COLLINEAR_BETWEEN_CASES : thm
val COLLINEAR_DIST_BETWEEN : thm
val COLLINEAR_DIST_IN_CLOSED_SEGMENT : thm
val COLLINEAR_DIST_IN_OPEN_SEGMENT : thm
val COLLINEAR_EMPTY : thm
val COLLINEAR_LEMMA : thm
val COLLINEAR_LEMMA_ALT : thm
val COLLINEAR_MIDPOINT : thm
val COLLINEAR_SING : thm
val COLLINEAR_SMALL : thm
val COLLINEAR_SUBSET : thm
val COLLINEAR_TRIPLES : thm
val COMPACT_AFFINITY : thm
val COMPACT_ATTAINS_INF : thm
val COMPACT_ATTAINS_SUP : thm
val COMPACT_BIGINTER : thm
val COMPACT_BIGUNION : thm
val COMPACT_CBALL : thm
val COMPACT_CHAIN : thm
val COMPACT_CLOSED_DIFFERENCES : thm
val COMPACT_CLOSED_SUMS : thm
val COMPACT_CLOSURE : thm
val COMPACT_COMPONENTS : thm
val COMPACT_CONTINUOUS_IMAGE : thm
val COMPACT_CONTINUOUS_IMAGE_EQ : thm
val COMPACT_DIFF : thm
val COMPACT_EMPTY : thm
val COMPACT_EQ_BOLZANO_WEIERSTRASS : thm
val COMPACT_EQ_BOUNDED_CLOSED : thm
val COMPACT_EQ_HEINE_BOREL : thm
val COMPACT_EQ_HEINE_BOREL_SUBTOPOLOGY : thm
val COMPACT_FIP : thm
val COMPACT_FRONTIER : thm
val COMPACT_FRONTIER_BOUNDED : thm
val COMPACT_IMP_BOUNDED : thm
val COMPACT_IMP_CLOSED : thm
val COMPACT_IMP_COMPLETE : thm
val COMPACT_IMP_FIP : thm
val COMPACT_IMP_HEINE_BOREL : thm
val COMPACT_IMP_TOTALLY_BOUNDED : thm
val COMPACT_INSERT : thm
val COMPACT_INTER : thm
val COMPACT_INTERVAL : thm
val COMPACT_INTERVAL_EQ : thm
val COMPACT_INTER_CLOSED : thm
val COMPACT_LEMMA : thm
val COMPACT_LINEAR_IMAGE : thm
val COMPACT_NEGATIONS : thm
val COMPACT_NEST : thm
val COMPACT_REAL_LEMMA : thm
val COMPACT_SCALING : thm
val COMPACT_SEQUENCE_WITH_LIMIT : thm
val COMPACT_SING : thm
val COMPACT_SPHERE : thm
val COMPACT_TRANSLATION : thm
val COMPACT_TRANSLATION_EQ : thm
val COMPACT_UNIFORMLY_CONTINUOUS : thm
val COMPACT_UNIFORMLY_EQUICONTINUOUS : thm
val COMPACT_UNION : thm
val COMPLEMENT_CONNECTED_COMPONENT_BIGUNION : thm
val COMPLETE_EQ_CLOSED : thm
val COMPLETE_INJECTIVE_LINEAR_IMAGE : thm
val COMPLETE_INJECTIVE_LINEAR_IMAGE_EQ : thm
val COMPLETE_ISOMETRIC_IMAGE : thm
val COMPLETE_UNIV : thm
val COMPONENTS_EMPTY : thm
val COMPONENTS_EQ : thm
val COMPONENTS_EQ_EMPTY : thm
val COMPONENTS_EQ_SING : thm
val COMPONENTS_EQ_SING_EXISTS : thm
val COMPONENTS_EQ_SING_N_EXISTS : thm
val COMPONENTS_INTERMEDIATE_SUBSET : thm
val COMPONENTS_MAXIMAL : thm
val COMPONENTS_NONOVERLAP : thm
val COMPONENTS_UNIQUE : thm
val COMPONENTS_UNIQUE_EQ : thm
val COMPONENTS_UNIV : thm
val CONDENSATION_POINT_IMP_LIMPT : thm
val CONDENSATION_POINT_INFINITE_BALL : thm
val CONDENSATION_POINT_INFINITE_BALL_CBALL : thm
val CONDENSATION_POINT_INFINITE_CBALL : thm
val CONDENSATION_POINT_OF_SUBSET : thm
val CONNECTED_BIGUNION : thm
val CONNECTED_CHAIN : thm
val CONNECTED_CHAIN_GEN : thm
val CONNECTED_CLOPEN : thm
val CONNECTED_CLOSED : thm
val CONNECTED_CLOSED_IN : thm
val CONNECTED_CLOSED_IN_EQ : thm
val CONNECTED_CLOSED_MONOTONE_PREIMAGE : thm
val CONNECTED_CLOSED_SET : thm
val CONNECTED_CLOSURE : thm
val CONNECTED_COMPONENT_BIGUNION : thm
val CONNECTED_COMPONENT_DISJOINT : thm
val CONNECTED_COMPONENT_EMPTY : thm
val CONNECTED_COMPONENT_EQ : thm
val CONNECTED_COMPONENT_EQUIVALENCE_RELATION : thm
val CONNECTED_COMPONENT_EQ_EMPTY : thm
val CONNECTED_COMPONENT_EQ_EQ : thm
val CONNECTED_COMPONENT_EQ_SELF : thm
val CONNECTED_COMPONENT_EQ_UNIV : thm
val CONNECTED_COMPONENT_IDEMP : thm
val CONNECTED_COMPONENT_IN : thm
val CONNECTED_COMPONENT_INTERMEDIATE_SUBSET : thm
val CONNECTED_COMPONENT_MAXIMAL : thm
val CONNECTED_COMPONENT_MONO : thm
val CONNECTED_COMPONENT_NONOVERLAP : thm
val CONNECTED_COMPONENT_OF_SUBSET : thm
val CONNECTED_COMPONENT_OVERLAP : thm
val CONNECTED_COMPONENT_REFL : thm
val CONNECTED_COMPONENT_REFL_EQ : thm
val CONNECTED_COMPONENT_SET : thm
val CONNECTED_COMPONENT_SUBSET : thm
val CONNECTED_COMPONENT_SYM : thm
val CONNECTED_COMPONENT_SYM_EQ : thm
val CONNECTED_COMPONENT_TRANS : thm
val CONNECTED_COMPONENT_UNIQUE : thm
val CONNECTED_COMPONENT_UNIV : thm
val CONNECTED_CONNECTED_COMPONENT : thm
val CONNECTED_CONNECTED_COMPONENT_SET : thm
val CONNECTED_CONTINUOUS_IMAGE : thm
val CONNECTED_DIFF_OPEN_FROM_CLOSED : thm
val CONNECTED_DISJOINT_BIGUNION_OPEN_UNIQUE : thm
val CONNECTED_EMPTY : thm
val CONNECTED_EQUIVALENCE_RELATION : thm
val CONNECTED_EQUIVALENCE_RELATION_GEN : thm
val CONNECTED_EQ_COMPONENTS_SUBSET_SING : thm
val CONNECTED_EQ_COMPONENTS_SUBSET_SING_EXISTS : thm
val CONNECTED_EQ_CONNECTED_COMPONENTS_EQ : thm
val CONNECTED_EQ_CONNECTED_COMPONENT_EQ : thm
val CONNECTED_FROM_CLOSED_UNION_AND_INTER : thm
val CONNECTED_FROM_OPEN_UNION_AND_INTER : thm
val CONNECTED_IFF_CONNECTABLE_POINTS : thm
val CONNECTED_IFF_CONNECTED_COMPONENT : thm
val CONNECTED_IMP_PERFECT : thm
val CONNECTED_IMP_PERFECT_CLOSED : thm
val CONNECTED_INDUCTION : thm
val CONNECTED_INDUCTION_SIMPLE : thm
val CONNECTED_INTERMEDIATE_CLOSURE : thm
val CONNECTED_INTER_FRONTIER : thm
val CONNECTED_IVT_COMPONENT : thm
val CONNECTED_IVT_HYPERPLANE : thm
val CONNECTED_LINEAR_IMAGE : thm
val CONNECTED_MONOTONE_QUOTIENT_PREIMAGE : thm
val CONNECTED_MONOTONE_QUOTIENT_PREIMAGE_GEN : thm
val CONNECTED_NEGATIONS : thm
val CONNECTED_NEST : thm
val CONNECTED_NEST_GEN : thm
val CONNECTED_OPEN_IN : thm
val CONNECTED_OPEN_IN_EQ : thm
val CONNECTED_OPEN_MONOTONE_PREIMAGE : thm
val CONNECTED_OPEN_SET : thm
val CONNECTED_REAL_LEMMA : thm
val CONNECTED_SCALING : thm
val CONNECTED_SEGMENT : thm
val CONNECTED_SING : thm
val CONNECTED_SUBSET_CLOPEN : thm
val CONNECTED_TRANSLATION : thm
val CONNECTED_TRANSLATION_EQ : thm
val CONNECTED_UNION : thm
val CONNECTED_UNION_STRONG : thm
val CONNECTED_UNIV : thm
val CONTENT_0_SUBSET : thm
val CONTENT_0_SUBSET_GEN : thm
val CONTENT_CLOSED_INTERVAL : thm
val CONTENT_CLOSED_INTERVAL_CASES : thm
val CONTENT_EMPTY : thm
val CONTENT_EQ_0 : thm
val CONTENT_EQ_0_1 : thm
val CONTENT_EQ_0_GEN : thm
val CONTENT_EQ_0_INTERIOR : thm
val CONTENT_LT_NZ : thm
val CONTENT_POS_LE : thm
val CONTENT_POS_LT : thm
val CONTENT_POS_LT_EQ : thm
val CONTENT_SUBSET : thm
val CONTENT_UNIT : thm
val CONTINUOUS_ABS : thm
val CONTINUOUS_ABS_COMPOSE : thm
val CONTINUOUS_ADD : thm
val CONTINUOUS_AGREE_ON_CLOSURE : thm
val CONTINUOUS_AT : thm
val CONTINUOUS_ATTAINS_INF : thm
val CONTINUOUS_ATTAINS_SUP : thm
val CONTINUOUS_AT_ABS : thm
val CONTINUOUS_AT_AVOID : thm
val CONTINUOUS_AT_BALL : thm
val CONTINUOUS_AT_COMPOSE : thm
val CONTINUOUS_AT_COMPOSE_EQ : thm
val CONTINUOUS_AT_DIST : thm
val CONTINUOUS_AT_DIST_CLOSEST_POINT : thm
val CONTINUOUS_AT_ID : thm
val CONTINUOUS_AT_IMP_CONTINUOUS_ON : thm
val CONTINUOUS_AT_INV : thm
val CONTINUOUS_AT_LIFT_DOT : thm
val CONTINUOUS_AT_OPEN : thm
val CONTINUOUS_AT_RANGE : thm
val CONTINUOUS_AT_SEQUENTIALLY : thm
val CONTINUOUS_AT_SETDIST : thm
val CONTINUOUS_AT_TRANSLATION : thm
val CONTINUOUS_AT_WITHIN : thm
val CONTINUOUS_AT_WITHIN_INV : thm
val CONTINUOUS_CLOSED_IMP_CAUCHY_CONTINUOUS : thm
val CONTINUOUS_CLOSED_IN_PREIMAGE : thm
val CONTINUOUS_CLOSED_IN_PREIMAGE_CONSTANT : thm
val CONTINUOUS_CLOSED_IN_PREIMAGE_EQ : thm
val CONTINUOUS_CLOSED_IN_PREIMAGE_GEN : thm
val CONTINUOUS_CLOSED_PREIMAGE : thm
val CONTINUOUS_CLOSED_PREIMAGE_CONSTANT : thm
val CONTINUOUS_CLOSED_PREIMAGE_UNIV : thm
val CONTINUOUS_CMUL : thm
val CONTINUOUS_COMPONENT_COMPOSE : thm
val CONTINUOUS_CONST : thm
val CONTINUOUS_CONSTANT_ON_CLOSURE : thm
val CONTINUOUS_DIAMETER : thm
val CONTINUOUS_DISCONNECTED_DISCRETE_FINITE_RANGE_CONSTANT_EQ : thm
val CONTINUOUS_DISCONNECTED_RANGE_CONSTANT : thm
val CONTINUOUS_DISCONNECTED_RANGE_CONSTANT_EQ : thm
val CONTINUOUS_DISCRETE_RANGE_CONSTANT : thm
val CONTINUOUS_DISCRETE_RANGE_CONSTANT_EQ : thm
val CONTINUOUS_DOT2 : thm
val CONTINUOUS_FINITE_RANGE_CONSTANT : thm
val CONTINUOUS_FINITE_RANGE_CONSTANT_EQ : thm
val CONTINUOUS_GE_ON_CLOSURE : thm
val CONTINUOUS_IMP_CLOSED_MAP : thm
val CONTINUOUS_IMP_QUOTIENT_MAP : thm
val CONTINUOUS_INV : thm
val CONTINUOUS_LEFT_INVERSE_IMP_QUOTIENT_MAP : thm
val CONTINUOUS_LEVELSET_OPEN : thm
val CONTINUOUS_LEVELSET_OPEN_IN : thm
val CONTINUOUS_LEVELSET_OPEN_IN_CASES : thm
val CONTINUOUS_LE_ON_CLOSURE : thm
val CONTINUOUS_MAP_CLOSURES : thm
val CONTINUOUS_MAX : thm
val CONTINUOUS_MIN : thm
val CONTINUOUS_MUL : thm
val CONTINUOUS_NEG : thm
val CONTINUOUS_ON : thm
val CONTINUOUS_ON_ABS : thm
val CONTINUOUS_ON_ABS_COMPOSE : thm
val CONTINUOUS_ON_ADD : thm
val CONTINUOUS_ON_AVOID : thm
val CONTINUOUS_ON_CASES : thm
val CONTINUOUS_ON_CASES_1 : thm
val CONTINUOUS_ON_CASES_LE : thm
val CONTINUOUS_ON_CASES_LOCAL : thm
val CONTINUOUS_ON_CASES_LOCAL_OPEN : thm
val CONTINUOUS_ON_CASES_OPEN : thm
val CONTINUOUS_ON_CLOSED : thm
val CONTINUOUS_ON_CLOSED_GEN : thm
val CONTINUOUS_ON_CLOSURE : thm
val CONTINUOUS_ON_CLOSURE_ABS_LE : thm
val CONTINUOUS_ON_CLOSURE_COMPONENT_GE : thm
val CONTINUOUS_ON_CLOSURE_COMPONENT_LE : thm
val CONTINUOUS_ON_CLOSURE_SEQUENTIALLY : thm
val CONTINUOUS_ON_CMUL : thm
val CONTINUOUS_ON_COMPONENTS_FINITE : thm
val CONTINUOUS_ON_COMPONENTS_GEN : thm
val CONTINUOUS_ON_COMPONENT_COMPOSE : thm
val CONTINUOUS_ON_COMPOSE : thm
val CONTINUOUS_ON_COMPOSE_QUOTIENT : thm
val CONTINUOUS_ON_CONST : thm
val CONTINUOUS_ON_DIST : thm
val CONTINUOUS_ON_DIST_CLOSEST_POINT : thm
val CONTINUOUS_ON_DOT2 : thm
val CONTINUOUS_ON_EMPTY : thm
val CONTINUOUS_ON_EQ : thm
val CONTINUOUS_ON_EQ_CONTINUOUS_AT : thm
val CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN : thm
val CONTINUOUS_ON_FINITE : thm
val CONTINUOUS_ON_ID : thm
val CONTINUOUS_ON_IMP_CLOSED_IN : thm
val CONTINUOUS_ON_IMP_OPEN_IN : thm
val CONTINUOUS_ON_INTERIOR : thm
val CONTINUOUS_ON_INV : thm
val CONTINUOUS_ON_INVERSE : thm
val CONTINUOUS_ON_INVERSE_CLOSED_MAP : thm
val CONTINUOUS_ON_INVERSE_OPEN_MAP : thm
val CONTINUOUS_ON_IVT : thm
val CONTINUOUS_ON_LIFT_DOT : thm
val CONTINUOUS_ON_MAX : thm
val CONTINUOUS_ON_MIN : thm
val CONTINUOUS_ON_MUL : thm
val CONTINUOUS_ON_NEG : thm
val CONTINUOUS_ON_NO_LIMPT : thm
val CONTINUOUS_ON_OPEN : thm
val CONTINUOUS_ON_OPEN_AVOID : thm
val CONTINUOUS_ON_OPEN_GEN : thm
val CONTINUOUS_ON_POW : thm
val CONTINUOUS_ON_PRODUCT : thm
val CONTINUOUS_ON_RANGE : thm
val CONTINUOUS_ON_SEQUENTIALLY : thm
val CONTINUOUS_ON_SETDIST : thm
val CONTINUOUS_ON_SING : thm
val CONTINUOUS_ON_SUB : thm
val CONTINUOUS_ON_SUBSET : thm
val CONTINUOUS_ON_SUM : thm
val CONTINUOUS_ON_UNION : thm
val CONTINUOUS_ON_UNION_LOCAL : thm
val CONTINUOUS_ON_UNION_LOCAL_OPEN : thm
val CONTINUOUS_ON_UNION_OPEN : thm
val CONTINUOUS_ON_VMUL : thm
val CONTINUOUS_OPEN_IN_PREIMAGE : thm
val CONTINUOUS_OPEN_IN_PREIMAGE_EQ : thm
val CONTINUOUS_OPEN_IN_PREIMAGE_GEN : thm
val CONTINUOUS_OPEN_PREIMAGE : thm
val CONTINUOUS_OPEN_PREIMAGE_UNIV : thm
val CONTINUOUS_POW : thm
val CONTINUOUS_PRODUCT : thm
val CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP : thm
val CONTINUOUS_SUB : thm
val CONTINUOUS_SUM : thm
val CONTINUOUS_TRANSFORM_AT : thm
val CONTINUOUS_TRANSFORM_WITHIN : thm
val CONTINUOUS_TRANSFORM_WITHIN_OPEN : thm
val CONTINUOUS_TRANSFORM_WITHIN_OPEN_IN : thm
val CONTINUOUS_TRANSFORM_WITHIN_SET_IMP : thm
val CONTINUOUS_TRIVIAL_LIMIT : thm
val CONTINUOUS_UNIFORM_LIMIT : thm
val CONTINUOUS_VMUL : thm
val CONTINUOUS_WITHIN : thm
val CONTINUOUS_WITHIN_AVOID : thm
val CONTINUOUS_WITHIN_BALL : thm
val CONTINUOUS_WITHIN_CLOSED_NONTRIVIAL : thm
val CONTINUOUS_WITHIN_COMPARISON : thm
val CONTINUOUS_WITHIN_COMPOSE : thm
val CONTINUOUS_WITHIN_ID : thm
val CONTINUOUS_WITHIN_OPEN : thm
val CONTINUOUS_WITHIN_SEQUENTIALLY : thm
val CONTINUOUS_WITHIN_SUBSET : thm
val CONTRACTION_IMP_CONTINUOUS_ON : thm
val CONVERGENT_BOUNDED_INCREASING : thm
val CONVERGENT_BOUNDED_MONOTONE : thm
val CONVERGENT_EQ_CAUCHY : thm
val CONVERGENT_IMP_BOUNDED : thm
val CONVERGENT_IMP_CAUCHY : thm
val COUNTABLE_OPEN_INTERVAL : thm
val DECREASING_CLOSED_NEST : thm
val DECREASING_CLOSED_NEST_SING : thm
val DENSE_IMP_PERFECT : thm
val DENSE_LIMIT_POINTS : thm
val DENSE_OPEN_INTER : thm
val DEPENDENT_CHOICE : thm
val DEPENDENT_CHOICE_FIXED : thm
val DEPENDENT_EXPLICIT : thm
val DEPENDENT_MONO : thm
val DIAMETER_BALL : thm
val DIAMETER_BOUNDED : thm
val DIAMETER_BOUNDED_BOUND : thm
val DIAMETER_CBALL : thm
val DIAMETER_CLOSURE : thm
val DIAMETER_EMPTY : thm
val DIAMETER_EQ_0 : thm
val DIAMETER_INTERVAL : thm
val DIAMETER_LE : thm
val DIAMETER_LINEAR_IMAGE : thm
val DIAMETER_POS_LE : thm
val DIAMETER_SING : thm
val DIAMETER_SUBSET : thm
val DIAMETER_SUBSET_CBALL : thm
val DIAMETER_SUBSET_CBALL_NONEMPTY : thm
val DIAMETER_SUMS : thm
val DIFF_CLOSURE_SUBSET : thm
val DIM_LE_CARD : thm
val DIM_SUBSET : thm
val DIM_SUBSET_UNIV : thm
val DIM_SUBSTANDARD : thm
val DIM_UNIQUE : thm
val DIM_UNIV : thm
val DINI : thm
val DISCRETE_BOUNDED_IMP_FINITE : thm
val DISCRETE_IMP_CLOSED : thm
val DISJOINT_INTERVAL : thm
val DISTANCE_ATTAINS_INF : thm
val DISTANCE_ATTAINS_SUP : thm
val DIST_0 : thm
val DIST_CLOSEST_POINT_LIPSCHITZ : thm
val DIST_EQ : thm
val DIST_EQ_0 : thm
val DIST_IN_CLOSED_SEGMENT : thm
val DIST_IN_OPEN_CLOSED_SEGMENT : thm
val DIST_IN_OPEN_SEGMENT : thm
val DIST_LE_0 : thm
val DIST_MIDPOINT : thm
val DIST_MUL : thm
val DIST_NZ : thm
val DIST_POS_LE : thm
val DIST_POS_LT : thm
val DIST_REFL : thm
val DIST_SYM : thm
val DIST_TRIANGLE : thm
val DIST_TRIANGLE_ADD : thm
val DIST_TRIANGLE_ADD_HALF : thm
val DIST_TRIANGLE_ALT : thm
val DIST_TRIANGLE_EQ : thm
val DIST_TRIANGLE_HALF_L : thm
val DIST_TRIANGLE_HALF_R : thm
val DIST_TRIANGLE_LE : thm
val DIST_TRIANGLE_LT : thm
val DORDER_NET : thm
val EMPTY_AS_INTERVAL : thm
val EMPTY_INTERIOR_FINITE : thm
val ENDS_IN_INTERVAL : thm
val ENDS_IN_SEGMENT : thm
val ENDS_IN_UNIT_INTERVAL : thm
val ENDS_NOT_IN_SEGMENT : thm
val EQ_BALLS : thm
val EQ_INTERVAL : thm
val EVENTUALLY_AND : thm
val EVENTUALLY_AT : thm
val EVENTUALLY_AT_INFINITY : thm
val EVENTUALLY_AT_INFINITY_POS : thm
val EVENTUALLY_AT_NEGINFINITY : thm
val EVENTUALLY_AT_POSINFINITY : thm
val EVENTUALLY_FALSE : thm
val EVENTUALLY_FORALL : thm
val EVENTUALLY_HAPPENS : thm
val EVENTUALLY_MONO : thm
val EVENTUALLY_MP : thm
val EVENTUALLY_SEQUENTIALLY : thm
val EVENTUALLY_TRUE : thm
val EVENTUALLY_WITHIN : thm
val EVENTUALLY_WITHIN_INTERIOR : thm
val EVENTUALLY_WITHIN_LE : thm
val EXCHANGE_LEMMA : thm
val EXISTS_COMPONENT_SUPERSET : thm
val EXISTS_DIFF : thm
val EXISTS_IN_INSERT : thm
val EXTENSION_FROM_CLOPEN : thm
val FINITE_BALL : thm
val FINITE_CBALL : thm
val FINITE_IMP_BOUNDED : thm
val FINITE_IMP_CLOSED : thm
val FINITE_IMP_CLOSED_IN : thm
val FINITE_IMP_COMPACT : thm
val FINITE_IMP_NOT_OPEN : thm
val FINITE_INTERVAL : thm
val FINITE_INTER_NUMSEG : thm
val FINITE_SET_AVOID : thm
val FINITE_SPHERE : thm
val FORALL_EVENTUALLY : thm
val FORALL_IN_CLOSURE : thm
val FORALL_IN_CLOSURE_EQ : thm
val FORALL_POS_MONO_1 : thm
val FRONTIER_BALL : thm
val FRONTIER_CBALL : thm
val FRONTIER_CLOSED : thm
val FRONTIER_CLOSED_INTERVAL : thm
val FRONTIER_CLOSURES : thm
val FRONTIER_CLOSURE_SUBSET : thm
val FRONTIER_COMPLEMENT : thm
val FRONTIER_DISJOINT_EQ : thm
val FRONTIER_EMPTY : thm
val FRONTIER_FRONTIER : thm
val FRONTIER_FRONTIER_FRONTIER : thm
val FRONTIER_FRONTIER_SUBSET : thm
val FRONTIER_HALFSPACE_GE : thm
val FRONTIER_HALFSPACE_GT : thm
val FRONTIER_HALFSPACE_LE : thm
val FRONTIER_HALFSPACE_LT : thm
val FRONTIER_INTERIORS : thm
val FRONTIER_INTERIOR_SUBSET : thm
val FRONTIER_INTER_SUBSET : thm
val FRONTIER_INTER_SUBSET_INTER : thm
val FRONTIER_OPEN_INTERVAL : thm
val FRONTIER_SING : thm
val FRONTIER_STRADDLE : thm
val FRONTIER_SUBSET_CLOSED : thm
val FRONTIER_SUBSET_COMPACT : thm
val FRONTIER_SUBSET_EQ : thm
val FRONTIER_UNION : thm
val FRONTIER_UNION_SUBSET : thm
val FRONTIER_UNIV : thm
val FUNCTION_FACTORS_LEFT_GEN : thm
val GDELTA_COMPLEMENT : thm
val GREATER_EQ_REFL : thm
val HAS_SIZE_STDBASIS : thm
val HAUSDIST_ALT : thm
val HAUSDIST_BALLS : thm
val HAUSDIST_CLOSURE : thm
val HAUSDIST_COMPACT_EXISTS : thm
val HAUSDIST_COMPACT_NONTRIVIAL : thm
val HAUSDIST_COMPACT_SUMS : thm
val HAUSDIST_EMPTY : thm
val HAUSDIST_EQ : thm
val HAUSDIST_EQ_0 : thm
val HAUSDIST_INSERT_LE : thm
val HAUSDIST_LINEAR_IMAGE : thm
val HAUSDIST_NONTRIVIAL : thm
val HAUSDIST_NONTRIVIAL_ALT : thm
val HAUSDIST_POS_LE : thm
val HAUSDIST_REFL : thm
val HAUSDIST_SETDIST_TRIANGLE : thm
val HAUSDIST_SINGS : thm
val HAUSDIST_SYM : thm
val HAUSDIST_TRANS : thm
val HAUSDIST_TRANSLATION : thm
val HAUSDIST_TRIANGLE : thm
val HAUSDIST_UNION_LE : thm
val HEINE_BOREL_IMP_BOLZANO_WEIERSTRASS : thm
val HEINE_BOREL_LEMMA : thm
val HOMEOMORPHIC_AFFINITY : thm
val HOMEOMORPHIC_BALLS : thm
val HOMEOMORPHIC_BALLS_CBALL_SPHERE : thm
val HOMEOMORPHIC_CBALL : thm
val HOMEOMORPHIC_COMPACT : thm
val HOMEOMORPHIC_COMPACTNESS : thm
val HOMEOMORPHIC_CONNECTEDNESS : thm
val HOMEOMORPHIC_EMPTY : thm
val HOMEOMORPHIC_FINITE : thm
val HOMEOMORPHIC_FINITENESS : thm
val HOMEOMORPHIC_FINITE_STRONG : thm
val HOMEOMORPHIC_HYPERPLANES : thm
val HOMEOMORPHIC_HYPERPLANE_STANDARD_HYPERPLANE : thm
val HOMEOMORPHIC_IMP_CARD_EQ : thm
val HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ : thm
val HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ : thm
val HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF : thm
val HOMEOMORPHIC_LOCALLY : thm
val HOMEOMORPHIC_LOCAL_COMPACTNESS : thm
val HOMEOMORPHIC_MINIMAL : thm
val HOMEOMORPHIC_ONE_POINT_COMPACTIFICATIONS : thm
val HOMEOMORPHIC_OPEN_INTERVALS : thm
val HOMEOMORPHIC_OPEN_INTERVAL_UNIV : thm
val HOMEOMORPHIC_REFL : thm
val HOMEOMORPHIC_SCALING : thm
val HOMEOMORPHIC_SCALING_LEFT : thm
val HOMEOMORPHIC_SCALING_RIGHT : thm
val HOMEOMORPHIC_SING : thm
val HOMEOMORPHIC_SPHERE : thm
val HOMEOMORPHIC_STANDARD_HYPERPLANE_HYPERPLANE : thm
val HOMEOMORPHIC_SYM : thm
val HOMEOMORPHIC_TRANS : thm
val HOMEOMORPHIC_TRANSLATION : thm
val HOMEOMORPHIC_TRANSLATION_LEFT_EQ : thm
val HOMEOMORPHIC_TRANSLATION_RIGHT_EQ : thm
val HOMEOMORPHIC_TRANSLATION_SELF : thm
val HOMEOMORPHISM : thm
val HOMEOMORPHISM_COMPACT : thm
val HOMEOMORPHISM_COMPOSE : thm
val HOMEOMORPHISM_FROM_COMPOSITION_INJECTIVE : thm
val HOMEOMORPHISM_FROM_COMPOSITION_SURJECTIVE : thm
val HOMEOMORPHISM_ID : thm
val HOMEOMORPHISM_IMP_CLOSED_MAP : thm
val HOMEOMORPHISM_IMP_OPEN_MAP : thm
val HOMEOMORPHISM_IMP_QUOTIENT_MAP : thm
val HOMEOMORPHISM_INJECTIVE_CLOSED_MAP : thm
val HOMEOMORPHISM_INJECTIVE_CLOSED_MAP_EQ : thm
val HOMEOMORPHISM_INJECTIVE_OPEN_MAP : thm
val HOMEOMORPHISM_INJECTIVE_OPEN_MAP_EQ : thm
val HOMEOMORPHISM_LOCALLY : thm
val HOMEOMORPHISM_OF_SUBSETS : thm
val HOMEOMORPHISM_SYM : thm
val IMAGE_AFFINITY_INTERVAL : thm
val IMAGE_CLOSURE_SUBSET : thm
val IMAGE_STRETCH_INTERVAL : thm
val IMAGE_TWIZZLE_INTERVAL : thm
val INDEPENDENT_BOUND : thm
val INDEPENDENT_CARD_LE_DIM : thm
val INDEPENDENT_EMPTY : thm
val INDEPENDENT_INJECTIVE_IMAGE : thm
val INDEPENDENT_INJECTIVE_IMAGE_GEN : thm
val INDEPENDENT_INSERT : thm
val INDEPENDENT_MONO : thm
val INDEPENDENT_NONZERO : thm
val INDEPENDENT_SING : thm
val INDEPENDENT_SPAN_BOUND : thm
val INDEPENDENT_STDBASIS : thm
val INFINITE_OPEN_IN : thm
val INFINITE_SUPERSET : thm
val INFSUM_0 : thm
val INFSUM_ADD : thm
val INFSUM_CMUL : thm
val INFSUM_EQ : thm
val INFSUM_LINEAR : thm
val INFSUM_NEG : thm
val INFSUM_RESTRICT : thm
val INFSUM_SUB : thm
val INFSUM_UNIQUE : thm
val INF_INSERT : thm
val INJECTIVE_IMP_ISOMETRIC : thm
val INJECTIVE_MAP_OPEN_IFF_CLOSED : thm
val INTERIOR_BALL : thm
val INTERIOR_BIGINTER_SUBSET : thm
val INTERIOR_BIJECTIVE_LINEAR_IMAGE : thm
val INTERIOR_CBALL : thm
val INTERIOR_CLOSED_EQ_EMPTY_AS_FRONTIER : thm
val INTERIOR_CLOSED_INTERVAL : thm
val INTERIOR_CLOSED_UNION_EMPTY_INTERIOR : thm
val INTERIOR_CLOSURE : thm
val INTERIOR_CLOSURE_IDEMP : thm
val INTERIOR_CLOSURE_INTER_OPEN : thm
val INTERIOR_COMPLEMENT : thm
val INTERIOR_DIFF : thm
val INTERIOR_EMPTY : thm
val INTERIOR_EQ : thm
val INTERIOR_EQ_EMPTY : thm
val INTERIOR_EQ_EMPTY_ALT : thm
val INTERIOR_FINITE_BIGINTER : thm
val INTERIOR_FRONTIER : thm
val INTERIOR_FRONTIER_EMPTY : thm
val INTERIOR_HALFSPACE_COMPONENT_GE : thm
val INTERIOR_HALFSPACE_COMPONENT_LE : thm
val INTERIOR_HALFSPACE_GE : thm
val INTERIOR_HALFSPACE_LE : thm
val INTERIOR_HYPERPLANE : thm
val INTERIOR_IMAGE_SUBSET : thm
val INTERIOR_INJECTIVE_LINEAR_IMAGE : thm
val INTERIOR_INTER : thm
val INTERIOR_INTERIOR : thm
val INTERIOR_INTERVAL : thm
val INTERIOR_LIMIT_POINT : thm
val INTERIOR_MAXIMAL : thm
val INTERIOR_MAXIMAL_EQ : thm
val INTERIOR_NEGATIONS : thm
val INTERIOR_OPEN : thm
val INTERIOR_SING : thm
val INTERIOR_STANDARD_HYPERPLANE : thm
val INTERIOR_SUBSET : thm
val INTERIOR_TRANSLATION : thm
val INTERIOR_UNIONS_OPEN_SUBSETS : thm
val INTERIOR_UNION_EQ_EMPTY : thm
val INTERIOR_UNIQUE : thm
val INTERIOR_UNIV : thm
val INTERVAL : thm
val INTERVAL_BOUNDS_EMPTY : thm
val INTERVAL_BOUNDS_NULL : thm
val INTERVAL_CASES : thm
val INTERVAL_CONTAINS_COMPACT_NEIGHBOURHOOD : thm
val INTERVAL_EQ_EMPTY : thm
val INTERVAL_IMAGE_STRETCH_INTERVAL : thm
val INTERVAL_LOWERBOUND : thm
val INTERVAL_LOWERBOUND_NONEMPTY : thm
val INTERVAL_NE_EMPTY : thm
val INTERVAL_OPEN_SUBSET_CLOSED : thm
val INTERVAL_SING : thm
val INTERVAL_SUBSET_IS_INTERVAL : thm
val INTERVAL_TRANSLATION : thm
val INTERVAL_UPPERBOUND : thm
val INTERVAL_UPPERBOUND_NONEMPTY : thm
val INTER_BALLS_EQ_EMPTY : thm
val INTER_INTERVAL : thm
val INTER_INTERVAL_MIXED_EQ_EMPTY : thm
val IN_BALL : thm
val IN_BALL_0 : thm
val IN_CBALL : thm
val IN_CBALL_0 : thm
val IN_CLOSURE_DELETE : thm
val IN_COMPONENTS : thm
val IN_COMPONENTS_BIGUNION_COMPLEMENT : thm
val IN_COMPONENTS_CONNECTED : thm
val IN_COMPONENTS_MAXIMAL : thm
val IN_COMPONENTS_NONEMPTY : thm
val IN_COMPONENTS_SELF : thm
val IN_COMPONENTS_SUBSET : thm
val IN_DIRECTION : thm
val IN_INTERIOR : thm
val IN_INTERIOR_CBALL : thm
val IN_INTERIOR_LINEAR_IMAGE : thm
val IN_INTERVAL : thm
val IN_INTERVAL_REFLECT : thm
val IN_OPEN_SEGMENT : thm
val IN_OPEN_SEGMENT_ALT : thm
val IN_SEGMENT : thm
val IN_SEGMENT_COMPONENT : thm
val IN_SPAN_DELETE : thm
val IN_SPAN_INSERT : thm
val IN_SPHERE : thm
val IN_SPHERE_0 : thm
val ISOMETRY_IMP_EMBEDDING : thm
val ISOMETRY_IMP_HOMEOMORPHISM_COMPACT : thm
val ISOMETRY_IMP_OPEN_MAP : thm
val ISOMETRY_ON_IMP_CONTINUOUS_ON : thm
val IS_INTERVAL : thm
val IS_INTERVAL_CASES : thm
val IS_INTERVAL_COMPACT : thm
val IS_INTERVAL_EMPTY : thm
val IS_INTERVAL_IMP_LOCALLY_COMPACT : thm
val IS_INTERVAL_INTER : thm
val IS_INTERVAL_INTERVAL : thm
val IS_INTERVAL_POINTWISE : thm
val IS_INTERVAL_SCALING : thm
val IS_INTERVAL_SCALING_EQ : thm
val IS_INTERVAL_SING : thm
val IS_INTERVAL_SUMS : thm
val IS_INTERVAL_UNIV : thm
val JOINABLE_COMPONENTS_EQ : thm
val JOINABLE_CONNECTED_COMPONENT_EQ : thm
val LEBESGUE_COVERING_LEMMA : thm
val LE_1 : thm
val LIFT_TO_QUOTIENT_SPACE : thm
val LIFT_TO_QUOTIENT_SPACE_UNIQUE : thm
val LIMIT_POINT_FINITE : thm
val LIMIT_POINT_UNION : thm
val LIMPT_APPROACHABLE : thm
val LIMPT_APPROACHABLE_LE : thm
val LIMPT_BALL : thm
val LIMPT_EMPTY : thm
val LIMPT_INFINITE_BALL : thm
val LIMPT_INFINITE_CBALL : thm
val LIMPT_INFINITE_OPEN : thm
val LIMPT_INFINITE_OPEN_BALL_CBALL : thm
val LIMPT_INJECTIVE_LINEAR_IMAGE_EQ : thm
val LIMPT_INSERT : thm
val LIMPT_OF_CLOSURE : thm
val LIMPT_OF_LIMPTS : thm
val LIMPT_OF_OPEN : thm
val LIMPT_OF_OPEN_IN : thm
val LIMPT_OF_SEQUENCE_SUBSEQUENCE : thm
val LIMPT_OF_UNIV : thm
val LIMPT_SEQUENTIAL : thm
val LIMPT_SEQUENTIAL_INJ : thm
val LIMPT_SING : thm
val LIMPT_SUBSET : thm
val LIMPT_UNIV : thm
val LIM_ABS : thm
val LIM_ABS_LBOUND : thm
val LIM_ABS_UBOUND : thm
val LIM_ADD : thm
val LIM_AT : thm
val LIM_AT_ID : thm
val LIM_AT_INFINITY : thm
val LIM_AT_INFINITY_POS : thm
val LIM_AT_LE : thm
val LIM_AT_NEGINFINITY : thm
val LIM_AT_POSINFINITY : thm
val LIM_AT_WITHIN : thm
val LIM_AT_ZERO : thm
val LIM_BILINEAR : thm
val LIM_CASES_COFINITE_SEQUENTIALLY : thm
val LIM_CASES_FINITE_SEQUENTIALLY : thm
val LIM_CASES_SEQUENTIALLY : thm
val LIM_CMUL : thm
val LIM_CMUL_EQ : thm
val LIM_COMPONENT : thm
val LIM_COMPONENT_EQ : thm
val LIM_COMPONENT_LBOUND : thm
val LIM_COMPONENT_LE : thm
val LIM_COMPONENT_UBOUND : thm
val LIM_COMPOSE_AT : thm
val LIM_COMPOSE_WITHIN : thm
val LIM_CONG_AT : thm
val LIM_CONG_WITHIN : thm
val LIM_CONST : thm
val LIM_CONST_EQ : thm
val LIM_CONTINUOUS_FUNCTION : thm
val LIM_DEF : thm
val LIM_DROP_LBOUND : thm
val LIM_DROP_LE : thm
val LIM_DROP_UBOUND : thm
val LIM_EVENTUALLY : thm
val LIM_INFINITY_POSINFINITY : thm
val LIM_INV : thm
val LIM_IN_CLOSED_SET : thm
val LIM_LIFT_DOT : thm
val LIM_LINEAR : thm
val LIM_MAX : thm
val LIM_MIN : thm
val LIM_MUL : thm
val LIM_NEG : thm
val LIM_NEG_EQ : thm
val LIM_NULL : thm
val LIM_NULL_ABS : thm
val LIM_NULL_ADD : thm
val LIM_NULL_CMUL : thm
val LIM_NULL_CMUL_BOUNDED : thm
val LIM_NULL_CMUL_EQ : thm
val LIM_NULL_COMPARISON : thm
val LIM_NULL_SUB : thm
val LIM_NULL_SUM : thm
val LIM_POSINFINITY_SEQUENTIALLY : thm
val LIM_SEQUENTIALLY : thm
val LIM_SUB : thm
val LIM_SUBSEQUENCE : thm
val LIM_SUBSEQUENCE_WEAK : thm
val LIM_SUM : thm
val LIM_TRANSFORM : thm
val LIM_TRANSFORM_AT : thm
val LIM_TRANSFORM_AWAY_AT : thm
val LIM_TRANSFORM_AWAY_WITHIN : thm
val LIM_TRANSFORM_BOUND : thm
val LIM_TRANSFORM_EQ : thm
val LIM_TRANSFORM_EVENTUALLY : thm
val LIM_TRANSFORM_WITHIN : thm
val LIM_TRANSFORM_WITHIN_OPEN : thm
val LIM_TRANSFORM_WITHIN_OPEN_IN : thm
val LIM_TRANSFORM_WITHIN_SET : thm
val LIM_TRANSFORM_WITHIN_SET_IMP : thm
val LIM_UNION : thm
val LIM_UNION_UNIV : thm
val LIM_UNIQUE : thm
val LIM_VMUL : thm
val LIM_WITHIN : thm
val LIM_WITHIN_CLOSED_TRIVIAL : thm
val LIM_WITHIN_EMPTY : thm
val LIM_WITHIN_ID : thm
val LIM_WITHIN_INTERIOR : thm
val LIM_WITHIN_LE : thm
val LIM_WITHIN_OPEN : thm
val LIM_WITHIN_SUBSET : thm
val LIM_WITHIN_UNION : thm
val LINEAR_0 : thm
val LINEAR_ADD : thm
val LINEAR_BOUNDED : thm
val LINEAR_BOUNDED_POS : thm
val LINEAR_CMUL : thm
val LINEAR_COMPOSE : thm
val LINEAR_COMPOSE_ADD : thm
val LINEAR_COMPOSE_CMUL : thm
val LINEAR_COMPOSE_NEG : thm
val LINEAR_COMPOSE_SUB : thm
val LINEAR_COMPOSE_SUM : thm
val LINEAR_CONTINUOUS_AT : thm
val LINEAR_CONTINUOUS_COMPOSE : thm
val LINEAR_CONTINUOUS_ON : thm
val LINEAR_CONTINUOUS_ON_COMPOSE : thm
val LINEAR_CONTINUOUS_WITHIN : thm
val LINEAR_EQ : thm
val LINEAR_EQ_0 : thm
val LINEAR_EQ_0_SPAN : thm
val LINEAR_EQ_STDBASIS : thm
val LINEAR_ID : thm
val LINEAR_IMAGE_SUBSET_INTERIOR : thm
val LINEAR_INDEPENDENT_EXTEND : thm
val LINEAR_INDEPENDENT_EXTEND_LEMMA : thm
val LINEAR_INJECTIVE_0_SUBSPACE : thm
val LINEAR_INJECTIVE_BOUNDED_BELOW_POS : thm
val LINEAR_INJECTIVE_IMP_SURJECTIVE : thm
val LINEAR_INJECTIVE_LEFT_INVERSE : thm
val LINEAR_INTERIOR_IMAGE_SUBSET : thm
val LINEAR_INVERTIBLE_BOUNDED_BELOW : thm
val LINEAR_INVERTIBLE_BOUNDED_BELOW_POS : thm
val LINEAR_LIM_0 : thm
val LINEAR_MUL_COMPONENT : thm
val LINEAR_NEG : thm
val LINEAR_NEGATION : thm
val LINEAR_OPEN_MAPPING : thm
val LINEAR_SCALING : thm
val LINEAR_SUB : thm
val LINEAR_SUM : thm
val LINEAR_SUM_MUL : thm
val LINEAR_UNIFORMLY_CONTINUOUS_ON : thm
val LINEAR_ZERO : thm
val LOCALLY_CLOSED : thm
val LOCALLY_COMPACT : thm
val LOCALLY_COMPACT_ALT : thm
val LOCALLY_COMPACT_CLOSED_IN : thm
val LOCALLY_COMPACT_CLOSED_INTER_OPEN : thm
val LOCALLY_COMPACT_CLOSED_IN_OPEN : thm
val LOCALLY_COMPACT_CLOSED_UNION : thm
val LOCALLY_COMPACT_COMPACT : thm
val LOCALLY_COMPACT_COMPACT_ALT : thm
val LOCALLY_COMPACT_COMPACT_SUBOPEN : thm
val LOCALLY_COMPACT_DELETE : thm
val LOCALLY_COMPACT_INTER : thm
val LOCALLY_COMPACT_INTER_CBALL : thm
val LOCALLY_COMPACT_INTER_CBALLS : thm
val LOCALLY_COMPACT_OPEN_IN : thm
val LOCALLY_COMPACT_OPEN_INTER_CLOSURE : thm
val LOCALLY_COMPACT_OPEN_UNION : thm
val LOCALLY_COMPACT_PROPER_IMAGE : thm
val LOCALLY_COMPACT_PROPER_IMAGE_EQ : thm
val LOCALLY_COMPACT_TRANSLATION_EQ : thm
val LOCALLY_COMPACT_UNIV : thm
val LOCALLY_DIFF_CLOSED : thm
val LOCALLY_EMPTY : thm
val LOCALLY_INJECTIVE_LINEAR_IMAGE : thm
val LOCALLY_INTER : thm
val LOCALLY_MONO : thm
val LOCALLY_OPEN_MAP_IMAGE : thm
val LOCALLY_OPEN_SUBSET : thm
val LOCALLY_SING : thm
val LOCALLY_TRANSLATION : thm
val LOWER_HEMICONTINUOUS : thm
val MAPPING_CONNECTED_ONTO_SEGMENT : thm
val MAXIMAL_INDEPENDENT_SUBSET : thm
val MAXIMAL_INDEPENDENT_SUBSET_EXTEND : thm
val METRIZABLE_SPACE_EUCLIDEAN : thm
val MIDPOINT_COLLINEAR : thm
val MIDPOINT_EQ_ENDPOINT : thm
val MIDPOINT_IN_SEGMENT : thm
val MIDPOINT_LINEAR_IMAGE : thm
val MIDPOINT_REFL : thm
val MIDPOINT_SYM : thm
val MONOTONE_BIGGER : thm
val MONOTONE_SUBSEQUENCE : thm
val MUL_CAUCHY_SCHWARZ_EQUAL : thm
val MUMFORD_LEMMA : thm
val NEGATIONS_BALL : thm
val NEGATIONS_CBALL : thm
val NEGATIONS_SPHERE : thm
val NET : thm
val NETLIMIT_AT : thm
val NETLIMIT_WITHIN : thm
val NETLIMIT_WITHIN_INTERIOR : thm
val NET_DILEMMA : thm
val NONTRIVIAL_LIMIT_WITHIN : thm
val NOT_BOUNDED_UNIV : thm
val NOT_EVENTUALLY : thm
val NOT_INTERVAL_UNIV : thm
val NOWHERE_DENSE : thm
val NOWHERE_DENSE_COUNTABLE_BIGUNION : thm
val NOWHERE_DENSE_COUNTABLE_BIGUNION_CLOSED : thm
val NOWHERE_DENSE_UNION : thm
val NO_LIMIT_POINT_IMP_CLOSED : thm
val OLDNET : thm
val OPEN : thm
val OPEN_AFFINITY : thm
val OPEN_BALL : thm
val OPEN_BIGINTER : thm
val OPEN_BIGUNION : thm
val OPEN_BIJECTIVE_LINEAR_IMAGE_EQ : thm
val OPEN_CLOSED : thm
val OPEN_CLOSED_INTERVAL : thm
val OPEN_CLOSED_INTERVAL_CONVEX : thm
val OPEN_CONTAINS_BALL : thm
val OPEN_CONTAINS_BALL_EQ : thm
val OPEN_CONTAINS_CBALL : thm
val OPEN_CONTAINS_CBALL_EQ : thm
val OPEN_CONTAINS_INTERVAL : thm
val OPEN_CONTAINS_INTERVAL_OPEN_INTERVAL : thm
val OPEN_CONTAINS_OPEN_INTERVAL : thm
val OPEN_DELETE : thm
val OPEN_DIFF : thm
val OPEN_EMPTY : thm
val OPEN_EXISTS : thm
val OPEN_EXISTS_IN : thm
val OPEN_HALFSPACE_COMPONENT_GT : thm
val OPEN_HALFSPACE_COMPONENT_LT : thm
val OPEN_HALFSPACE_GT : thm
val OPEN_HALFSPACE_LT : thm
val OPEN_IMP_INFINITE : thm
val OPEN_IMP_LOCALLY_COMPACT : thm
val OPEN_IN : thm
val OPEN_INTER : thm
val OPEN_INTERIOR : thm
val OPEN_INTERVAL : thm
val OPEN_INTERVAL_EQ : thm
val OPEN_INTERVAL_LEFT : thm
val OPEN_INTERVAL_LEMMA : thm
val OPEN_INTERVAL_LOWERBOUND : thm
val OPEN_INTERVAL_MIDPOINT : thm
val OPEN_INTERVAL_RIGHT : thm
val OPEN_INTERVAL_UPPERBOUND : thm
val OPEN_INTER_CLOSURE_EQ_EMPTY : thm
val OPEN_INTER_CLOSURE_SUBSET : thm
val OPEN_IN_CONNECTED_COMPONENT : thm
val OPEN_IN_CONTAINS_BALL : thm
val OPEN_IN_CONTAINS_CBALL : thm
val OPEN_IN_DELETE : thm
val OPEN_IN_INTER_OPEN : thm
val OPEN_IN_LOCALLY_COMPACT : thm
val OPEN_IN_OPEN : thm
val OPEN_IN_OPEN_EQ : thm
val OPEN_IN_OPEN_INTER : thm
val OPEN_IN_OPEN_TRANS : thm
val OPEN_IN_REFL : thm
val OPEN_IN_SING : thm
val OPEN_IN_SUBSET_TRANS : thm
val OPEN_IN_SUBTOPOLOGY_INTER_SUBSET : thm
val OPEN_IN_TRANS : thm
val OPEN_IN_TRANS_EQ : thm
val OPEN_MAP_CLOSED_SUPERSET_PREIMAGE : thm
val OPEN_MAP_CLOSED_SUPERSET_PREIMAGE_EQ : thm
val OPEN_MAP_FROM_COMPOSITION_INJECTIVE : thm
val OPEN_MAP_FROM_COMPOSITION_SURJECTIVE : thm
val OPEN_MAP_IFF_LOWER_HEMICONTINUOUS_PREIMAGE : thm
val OPEN_MAP_IMP_CLOSED_MAP : thm
val OPEN_MAP_IMP_QUOTIENT_MAP : thm
val OPEN_MAP_INTERIORS : thm
val OPEN_MAP_RESTRICT : thm
val OPEN_NEGATIONS : thm
val OPEN_OPEN_IN_TRANS : thm
val OPEN_POSITIVE_MULTIPLES : thm
val OPEN_POSITIVE_ORTHANT : thm
val OPEN_SCALING : thm
val OPEN_SEGMENT : thm
val OPEN_SEGMENT_ALT : thm
val OPEN_SEGMENT_LINEAR_IMAGE : thm
val OPEN_SUBSET : thm
val OPEN_SUBSET_INTERIOR : thm
val OPEN_SUB_OPEN : thm
val OPEN_SUMS : thm
val OPEN_SURJECTIVE_LINEAR_IMAGE : thm
val OPEN_TRANSLATION : thm
val OPEN_TRANSLATION_EQ : thm
val OPEN_UNION : thm
val OPEN_UNION_COMPACT_SUBSETS : thm
val OPEN_UNIV : thm
val PAIRWISE_DISJOINT_COMPONENTS : thm
val PARTIAL_SUMS_COMPONENT_LE_INFSUM : thm
val PARTIAL_SUMS_DROP_LE_INFSUM : thm
val PASTING_LEMMA : thm
val PASTING_LEMMA_CLOSED : thm
val PASTING_LEMMA_EXISTS : thm
val PASTING_LEMMA_EXISTS_CLOSED : thm
val PROPER_MAP : thm
val PROPER_MAP_COMPOSE : thm
val PROPER_MAP_FROM_COMPACT : thm
val PROPER_MAP_FROM_COMPOSITION_LEFT : thm
val PROPER_MAP_FROM_COMPOSITION_RIGHT : thm
val QUASICOMPACT_OPEN_CLOSED : thm
val QUOTIENT_MAP_CLOSED_MAP_EQ : thm
val QUOTIENT_MAP_COMPOSE : thm
val QUOTIENT_MAP_FROM_COMPOSITION : thm
val QUOTIENT_MAP_FROM_SUBSET : thm
val QUOTIENT_MAP_IMP_CONTINUOUS_CLOSED : thm
val QUOTIENT_MAP_IMP_CONTINUOUS_OPEN : thm
val QUOTIENT_MAP_OPEN_CLOSED : thm
val QUOTIENT_MAP_OPEN_MAP_EQ : thm
val QUOTIENT_MAP_RESTRICT : thm
val REAL_AFFINITY_EQ : thm
val REAL_AFFINITY_LE : thm
val REAL_AFFINITY_LT : thm
val REAL_ARCH_RDIV_EQ_0 : thm
val REAL_CHOOSE_DIST : thm
val REAL_CHOOSE_SIZE : thm
val REAL_CONVEX_BOUND_LE : thm
val REAL_EQ_AFFINITY : thm
val REAL_EQ_LINV : thm
val REAL_EQ_RINV : thm
val REAL_HAUSDIST_LE : thm
val REAL_HAUSDIST_LE_EQ : thm
val REAL_HAUSDIST_LE_SUMS : thm
val REAL_LE_AFFINITY : thm
val REAL_LE_HAUSDIST : thm
val REAL_LE_SETDIST : thm
val REAL_LE_SETDIST_EQ : thm
val REAL_LT_AFFINITY : thm
val REAL_LT_HAUSDIST_POINT_EXISTS : thm
val REAL_SETDIST_LT_EXISTS : thm
val REFLECT_INTERVAL : thm
val REGULAR_CLOSED_BIGUNION : thm
val REGULAR_CLOSED_UNION : thm
val REGULAR_OPEN_INTER : thm
val SEGMENT : thm
val SEGMENT_CLOSED_OPEN : thm
val SEGMENT_OPEN_SUBSET_CLOSED : thm
val SEGMENT_REFL : thm
val SEGMENT_SCALAR_MULTIPLE : thm
val SEGMENT_SYM : thm
val SEGMENT_TO_CLOSEST_POINT : thm
val SEGMENT_TO_POINT_EXISTS : thm
val SEGMENT_TRANSLATION : thm
val SEPARATE_CLOSED_COMPACT : thm
val SEPARATE_COMPACT_CLOSED : thm
val SEPARATE_POINT_CLOSED : thm
val SEPARATION_CLOSURES : thm
val SEPARATION_HAUSDORFF : thm
val SEPARATION_NORMAL : thm
val SEPARATION_NORMAL_COMPACT : thm
val SEPARATION_NORMAL_LOCAL : thm
val SEPARATION_T0 : thm
val SEPARATION_T1 : thm
val SEPARATION_T2 : thm
val SEQUENCE_CAUCHY_WLOG : thm
val SEQUENCE_INFINITE_LEMMA : thm
val SEQUENCE_UNIQUE_LIMPT : thm
val SEQUENTIALLY : thm
val SEQ_HARMONIC : thm
val SEQ_HARMONIC_OFFSET : thm
val SEQ_OFFSET : thm
val SEQ_OFFSET_NEG : thm
val SEQ_OFFSET_REV : thm
val SERIES_0 : thm
val SERIES_ABSCONV_IMP_CONV : thm
val SERIES_ADD : thm
val SERIES_BOUND : thm
val SERIES_CAUCHY : thm
val SERIES_CAUCHY_UNIFORM : thm
val SERIES_CMUL : thm
val SERIES_COMPARISON : thm
val SERIES_COMPARISON_BOUND : thm
val SERIES_COMPARISON_UNIFORM : thm
val SERIES_COMPONENT : thm
val SERIES_DIFFS : thm
val SERIES_DIRICHLET : thm
val SERIES_DIRICHLET_BILINEAR : thm
val SERIES_DROP_LE : thm
val SERIES_DROP_POS : thm
val SERIES_FINITE : thm
val SERIES_FINITE_SUPPORT : thm
val SERIES_FROM : thm
val SERIES_GOESTOZERO : thm
val SERIES_INJECTIVE_IMAGE : thm
val SERIES_INJECTIVE_IMAGE_STRONG : thm
val SERIES_LINEAR : thm
val SERIES_NEG : thm
val SERIES_RATIO : thm
val SERIES_REARRANGE : thm
val SERIES_REARRANGE_EQ : thm
val SERIES_RESTRICT : thm
val SERIES_SUB : thm
val SERIES_SUBSET : thm
val SERIES_SUM : thm
val SERIES_TERMS_TOZERO : thm
val SERIES_TRIVIAL : thm
val SERIES_UNIQUE : thm
val SETDIST_BALLS : thm
val SETDIST_CLOSED_COMPACT : thm
val SETDIST_CLOSEST_POINT : thm
val SETDIST_CLOSURE : thm
val SETDIST_COMPACT_CLOSED : thm
val SETDIST_DIFFERENCES : thm
val SETDIST_EMPTY : thm
val SETDIST_EQ_0_BOUNDED : thm
val SETDIST_EQ_0_CLOSED : thm
val SETDIST_EQ_0_CLOSED_COMPACT : thm
val SETDIST_EQ_0_CLOSED_IN : thm
val SETDIST_EQ_0_COMPACT_CLOSED : thm
val SETDIST_EQ_0_SING : thm
val SETDIST_FRONTIER : thm
val SETDIST_FRONTIERS : thm
val SETDIST_HAUSDIST_TRIANGLE : thm
val SETDIST_LE_DIST : thm
val SETDIST_LE_HAUSDIST : thm
val SETDIST_LE_SING : thm
val SETDIST_LINEAR_IMAGE : thm
val SETDIST_LIPSCHITZ : thm
val SETDIST_POS_LE : thm
val SETDIST_REFL : thm
val SETDIST_SINGS : thm
val SETDIST_SING_FRONTIER : thm
val SETDIST_SING_FRONTIER_CASES : thm
val SETDIST_SING_IN_SET : thm
val SETDIST_SING_LE_HAUSDIST : thm
val SETDIST_SING_TRIANGLE : thm
val SETDIST_SUBSETS_EQ : thm
val SETDIST_SUBSET_LEFT : thm
val SETDIST_SUBSET_RIGHT : thm
val SETDIST_SYM : thm
val SETDIST_TRANSLATION : thm
val SETDIST_TRIANGLE : thm
val SETDIST_UNIQUE : thm
val SETDIST_UNIV : thm
val SETDIST_ZERO : thm
val SETDIST_ZERO_STRONG : thm
val SET_DIFF_FRONTIER : thm
val SPANNING_SUBSET_INDEPENDENT : thm
val SPAN_0 : thm
val SPAN_ADD : thm
val SPAN_ADD_EQ : thm
val SPAN_BREAKDOWN : thm
val SPAN_BREAKDOWN_EQ : thm
val SPAN_CARD_GE_DIM : thm
val SPAN_CLAUSES : thm
val SPAN_EMPTY : thm
val SPAN_EQ_SELF : thm
val SPAN_EXPLICIT : thm
val SPAN_INC : thm
val SPAN_INDUCT : thm
val SPAN_INDUCT_ALT : thm
val SPAN_LINEAR_IMAGE : thm
val SPAN_MONO : thm
val SPAN_MUL : thm
val SPAN_MUL_EQ : thm
val SPAN_NEG : thm
val SPAN_NEG_EQ : thm
val SPAN_SPAN : thm
val SPAN_STDBASIS : thm
val SPAN_SUB : thm
val SPAN_SUBSET_SUBSPACE : thm
val SPAN_SUBSPACE : thm
val SPAN_SUM : thm
val SPAN_SUPERSET : thm
val SPAN_TRANS : thm
val SPAN_UNION : thm
val SPAN_UNION_SUBSET : thm
val SPAN_UNIV : thm
val SPHERE : thm
val SPHERE_EMPTY : thm
val SPHERE_EQ_EMPTY : thm
val SPHERE_EQ_SING : thm
val SPHERE_LINEAR_IMAGE : thm
val SPHERE_SING : thm
val SPHERE_SUBSET_CBALL : thm
val SPHERE_TRANSLATION : thm
val SPHERE_UNION_BALL : thm
val SUBORDINATE_PARTITION_OF_UNITY : thm
val SUBSET_BALL : thm
val SUBSET_BALLS : thm
val SUBSET_CBALL : thm
val SUBSET_CLOSURE : thm
val SUBSET_INTERIOR : thm
val SUBSET_INTERIOR_EQ : thm
val SUBSET_INTERVAL : thm
val SUBSET_INTERVAL_IMP : thm
val SUBSPACE_0 : thm
val SUBSPACE_ADD : thm
val SUBSPACE_BIGINTER : thm
val SUBSPACE_BOUNDED_EQ_TRIVIAL : thm
val SUBSPACE_IMP_NONEMPTY : thm
val SUBSPACE_INTER : thm
val SUBSPACE_KERNEL : thm
val SUBSPACE_LINEAR_IMAGE : thm
val SUBSPACE_LINEAR_PREIMAGE : thm
val SUBSPACE_MUL : thm
val SUBSPACE_NEG : thm
val SUBSPACE_SPAN : thm
val SUBSPACE_SUB : thm
val SUBSPACE_SUBSTANDARD : thm
val SUBSPACE_SUM : thm
val SUBSPACE_SUMS : thm
val SUBSPACE_TRANSLATION_SELF : thm
val SUBSPACE_TRANSLATION_SELF_EQ : thm
val SUBSPACE_TRIVIAL : thm
val SUBSPACE_UNION_CHAIN : thm
val SUBSPACE_UNIV : thm
val SUMMABLE_0 : thm
val SUMMABLE_ADD : thm
val SUMMABLE_BILINEAR_PARTIAL_PRE : thm
val SUMMABLE_CAUCHY : thm
val SUMMABLE_CMUL : thm
val SUMMABLE_COMPARISON : thm
val SUMMABLE_COMPONENT : thm
val SUMMABLE_EQ : thm
val SUMMABLE_EQ_COFINITE : thm
val SUMMABLE_EQ_EVENTUALLY : thm
val SUMMABLE_FROM_ELSEWHERE : thm
val SUMMABLE_IFF : thm
val SUMMABLE_IFF_COFINITE : thm
val SUMMABLE_IFF_EVENTUALLY : thm
val SUMMABLE_IMP_BOUNDED : thm
val SUMMABLE_IMP_SUMS_BOUNDED : thm
val SUMMABLE_IMP_TOZERO : thm
val SUMMABLE_LINEAR : thm
val SUMMABLE_NEG : thm
val SUMMABLE_REARRANGE : thm
val SUMMABLE_REINDEX : thm
val SUMMABLE_RESTRICT : thm
val SUMMABLE_SUB : thm
val SUMMABLE_SUBSET : thm
val SUMMABLE_SUBSET_ABSCONV : thm
val SUMMABLE_TRIVIAL : thm
val SUMS_0 : thm
val SUMS_EQ : thm
val SUMS_FINITE_DIFF : thm
val SUMS_FINITE_UNION : thm
val SUMS_IFF : thm
val SUMS_INFSUM : thm
val SUMS_INTERVALS : thm
val SUMS_LIM : thm
val SUMS_OFFSET : thm
val SUMS_OFFSET_REV : thm
val SUMS_REINDEX : thm
val SUMS_REINDEX_GEN : thm
val SUMS_SUMMABLE : thm
val SUM_DIFF_LEMMA : thm
val SUP_INSERT : thm
val SURJECTIVE_IMAGE_EQ : thm
val SYMMETRIC_CLOSURE : thm
val SYMMETRIC_INTERIOR : thm
val SYMMETRIC_LINEAR_IMAGE : thm
val TENDSTO_LIM : thm
val TOPSPACE_EUCLIDEAN : thm
val TOPSPACE_EUCLIDEAN_SUBTOPOLOGY : thm
val TRANSITIVE_STEPWISE_LT : thm
val TRANSITIVE_STEPWISE_LT_EQ : thm
val TRANSLATION_DIFF : thm
val TRIVIAL_LIMIT_AT : thm
val TRIVIAL_LIMIT_AT_INFINITY : thm
val TRIVIAL_LIMIT_AT_NEGINFINITY : thm
val TRIVIAL_LIMIT_AT_POSINFINITY : thm
val TRIVIAL_LIMIT_SEQUENTIALLY : thm
val TRIVIAL_LIMIT_WITHIN : thm
val UNBOUNDED_HALFSPACE_COMPONENT_GE : thm
val UNBOUNDED_HALFSPACE_COMPONENT_GT : thm
val UNBOUNDED_HALFSPACE_COMPONENT_LE : thm
val UNBOUNDED_HALFSPACE_COMPONENT_LT : thm
val UNBOUNDED_INTER_COBOUNDED : thm
val UNCOUNTABLE_EUCLIDEAN : thm
val UNCOUNTABLE_INTERVAL : thm
val UNCOUNTABLE_OPEN : thm
val UNCOUNTABLE_REAL : thm
val UNIFORMLY_CAUCHY_IMP_UNIFORMLY_CONVERGENT : thm
val UNIFORMLY_CONTINUOUS_EXTENDS_TO_CLOSURE : thm
val UNIFORMLY_CONTINUOUS_IMP_CAUCHY_CONTINUOUS : thm
val UNIFORMLY_CONTINUOUS_IMP_CONTINUOUS : thm
val UNIFORMLY_CONTINUOUS_ON_ADD : thm
val UNIFORMLY_CONTINUOUS_ON_CLOSURE : thm
val UNIFORMLY_CONTINUOUS_ON_CMUL : thm
val UNIFORMLY_CONTINUOUS_ON_COMPOSE : thm
val UNIFORMLY_CONTINUOUS_ON_CONST : thm
val UNIFORMLY_CONTINUOUS_ON_DIST_CLOSEST_POINT : thm
val UNIFORMLY_CONTINUOUS_ON_EQ : thm
val UNIFORMLY_CONTINUOUS_ON_ID : thm
val UNIFORMLY_CONTINUOUS_ON_MUL : thm
val UNIFORMLY_CONTINUOUS_ON_NEG : thm
val UNIFORMLY_CONTINUOUS_ON_SEQUENTIALLY : thm
val UNIFORMLY_CONTINUOUS_ON_SETDIST : thm
val UNIFORMLY_CONTINUOUS_ON_SUB : thm
val UNIFORMLY_CONTINUOUS_ON_SUBSET : thm
val UNIFORMLY_CONTINUOUS_ON_SUM : thm
val UNIFORMLY_CONTINUOUS_ON_VMUL : thm
val UNIFORMLY_CONVERGENT_EQ_CAUCHY : thm
val UNIFORMLY_CONVERGENT_EQ_CAUCHY_ALT : thm
val UNIFORM_LIM_ADD : thm
val UNIFORM_LIM_BILINEAR : thm
val UNIFORM_LIM_SUB : thm
val UNION_FRONTIER : thm
val UNION_INTERIOR_SUBSET : thm
val UNIT_INTERVAL_NONEMPTY : thm
val UPPER_HEMICONTINUOUS : thm
val UPPER_LOWER_HEMICONTINUOUS : thm
val UPPER_LOWER_HEMICONTINUOUS_EXPLICIT : thm
val URYSOHN : thm
val URYSOHN_LOCAL : thm
val URYSOHN_LOCAL_STRONG : thm
val URYSOHN_STRONG : thm
val WITHIN : thm
val WITHIN_UNIV : thm
val WITHIN_WITHIN : thm
val at : thm
val ball : thm
val closed_def : thm
val continuous_at : thm
val continuous_within : thm
val diameter : thm
val dist : thm
val euclidean : thm
val fsigma : thm
val gdelta : thm
val interval : thm
val limit_point_of : thm
val linear_alt_cmul : thm
val linear_repr : thm
val net_tybij : thm
val open_def : thm
val open_in : thm
val segment : thm
val setdist : thm
val real_topology_grammars : type_grammar.grammar * term_grammar.grammar
(*
[nets] Parent theory of "real_topology"
[real_sigma] Parent theory of "real_topology"
[CLOSED_interval] Definition
⊢ ∀l. interval l = {x | FST (HD l) ≤ x ∧ x ≤ SND (HD l)}
[OPEN_interval] Definition
⊢ ∀a b. interval (a,b) = {x | a < x ∧ x < b}
[at_def] Definition
⊢ ∀z. at z = mk_net (tendsto (mr1,z))
[at_infinity] Definition
⊢ at_infinity = mk_net (λx y. abs x ≥ abs y)
[at_neginfinity] Definition
⊢ at_neginfinity = mk_net (λx y. x ≤ y)
[at_posinfinity] Definition
⊢ at_posinfinity = mk_net (λx y. x ≥ y)
[ball_def] Definition
⊢ ball = mball mr1
[between] Definition
⊢ ∀x a b. between x (a,b) ⇔ dist (a,b) = dist (a,x) + dist (x,b)
[bilinear] Definition
⊢ ∀f. bilinear f ⇔ (∀x. linear (λy. f x y)) ∧ ∀y. linear (λx. f x y)
[bounded_def] Definition
⊢ ∀s. bounded s ⇔ ∃a. ∀x. x ∈ s ⇒ abs x ≤ a
[cauchy_def] Definition
⊢ ∀s. cauchy s ⇔
∀e. 0 < e ⇒ ∃N. ∀m n. m ≥ N ∧ n ≥ N ⇒ dist (s m,s n) < e
[cball] Definition
⊢ ∀x e. cball (x,e) = {y | dist (x,y) ≤ e}
[closed_segment] Definition
⊢ ∀l. segment l =
{(1 − u) * FST (HD l) + u * SND (HD l) | 0 ≤ u ∧ u ≤ 1}
[closest_point] Definition
⊢ ∀s a.
closest_point s a =
@x. x ∈ s ∧ ∀y. y ∈ s ⇒ dist (a,x) ≤ dist (a,y)
[closure] Definition
⊢ ∀s. closure s = s ∪ {x | x limit_point_of s}
[collinear] Definition
⊢ ∀s. collinear s ⇔ ∃u. ∀x y. x ∈ s ∧ y ∈ s ⇒ ∃c. x − y = c * u
[compact] Definition
⊢ ∀s. compact s ⇔
∀f. (∀n. f n ∈ s) ⇒
∃l r.
l ∈ s ∧ (∀m n. m < n ⇒ r m < r n) ∧
(f ∘ r ⟶ l) sequentially
[complete] Definition
⊢ ∀s. complete s ⇔
∀f. (∀n. f n ∈ s) ∧ cauchy f ⇒ ∃l. l ∈ s ∧ (f ⟶ l) sequentially
[components] Definition
⊢ ∀s. components s = {connected_component s x | x | x ∈ s}
[condensation_point_of] Definition
⊢ ∀x s.
x condensation_point_of s ⇔
∀t. x ∈ t ∧ open t ⇒ uncountable (s ∩ t)
[connected] Definition
⊢ ∀s. connected s ⇔
¬∃e1 e2.
open e1 ∧ open e2 ∧ s ⊆ e1 ∪ e2 ∧ e1 ∩ e2 ∩ s = ∅ ∧
e1 ∩ s ≠ ∅ ∧ e2 ∩ s ≠ ∅
[connected_component] Definition
⊢ ∀s x y.
connected_component s x y ⇔
∃t. connected t ∧ t ⊆ s ∧ x ∈ t ∧ y ∈ t
[content] Definition
⊢ ∀s. content s =
if s = ∅ then 0
else interval_upperbound s − interval_lowerbound s
[continuous] Definition
⊢ ∀f net. f continuous net ⇔ (f ⟶ f (netlimit net)) net
[continuous_on] Definition
⊢ ∀f s.
f continuous_on s ⇔
∀x. x ∈ s ⇒
∀e. 0 < e ⇒
∃d. 0 < d ∧
∀x'. x' ∈ s ∧ dist (x',x) < d ⇒ dist (f x',f x) < e
[dependent] Definition
⊢ ∀s. dependent s ⇔ ∃a. a ∈ s ∧ a ∈ span (s DELETE a)
[dim] Definition
⊢ ∀v. dim v =
@n. ∃b. b ⊆ v ∧ independent b ∧ v ⊆ span b ∧ b HAS_SIZE n
[dist_def] Definition
⊢ dist = dist mr1
[euclidean_closed_def] Definition
⊢ closed = closed_in euclidean
[euclidean_def] Definition
⊢ euclidean = mtop mr1
[euclidean_open_def] Definition
⊢ open = open_in euclidean
[eventually] Definition
⊢ ∀p net.
eventually p net ⇔
trivial_limit net ∨
∃y. (∃x. netord net x y) ∧ ∀x. netord net x y ⇒ p x
[frontier] Definition
⊢ ∀s. frontier s = closure s DIFF interior s
[hausdist] Definition
⊢ ∀s t.
hausdist (s,t) =
if
{setdist ({x},t) | x ∈ s} ∪ {setdist ({y},s) | y ∈ t} ≠ ∅ ∧
∃b. ∀d.
d ∈ {setdist ({x},t) | x ∈ s} ∪ {setdist ({y},s) | y ∈ t} ⇒
d ≤ b
then
sup ({setdist ({x},t) | x ∈ s} ∪ {setdist ({y},s) | y ∈ t})
else 0
[homeomorphic] Definition
⊢ ∀s t. s homeomorphic t ⇔ ∃f g. homeomorphism (s,t) (f,g)
[homeomorphism] Definition
⊢ ∀s t f g.
homeomorphism (s,t) (f,g) ⇔
(∀x. x ∈ s ⇒ g (f x) = x) ∧ IMAGE f s = t ∧ f continuous_on s ∧
(∀y. y ∈ t ⇒ f (g y) = y) ∧ IMAGE g t = s ∧ g continuous_on t
[in_direction] Definition
⊢ ∀a v.
(a in_direction v) =
(at a within {b | ∃c. 0 ≤ c ∧ b − a = c * v})
[independent] Definition
⊢ ∀s. independent s ⇔ ¬dependent s
[interior] Definition
⊢ ∀s. interior s = {x | ∃t. open t ∧ x ∈ t ∧ t ⊆ s}
[interval_lowerbound] Definition
⊢ ∀s. interval_lowerbound s = if s = ∅ then 0 else inf s
[interval_upperbound] Definition
⊢ ∀s. interval_upperbound s = if s = ∅ then 0 else sup s
[is_interval] Definition
⊢ ∀s. is_interval s ⇔
∀a b x. a ∈ s ∧ b ∈ s ⇒ a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a ⇒ x ∈ s
[isnet] Definition
⊢ ∀g. isnet g ⇔ ∀x y. (∀z. g z x ⇒ g z y) ∨ ∀z. g z y ⇒ g z x
[limit_point_of_def] Definition
⊢ ∀x s. x limit_point_of s ⇔ limpt euclidean x s
[linear] Definition
⊢ ∀f. linear f ⇔
(∀x y. f (x + y) = f x + f y) ∧ ∀c x. f (c * x) = c * f x
[locally] Definition
⊢ ∀P s.
locally P s ⇔
∀w x.
open_in (subtopology euclidean s) w ∧ x ∈ w ⇒
∃u v.
open_in (subtopology euclidean s) u ∧ P v ∧ x ∈ u ∧ u ⊆ v ∧
v ⊆ w
[midpoint] Definition
⊢ ∀a b. midpoint (a,b) = 2⁻¹ * (a + b)
[net_TY_DEF] Definition
⊢ ∃rep. TYPE_DEFINITION isnet rep
[netlimit] Definition
⊢ ∀net. netlimit net = @a. ∀x. ¬netord net x a
[open_segment] Definition
⊢ ∀a b. segment (a,b) = segment [(a,b)] DIFF {a; b}
[reallim] Definition
⊢ ∀net f. lim net f = @l. (f ⟶ l) net
[sequentially] Definition
⊢ sequentially = mk_net (λm n. m ≥ n)
[set_diameter_def] Definition
⊢ ∀d s.
set_diameter d s =
if s = ∅ then 0 else sup {dist d (x,y) | x ∈ s ∧ y ∈ s}
[set_dist_def] Definition
⊢ ∀d s t.
set_dist d (s,t) =
if s = ∅ ∨ t = ∅ then 0 else inf {dist d (x,y) | x ∈ s ∧ y ∈ t}
[span] Definition
⊢ ∀s. span s = subspace hull s
[sphere] Definition
⊢ ∀x e. sphere (x,e) = {y | dist (x,y) = e}
[subspace] Definition
⊢ ∀s. subspace s ⇔
0 ∈ s ∧ (∀x y. x ∈ s ∧ y ∈ s ⇒ x + y ∈ s) ∧
∀c x. x ∈ s ⇒ c * x ∈ s
[suminf_def] Definition
⊢ ∀s f. suminf s f = @l. (f sums l) s
[summable_def] Definition
⊢ ∀s f. summable s f ⇔ ∃l. (f sums l) s
[sums_def] Definition
⊢ ∀f l s.
(f sums l) s ⇔ ((λn. sum (s ∩ {0 .. n}) f) ⟶ l) sequentially
[tendsto_real] Definition
⊢ ∀f l net.
(f ⟶ l) net ⇔ ∀e. 0 < e ⇒ eventually (λx. dist (f x,l) < e) net
[trivial_limit] Definition
⊢ ∀net.
trivial_limit net ⇔
(∀a b. a = b) ∨
∃a b. a ≠ b ∧ ∀x. ¬netord net x a ∧ ¬netord net x b
[uniformly_continuous_on] Definition
⊢ ∀f s.
f uniformly_continuous_on s ⇔
∀e. 0 < e ⇒
∃d. 0 < d ∧
∀x x'.
x ∈ s ∧ x' ∈ s ∧ dist (x',x) < d ⇒ dist (f x',f x) < e
[within] Definition
⊢ ∀net s. (net within s) = mk_net (λx y. netord net x y ∧ x ∈ s)
[ABS_CAUCHY_SCHWARZ_ABS_EQ] Theorem
⊢ ∀x y.
abs (x * y) = abs x * abs y ⇔
abs x * y = abs y * x ∨ abs x * y = -abs y * x
[ABS_CAUCHY_SCHWARZ_EQ] Theorem
⊢ ∀x y. x * y = abs x * abs y ⇔ abs x * y = abs y * x
[ABS_CAUCHY_SCHWARZ_EQUAL] Theorem
⊢ ∀x y. abs (x * y) = abs x * abs y ⇔ collinear {0; x; y}
[ABS_SUM_TRIVIAL_LEMMA] Theorem
⊢ ∀e. 0 < e ⇒
(P ⇒ abs (sum (s ∩ {m .. n}) f) < e ⇔
P ⇒ n < m ∨ abs (sum (s ∩ {m .. n}) f) < e)
[ABS_TRIANGLE_EQ] Theorem
⊢ ∀x y. abs (x + y) = abs x + abs y ⇔ abs x * y = abs y * x
[ABS_TRIANGLE_LE] Theorem
⊢ ∀x y. abs x + abs y ≤ e ⇒ abs (x + y) ≤ e
[AFFINITY_INVERSES] Theorem
⊢ ∀m c.
m ≠ 0 ⇒
(λx. m * x + c) ∘ (λx. m⁻¹ * x + -(m⁻¹ * c)) = (λx. x) ∧
(λx. m⁻¹ * x + -(m⁻¹ * c)) ∘ (λx. m * x + c) = (λx. x)
[ALWAYS_EVENTUALLY] Theorem
⊢ (∀x. p x) ⇒ eventually p net
[APPROACHABLE_LT_LE] Theorem
⊢ ∀P f.
(∃d. 0 < d ∧ ∀x. f x < d ⇒ P x) ⇔ ∃d. 0 < d ∧ ∀x. f x ≤ d ⇒ P x
[AT] Theorem
⊢ ∀a x y.
netord (at a) x y ⇔ 0 < dist (x,a) ∧ dist (x,a) ≤ dist (y,a)
[AT_INFINITY] Theorem
⊢ ∀x y. netord at_infinity x y ⇔ abs x ≥ abs y
[AT_NEGINFINITY] Theorem
⊢ ∀x y. netord at_neginfinity x y ⇔ x ≤ y
[AT_POSINFINITY] Theorem
⊢ ∀x y. netord at_posinfinity x y ⇔ x ≥ y
[BAIRE] Theorem
⊢ ∀g s.
locally compact s ∧ countable g ∧
(∀t. t ∈ g ⇒ open_in (subtopology euclidean s) t ∧ s ⊆ closure t) ⇒
s ⊆ closure (BIGINTER g)
[BAIRE_ALT] Theorem
⊢ ∀g s.
locally compact s ∧ s ≠ ∅ ∧ countable g ∧ BIGUNION g = s ⇒
∃t u. t ∈ g ∧ open_in (subtopology euclidean s) u ∧ u ⊆ closure t
[BALL] Theorem
⊢ ∀x r.
cball (x,r) = interval [(x − r,x + r)] ∧
ball (x,r) = interval (x − r,x + r)
[BALL_EMPTY] Theorem
⊢ ∀x e. e ≤ 0 ⇒ ball (x,e) = ∅
[BALL_EQ_EMPTY] Theorem
⊢ ∀x e. ball (x,e) = ∅ ⇔ e ≤ 0
[BALL_INTERVAL] Theorem
⊢ ∀x e. ball (x,e) = interval (x − e,x + e)
[BALL_INTERVAL_0] Theorem
⊢ ∀e. ball (0,e) = interval (-e,e)
[BALL_LINEAR_IMAGE] Theorem
⊢ ∀f x r.
linear f ∧ (∀y. ∃x. f x = y) ∧ (∀x. abs (f x) = abs x) ⇒
ball (f x,r) = IMAGE f (ball (x,r))
[BALL_MAX_UNION] Theorem
⊢ ∀a r s. ball (a,max r s) = ball (a,r) ∪ ball (a,s)
[BALL_MIN_INTER] Theorem
⊢ ∀a r s. ball (a,min r s) = ball (a,r) ∩ ball (a,s)
[BALL_SCALING] Theorem
⊢ ∀c. 0 < c ⇒
∀x r. ball (c * x,c * r) = IMAGE (λx. c * x) (ball (x,r))
[BALL_SUBSET_CBALL] Theorem
⊢ ∀x e. ball (x,e) ⊆ cball (x,e)
[BALL_TRANSLATION] Theorem
⊢ ∀a x r. ball (a + x,r) = IMAGE (λy. a + y) (ball (x,r))
[BALL_TRIVIAL] Theorem
⊢ ∀x. ball (x,0) = ∅
[BALL_UNION_SPHERE] Theorem
⊢ ∀a r. ball (a,r) ∪ sphere (a,r) = cball (a,r)
[BANACH_FIX] Theorem
⊢ ∀f s c.
complete s ∧ s ≠ ∅ ∧ 0 ≤ c ∧ c < 1 ∧ IMAGE f s ⊆ s ∧
(∀x y. x ∈ s ∧ y ∈ s ⇒ dist (f x,f y) ≤ c * dist (x,y)) ⇒
∃!x. x ∈ s ∧ f x = x
[BASIS_CARD_EQ_DIM] Theorem
⊢ ∀v b.
b ⊆ v ∧ v ⊆ span b ∧ independent b ⇒ FINITE b ∧ CARD b = dim v
[BASIS_EXISTS] Theorem
⊢ ∀v. ∃b. b ⊆ v ∧ independent b ∧ v ⊆ span b ∧ b HAS_SIZE dim v
[BASIS_HAS_SIZE_DIM] Theorem
⊢ ∀v b. independent b ∧ span b = v ⇒ b HAS_SIZE dim v
[BETWEEN_ABS] Theorem
⊢ ∀a b x.
between x (a,b) ⇔ abs (x − a) * (b − x) = abs (b − x) * (x − a)
[BETWEEN_ANTISYM] Theorem
⊢ ∀a b c. between a (b,c) ∧ between b (a,c) ⇒ a = b
[BETWEEN_IMP_COLLINEAR] Theorem
⊢ ∀a b x. between x (a,b) ⇒ collinear {a; x; b}
[BETWEEN_IN_SEGMENT] Theorem
⊢ ∀x a b. between x (a,b) ⇔ x ∈ segment [(a,b)]
[BETWEEN_MIDPOINT] Theorem
⊢ ∀a b.
between (midpoint (a,b)) (a,b) ∧ between (midpoint (a,b)) (b,a)
[BETWEEN_REFL] Theorem
⊢ ∀a b. between a (a,b) ∧ between b (a,b) ∧ between a (a,a)
[BETWEEN_REFL_EQ] Theorem
⊢ ∀a x. between x (a,a) ⇔ x = a
[BETWEEN_SYM] Theorem
⊢ ∀a b x. between x (a,b) ⇔ between x (b,a)
[BETWEEN_TRANS] Theorem
⊢ ∀a b c d. between a (b,c) ∧ between d (a,c) ⇒ between d (b,c)
[BETWEEN_TRANS_2] Theorem
⊢ ∀a b c d. between a (b,c) ∧ between d (a,b) ⇒ between a (c,d)
[BIGUNION_COMPONENTS] Theorem
⊢ ∀u. u = BIGUNION (components u)
[BIGUNION_CONNECTED_COMPONENT] Theorem
⊢ ∀s. BIGUNION {connected_component s x | x | x ∈ s} = s
[BIGUNION_DIFF] Theorem
⊢ ∀s t. BIGUNION s DIFF t = BIGUNION {x DIFF t | x ∈ s}
[BIGUNION_MONO] Theorem
⊢ (∀x. x ∈ s ⇒ ∃y. y ∈ t ∧ x ⊆ y) ⇒ BIGUNION s ⊆ BIGUNION t
[BIGUNION_MONO_IMAGE] Theorem
⊢ (∀x. x ∈ s ⇒ f x ⊆ g x) ⇒
BIGUNION (IMAGE f s) ⊆ BIGUNION (IMAGE g s)
[BILINEAR_BOUNDED] Theorem
⊢ ∀h. bilinear h ⇒ ∃B. ∀x y. abs (h x y) ≤ B * abs x * abs y
[BILINEAR_BOUNDED_POS] Theorem
⊢ ∀h. bilinear h ⇒ ∃B. 0 < B ∧ ∀x y. abs (h x y) ≤ B * abs x * abs y
[BILINEAR_CONTINUOUS_COMPOSE] Theorem
⊢ ∀net f g h.
f continuous net ∧ g continuous net ∧ bilinear h ⇒
(λx. h (f x) (g x)) continuous net
[BILINEAR_CONTINUOUS_ON_COMPOSE] Theorem
⊢ ∀f g h s.
f continuous_on s ∧ g continuous_on s ∧ bilinear h ⇒
(λx. h (f x) (g x)) continuous_on s
[BILINEAR_DOT] Theorem
⊢ bilinear (λx y. x * y)
[BILINEAR_LADD] Theorem
⊢ ∀h x y z. bilinear h ⇒ h (x + y) z = h x z + h y z
[BILINEAR_LMUL] Theorem
⊢ ∀h c x y. bilinear h ⇒ h (c * x) y = c * h x y
[BILINEAR_LNEG] Theorem
⊢ ∀h x y. bilinear h ⇒ h (-x) y = -h x y
[BILINEAR_LSUB] Theorem
⊢ ∀h x y z. bilinear h ⇒ h (x − y) z = h x z − h y z
[BILINEAR_LZERO] Theorem
⊢ ∀h x. bilinear h ⇒ h 0 x = 0
[BILINEAR_RADD] Theorem
⊢ ∀h x y z. bilinear h ⇒ h x (y + z) = h x y + h x z
[BILINEAR_RMUL] Theorem
⊢ ∀h c x y. bilinear h ⇒ h x (c * y) = c * h x y
[BILINEAR_RNEG] Theorem
⊢ ∀h x y. bilinear h ⇒ h x (-y) = -h x y
[BILINEAR_RSUB] Theorem
⊢ ∀h x y z. bilinear h ⇒ h x (y − z) = h x y − h x z
[BILINEAR_RZERO] Theorem
⊢ ∀h x. bilinear h ⇒ h x 0 = 0
[BILINEAR_SUM] Theorem
⊢ ∀h. bilinear h ∧ FINITE s ∧ FINITE t ⇒
h (sum s f) (sum t g) = sum (s × t) (λ(i,j). h (f i) (g j))
[BILINEAR_SUM_PARTIAL_PRE] Theorem
⊢ ∀f g h m n.
bilinear h ⇒
sum {m .. n} (λk. h (f k) (g k − g (k − 1))) =
if m ≤ n then
h (f (n + 1)) (g n) − h (f m) (g (m − 1)) −
sum {m .. n} (λk. h (f (k + 1) − f k) (g k))
else 0
[BILINEAR_SUM_PARTIAL_SUC] Theorem
⊢ ∀f g h m n.
bilinear h ⇒
sum {m .. n} (λk. h (f k) (g (k + 1) − g k)) =
if m ≤ n then
h (f (n + 1)) (g (n + 1)) − h (f m) (g m) −
sum {m .. n} (λk. h (f (k + 1) − f k) (g (k + 1)))
else 0
[BILINEAR_SWAP] Theorem
⊢ ∀op. bilinear (λx y. op y x) ⇔ bilinear op
[BILINEAR_UNIFORMLY_CONTINUOUS_ON_COMPOSE] Theorem
⊢ ∀f g h s.
f uniformly_continuous_on s ∧ g uniformly_continuous_on s ∧
bilinear h ∧ bounded (IMAGE f s) ∧ bounded (IMAGE g s) ⇒
(λx. h (f x) (g x)) uniformly_continuous_on s
[BOLZANO_WEIERSTRASS] Theorem
⊢ ∀s. bounded s ∧ INFINITE s ⇒ ∃x. x limit_point_of s
[BOLZANO_WEIERSTRASS_CONTRAPOS] Theorem
⊢ ∀s t.
compact s ∧ t ⊆ s ∧ (∀x. x ∈ s ⇒ ¬(x limit_point_of t)) ⇒
FINITE t
[BOLZANO_WEIERSTRASS_IMP_BOUNDED] Theorem
⊢ ∀s. (∀t. INFINITE t ∧ t ⊆ s ⇒ ∃x. x limit_point_of t) ⇒ bounded s
[BOLZANO_WEIERSTRASS_IMP_CLOSED] Theorem
⊢ ∀s. (∀t. INFINITE t ∧ t ⊆ s ⇒ ∃x. x ∈ s ∧ x limit_point_of t) ⇒
closed s
[BOUNDED_BALL] Theorem
⊢ ∀x e. bounded (ball (x,e))
[BOUNDED_BIGINTER] Theorem
⊢ ∀f. (∃s. s ∈ f ∧ bounded s) ⇒ bounded (BIGINTER f)
[BOUNDED_BIGUNION] Theorem
⊢ ∀f. FINITE f ∧ (∀s. s ∈ f ⇒ bounded s) ⇒ bounded (BIGUNION f)
[BOUNDED_CBALL] Theorem
⊢ ∀x e. bounded (cball (x,e))
[BOUNDED_CLOSED_CHAIN] Theorem
⊢ ∀f b.
(∀s. s ∈ f ⇒ closed s ∧ s ≠ ∅) ∧
(∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ∧ b ∈ f ∧ bounded b ⇒
BIGINTER f ≠ ∅
[BOUNDED_CLOSED_IMP_COMPACT] Theorem
⊢ ∀s. bounded s ∧ closed s ⇒ compact s
[BOUNDED_CLOSED_INTERVAL] Theorem
⊢ ∀a b. bounded (interval [(a,b)])
[BOUNDED_CLOSED_NEST] Theorem
⊢ ∀s. (∀n. closed (s n)) ∧ (∀n. s n ≠ ∅) ∧
(∀m n. m ≤ n ⇒ s n ⊆ s m) ∧ bounded (s 0) ⇒
∃a. ∀n. a ∈ s n
[BOUNDED_CLOSURE] Theorem
⊢ ∀s. bounded s ⇒ bounded (closure s)
[BOUNDED_CLOSURE_EQ] Theorem
⊢ ∀s. bounded (closure s) ⇔ bounded s
[BOUNDED_COMPONENTWISE] Theorem
⊢ ∀s. bounded s ⇔ bounded (IMAGE (λx. x) s)
[BOUNDED_DECREASING_CONVERGENT] Theorem
⊢ ∀s. bounded {s n | n ∈ 𝕌(:num)} ∧ (∀n. s (SUC n) ≤ s n) ⇒
∃l. (s ⟶ l) sequentially
[BOUNDED_DIFF] Theorem
⊢ ∀s t. bounded s ⇒ bounded (s DIFF t)
[BOUNDED_DIFFS] Theorem
⊢ ∀s t. bounded s ∧ bounded t ⇒ bounded {x − y | x ∈ s ∧ y ∈ t}
[BOUNDED_EMPTY] Theorem
⊢ bounded ∅
[BOUNDED_EQ_BOLZANO_WEIERSTRASS] Theorem
⊢ ∀s. bounded s ⇔ ∀t. t ⊆ s ∧ INFINITE t ⇒ ∃x. x limit_point_of t
[BOUNDED_FRONTIER] Theorem
⊢ ∀s. bounded s ⇒ bounded (frontier s)
[BOUNDED_HAS_INF] Theorem
⊢ ∀s. bounded s ∧ s ≠ ∅ ⇒
(∀x. x ∈ s ⇒ inf s ≤ x) ∧ ∀b. (∀x. x ∈ s ⇒ b ≤ x) ⇒ b ≤ inf s
[BOUNDED_HAS_SUP] Theorem
⊢ ∀s. bounded s ∧ s ≠ ∅ ⇒
(∀x. x ∈ s ⇒ x ≤ sup s) ∧ ∀b. (∀x. x ∈ s ⇒ x ≤ b) ⇒ sup s ≤ b
[BOUNDED_INCREASING_CONVERGENT] Theorem
⊢ ∀s. bounded {s n | n ∈ 𝕌(:num)} ∧ (∀n. s n ≤ s (SUC n)) ⇒
∃l. (s ⟶ l) sequentially
[BOUNDED_INSERT] Theorem
⊢ ∀x s. bounded (x INSERT s) ⇔ bounded s
[BOUNDED_INTER] Theorem
⊢ ∀s t. bounded s ∨ bounded t ⇒ bounded (s ∩ t)
[BOUNDED_INTERIOR] Theorem
⊢ ∀s. bounded s ⇒ bounded (interior s)
[BOUNDED_INTERVAL] Theorem
⊢ (∀a b. bounded (interval [(a,b)])) ∧ ∀a b. bounded (interval (a,b))
[BOUNDED_LINEAR_IMAGE] Theorem
⊢ ∀f s. bounded s ∧ linear f ⇒ bounded (IMAGE f s)
[BOUNDED_NEGATIONS] Theorem
⊢ ∀s. bounded s ⇒ bounded (IMAGE (λx. -x) s)
[BOUNDED_PARTIAL_SUMS] Theorem
⊢ ∀f k.
bounded {sum {k .. n} f | n ∈ 𝕌(:num)} ⇒
bounded {sum {m .. n} f | m ∈ 𝕌(:num) ∧ n ∈ 𝕌(:num)}
[BOUNDED_POS] Theorem
⊢ ∀s. bounded s ⇔ ∃b. 0 < b ∧ ∀x. x ∈ s ⇒ abs x ≤ b
[BOUNDED_POS_LT] Theorem
⊢ ∀s. bounded s ⇔ ∃b. 0 < b ∧ ∀x. x ∈ s ⇒ abs x < b
[BOUNDED_SCALING] Theorem
⊢ ∀c s. bounded s ⇒ bounded (IMAGE (λx. c * x) s)
[BOUNDED_SING] Theorem
⊢ ∀a. bounded {a}
[BOUNDED_SPHERE] Theorem
⊢ ∀a r. bounded (sphere (a,r))
[BOUNDED_SUBSET] Theorem
⊢ ∀s t. bounded t ∧ s ⊆ t ⇒ bounded s
[BOUNDED_SUBSET_BALL] Theorem
⊢ ∀s x. bounded s ⇒ ∃r. 0 < r ∧ s ⊆ ball (x,r)
[BOUNDED_SUBSET_CBALL] Theorem
⊢ ∀s x. bounded s ⇒ ∃r. 0 < r ∧ s ⊆ cball (x,r)
[BOUNDED_SUBSET_CLOSED_INTERVAL] Theorem
⊢ ∀s. bounded s ⇒ ∃a b. s ⊆ interval [(a,b)]
[BOUNDED_SUBSET_CLOSED_INTERVAL_SYMMETRIC] Theorem
⊢ ∀s. bounded s ⇒ ∃a. s ⊆ interval [(-a,a)]
[BOUNDED_SUBSET_OPEN_INTERVAL] Theorem
⊢ ∀s. bounded s ⇒ ∃a b. s ⊆ interval (a,b)
[BOUNDED_SUBSET_OPEN_INTERVAL_SYMMETRIC] Theorem
⊢ ∀s. bounded s ⇒ ∃a. s ⊆ interval (-a,a)
[BOUNDED_SUMS] Theorem
⊢ ∀s t. bounded s ∧ bounded t ⇒ bounded {x + y | x ∈ s ∧ y ∈ t}
[BOUNDED_SUMS_IMAGE] Theorem
⊢ ∀f g t.
bounded {f x | x ∈ t} ∧ bounded {g x | x ∈ t} ⇒
bounded {f x + g x | x ∈ t}
[BOUNDED_SUMS_IMAGES] Theorem
⊢ ∀f t s.
FINITE s ∧ (∀a. a ∈ s ⇒ bounded {f x a | x ∈ t}) ⇒
bounded {sum s (f x) | x ∈ t}
[BOUNDED_TRANSLATION] Theorem
⊢ ∀a s. bounded s ⇒ bounded (IMAGE (λx. a + x) s)
[BOUNDED_TRANSLATION_EQ] Theorem
⊢ ∀a s. bounded (IMAGE (λx. a + x) s) ⇔ bounded s
[BOUNDED_UNIFORMLY_CONTINUOUS_IMAGE] Theorem
⊢ ∀f s. f uniformly_continuous_on s ∧ bounded s ⇒ bounded (IMAGE f s)
[BOUNDED_UNION] Theorem
⊢ ∀s t. bounded (s ∪ t) ⇔ bounded s ∧ bounded t
[CARD_EQ_BALL] Theorem
⊢ ∀a r. 0 < r ⇒ ball (a,r) ≈ 𝕌(:real)
[CARD_EQ_CBALL] Theorem
⊢ ∀a r. 0 < r ⇒ cball (a,r) ≈ 𝕌(:real)
[CARD_EQ_EUCLIDEAN] Theorem
⊢ 𝕌(:real) ≈ 𝕌(:real)
[CARD_EQ_INTERVAL] Theorem
⊢ (∀a b. interval (a,b) ≠ ∅ ⇒ interval [(a,b)] ≈ 𝕌(:real)) ∧
∀a b. interval (a,b) ≠ ∅ ⇒ interval (a,b) ≈ 𝕌(:real)
[CARD_EQ_OPEN] Theorem
⊢ ∀s. open s ∧ s ≠ ∅ ⇒ s ≈ 𝕌(:real)
[CARD_EQ_REAL] Theorem
⊢ 𝕌(:real) ≈ 𝕌(:num -> bool)
[CARD_EQ_REAL_IMP_UNCOUNTABLE] Theorem
⊢ ∀s. s ≈ 𝕌(:real) ⇒ uncountable s
[CARD_FRONTIER_INTERVAL] Theorem
⊢ ∀s. is_interval s ⇒ FINITE (frontier s) ∧ CARD (frontier s) ≤ 2
[CARD_GE_DIM_INDEPENDENT] Theorem
⊢ ∀v b. b ⊆ v ∧ independent b ∧ dim v ≤ CARD b ⇒ v ⊆ span b
[CARD_STDBASIS] Theorem
⊢ CARD {1} = 1
[CAUCHY] Theorem
⊢ ∀s. cauchy s ⇔ ∀e. 0 < e ⇒ ∃N. ∀n. n ≥ N ⇒ dist (s n,s N) < e
[CAUCHY_CONTINUOUS_EXTENDS_TO_CLOSURE] Theorem
⊢ ∀f s.
(∀x. cauchy x ∧ (∀n. x n ∈ s) ⇒ cauchy (f ∘ x)) ⇒
∃g. g continuous_on closure s ∧ ∀x. x ∈ s ⇒ g x = f x
[CAUCHY_CONTINUOUS_IMP_CONTINUOUS] Theorem
⊢ ∀f s.
(∀x. cauchy x ∧ (∀n. x n ∈ s) ⇒ cauchy (f ∘ x)) ⇒
f continuous_on s
[CAUCHY_CONTINUOUS_UNIQUENESS_LEMMA] Theorem
⊢ ∀f s.
(∀x. cauchy x ∧ (∀n. x n ∈ s) ⇒ cauchy (f ∘ x)) ⇒
∀a x.
(∀n. x n ∈ s) ∧ (x ⟶ a) sequentially ⇒
∃l. (f ∘ x ⟶ l) sequentially ∧
∀y. (∀n. y n ∈ s) ∧ (y ⟶ a) sequentially ⇒
(f ∘ y ⟶ l) sequentially
[CAUCHY_IMP_BOUNDED] Theorem
⊢ ∀s. cauchy s ⇒ bounded {y | (∃n. y = s n)}
[CAUCHY_ISOMETRIC] Theorem
⊢ ∀f s e x.
0 < e ∧ subspace s ∧ linear f ∧
(∀x. x ∈ s ⇒ abs (f x) ≥ e * abs x) ∧ (∀n. x n ∈ s) ∧
cauchy (f ∘ x) ⇒
cauchy x
[CBALL_DIFF_BALL] Theorem
⊢ ∀a r. cball (a,r) DIFF ball (a,r) = sphere (a,r)
[CBALL_DIFF_SPHERE] Theorem
⊢ ∀a r. cball (a,r) DIFF sphere (a,r) = ball (a,r)
[CBALL_EMPTY] Theorem
⊢ ∀x e. e < 0 ⇒ cball (x,e) = ∅
[CBALL_EQ_EMPTY] Theorem
⊢ ∀x e. cball (x,e) = ∅ ⇔ e < 0
[CBALL_EQ_SING] Theorem
⊢ ∀x e. cball (x,e) = {x} ⇔ e = 0
[CBALL_INTERVAL] Theorem
⊢ ∀x e. cball (x,e) = interval [(x − e,x + e)]
[CBALL_INTERVAL_0] Theorem
⊢ ∀e. cball (0,e) = interval [(-e,e)]
[CBALL_LINEAR_IMAGE] Theorem
⊢ ∀f x r.
linear f ∧ (∀y. ∃x. f x = y) ∧ (∀x. abs (f x) = abs x) ⇒
cball (f x,r) = IMAGE f (cball (x,r))
[CBALL_MAX_UNION] Theorem
⊢ ∀a r s. cball (a,max r s) = cball (a,r) ∪ cball (a,s)
[CBALL_MIN_INTER] Theorem
⊢ ∀x d e. cball (x,min d e) = cball (x,d) ∩ cball (x,e)
[CBALL_SCALING] Theorem
⊢ ∀c. 0 < c ⇒
∀x r. cball (c * x,c * r) = IMAGE (λx. c * x) (cball (x,r))
[CBALL_SING] Theorem
⊢ ∀x e. e = 0 ⇒ cball (x,e) = {x}
[CBALL_TRANSLATION] Theorem
⊢ ∀a x r. cball (a + x,r) = IMAGE (λy. a + y) (cball (x,r))
[CBALL_TRIVIAL] Theorem
⊢ ∀x. cball (x,0) = {x}
[CENTRE_IN_BALL] Theorem
⊢ ∀x e. x ∈ ball (x,e) ⇔ 0 < e
[CENTRE_IN_CBALL] Theorem
⊢ ∀x e. x ∈ cball (x,e) ⇔ 0 ≤ e
[CLOPEN] Theorem
⊢ ∀s. closed s ∧ open s ⇔ s = ∅ ∨ s = 𝕌(:real)
[CLOPEN_BIGUNION_COMPONENTS] Theorem
⊢ ∀u s.
closed_in (subtopology euclidean u) s ∧
open_in (subtopology euclidean u) s ⇒
∃k. k ⊆ components u ∧ s = BIGUNION k
[CLOPEN_IN_COMPONENTS] Theorem
⊢ ∀u s.
closed_in (subtopology euclidean u) s ∧
open_in (subtopology euclidean u) s ∧ connected s ∧ s ≠ ∅ ⇒
s ∈ components u
[CLOSED] Theorem
⊢ ∀s. closed s ⇔
∀x. (∀e. 0 < e ⇒ ∃x'. x' ∈ s ∧ x' ≠ x ∧ abs (x' − x) < e) ⇒
x ∈ s
[CLOSED_APPROACHABLE] Theorem
⊢ ∀x s. closed s ⇒ ((∀e. 0 < e ⇒ ∃y. y ∈ s ∧ dist (y,x) < e) ⇔ x ∈ s)
[CLOSED_AS_GDELTA] Theorem
⊢ ∀s. closed s ⇒ gdelta s
[CLOSED_BIGINTER] Theorem
⊢ ∀f. (∀s. s ∈ f ⇒ closed s) ⇒ closed (BIGINTER f)
[CLOSED_BIGINTER_COMPACT] Theorem
⊢ ∀s. closed s ⇔ ∀e. compact (cball (0,e) ∩ s)
[CLOSED_BIGUNION] Theorem
⊢ ∀s. FINITE s ∧ (∀t. t ∈ s ⇒ closed t) ⇒ closed (BIGUNION s)
[CLOSED_CBALL] Theorem
⊢ ∀x e. closed (cball (x,e))
[CLOSED_CLOSURE] Theorem
⊢ ∀s. closed (closure s)
[CLOSED_COMPACT_DIFFERENCES] Theorem
⊢ ∀s t. closed s ∧ compact t ⇒ closed {x − y | x ∈ s ∧ y ∈ t}
[CLOSED_COMPACT_SUMS] Theorem
⊢ ∀s t. closed s ∧ compact t ⇒ closed {x + y | x ∈ s ∧ y ∈ t}
[CLOSED_COMPONENTS] Theorem
⊢ ∀s c. closed s ∧ c ∈ components s ⇒ closed c
[CLOSED_CONNECTED_COMPONENT] Theorem
⊢ ∀s x. closed s ⇒ closed (connected_component s x)
[CLOSED_CONTAINS_SEQUENTIAL_LIMIT] Theorem
⊢ ∀s x l. closed s ∧ (∀n. x n ∈ s) ∧ (x ⟶ l) sequentially ⇒ l ∈ s
[CLOSED_DIFF] Theorem
⊢ ∀s t. closed s ∧ open t ⇒ closed (s DIFF t)
[CLOSED_DIFF_OPEN_INTERVAL] Theorem
⊢ ∀a b.
interval [(a,b)] DIFF interval (a,b) =
if interval [(a,b)] = ∅ then ∅ else {a; b}
[CLOSED_EMPTY] Theorem
⊢ closed ∅
[CLOSED_FIP] Theorem
⊢ ∀f. (∀t. t ∈ f ⇒ closed t) ∧ (∃t. t ∈ f ∧ bounded t) ∧
(∀f'. FINITE f' ∧ f' ⊆ f ⇒ BIGINTER f' ≠ ∅) ⇒
BIGINTER f ≠ ∅
[CLOSED_FORALL] Theorem
⊢ ∀Q. (∀a. closed {x | Q a x}) ⇒ closed {x | (∀a. Q a x)}
[CLOSED_FORALL_IN] Theorem
⊢ ∀P Q.
(∀a. P a ⇒ closed {x | Q a x}) ⇒ closed {x | (∀a. P a ⇒ Q a x)}
[CLOSED_HALFSPACE_COMPONENT_GE] Theorem
⊢ ∀a. closed {x | x ≥ a}
[CLOSED_HALFSPACE_COMPONENT_LE] Theorem
⊢ ∀a. closed {x | x ≤ a}
[CLOSED_HALFSPACE_GE] Theorem
⊢ ∀a b. closed {x | a * x ≥ b}
[CLOSED_HALFSPACE_LE] Theorem
⊢ ∀a b. closed {x | a * x ≤ b}
[CLOSED_HYPERPLANE] Theorem
⊢ ∀a b. closed {x | a * x = b}
[CLOSED_IMP_FIP] Theorem
⊢ ∀s f.
closed s ∧ (∀t. t ∈ f ⇒ closed t) ∧ (∃t. t ∈ f ∧ bounded t) ∧
(∀f'. FINITE f' ∧ f' ⊆ f ⇒ s ∩ BIGINTER f' ≠ ∅) ⇒
s ∩ BIGINTER f ≠ ∅
[CLOSED_IMP_FIP_COMPACT] Theorem
⊢ ∀s f.
closed s ∧ (∀t. t ∈ f ⇒ compact t) ∧
(∀f'. FINITE f' ∧ f' ⊆ f ⇒ s ∩ BIGINTER f' ≠ ∅) ⇒
s ∩ BIGINTER f ≠ ∅
[CLOSED_IMP_LOCALLY_COMPACT] Theorem
⊢ ∀s. closed s ⇒ locally compact s
[CLOSED_IN] Theorem
⊢ ∀s. closed s ⇔ closed_in euclidean s
[CLOSED_INJECTIVE_IMAGE_SUBSPACE] Theorem
⊢ ∀f s.
subspace s ∧ linear f ∧ (∀x. x ∈ s ∧ f x = 0 ⇒ x = 0) ∧ closed s ⇒
closed (IMAGE f s)
[CLOSED_INJECTIVE_LINEAR_IMAGE] Theorem
⊢ ∀f. linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
∀s. closed s ⇒ closed (IMAGE f s)
[CLOSED_INJECTIVE_LINEAR_IMAGE_EQ] Theorem
⊢ ∀f s.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
(closed (IMAGE f s) ⇔ closed s)
[CLOSED_INSERT] Theorem
⊢ ∀a s. closed s ⇒ closed (a INSERT s)
[CLOSED_INTER] Theorem
⊢ ∀s t. closed s ∧ closed t ⇒ closed (s ∩ t)
[CLOSED_INTERVAL] Theorem
⊢ ∀a b. closed (interval [(a,b)])
[CLOSED_INTERVAL_EQ] Theorem
⊢ (∀a b. closed (interval [(a,b)])) ∧
∀a b. closed (interval (a,b)) ⇔ interval (a,b) = ∅
[CLOSED_INTERVAL_IMAGE_UNIT_INTERVAL] Theorem
⊢ ∀a b.
interval [(a,b)] ≠ ∅ ⇒
interval [(a,b)] =
IMAGE (λx. a + x)
(IMAGE (λx. @f. f = (b − a) * x) (interval [(0,1)]))
[CLOSED_INTERVAL_LEFT] Theorem
⊢ ∀b. closed {x | x ≤ b}
[CLOSED_INTERVAL_RIGHT] Theorem
⊢ ∀a. closed {x | a ≤ x}
[CLOSED_INTER_COMPACT] Theorem
⊢ ∀s t. closed s ∧ compact t ⇒ compact (s ∩ t)
[CLOSED_IN_CLOSED] Theorem
⊢ ∀s u.
closed_in (subtopology euclidean u) s ⇔ ∃t. closed t ∧ s = u ∩ t
[CLOSED_IN_CLOSED_EQ] Theorem
⊢ ∀s t.
closed s ⇒
(closed_in (subtopology euclidean s) t ⇔ closed t ∧ t ⊆ s)
[CLOSED_IN_CLOSED_INTER] Theorem
⊢ ∀u s. closed s ⇒ closed_in (subtopology euclidean u) (u ∩ s)
[CLOSED_IN_CLOSED_TRANS] Theorem
⊢ ∀s t. closed_in (subtopology euclidean t) s ∧ closed t ⇒ closed s
[CLOSED_IN_COMPACT] Theorem
⊢ ∀s t. compact s ∧ closed_in (subtopology euclidean s) t ⇒ compact t
[CLOSED_IN_COMPACT_EQ] Theorem
⊢ ∀s t.
compact s ⇒
(closed_in (subtopology euclidean s) t ⇔ compact t ∧ t ⊆ s)
[CLOSED_IN_COMPONENT] Theorem
⊢ ∀s c. c ∈ components s ⇒ closed_in (subtopology euclidean s) c
[CLOSED_IN_CONNECTED_COMPONENT] Theorem
⊢ ∀s x. closed_in (subtopology euclidean s) (connected_component s x)
[CLOSED_IN_INTER_CLOSED] Theorem
⊢ ∀s t u.
closed_in (subtopology euclidean u) s ∧ closed t ⇒
closed_in (subtopology euclidean u) (s ∩ t)
[CLOSED_IN_INTER_CLOSURE] Theorem
⊢ ∀s t. closed_in (subtopology euclidean s) t ⇔ s ∩ closure t = t
[CLOSED_IN_LIMPT] Theorem
⊢ ∀s t.
closed_in (subtopology euclidean t) s ⇔
s ⊆ t ∧ ∀x. x limit_point_of s ∧ x ∈ t ⇒ x ∈ s
[CLOSED_IN_REFL] Theorem
⊢ ∀s. closed_in (subtopology euclidean s) s
[CLOSED_IN_SING] Theorem
⊢ ∀u x. closed_in (subtopology euclidean u) {x} ⇔ x ∈ u
[CLOSED_IN_SUBSET_TRANS] Theorem
⊢ ∀s t u.
closed_in (subtopology euclidean u) s ∧ s ⊆ t ∧ t ⊆ u ⇒
closed_in (subtopology euclidean t) s
[CLOSED_IN_TRANS] Theorem
⊢ ∀s t u.
closed_in (subtopology euclidean t) s ∧
closed_in (subtopology euclidean u) t ⇒
closed_in (subtopology euclidean u) s
[CLOSED_IN_TRANS_EQ] Theorem
⊢ ∀s t.
(∀u. closed_in (subtopology euclidean t) u ⇒
closed_in (subtopology euclidean s) t) ⇔
closed_in (subtopology euclidean s) t
[CLOSED_LIMPT] Theorem
⊢ ∀s. closed s ⇔ ∀x. x limit_point_of s ⇒ x ∈ s
[CLOSED_LIMPTS] Theorem
⊢ ∀s. closed {x | x limit_point_of s}
[CLOSED_MAP_CLOSURES] Theorem
⊢ ∀f. (∀s. closed s ⇒ closed (IMAGE f s)) ⇔
∀s. closure (IMAGE f s) ⊆ IMAGE f (closure s)
[CLOSED_MAP_FROM_COMPOSITION_INJECTIVE] Theorem
⊢ ∀f g s t u.
IMAGE f s ⊆ t ∧ IMAGE g t ⊆ u ∧ g continuous_on t ∧
(∀x y. x ∈ t ∧ y ∈ t ∧ g x = g y ⇒ x = y) ∧
(∀k. closed_in (subtopology euclidean s) k ⇒
closed_in (subtopology euclidean u) (IMAGE (g ∘ f) k)) ⇒
∀k. closed_in (subtopology euclidean s) k ⇒
closed_in (subtopology euclidean t) (IMAGE f k)
[CLOSED_MAP_FROM_COMPOSITION_SURJECTIVE] Theorem
⊢ ∀f g s t u.
f continuous_on s ∧ IMAGE f s = t ∧ IMAGE g t ⊆ u ∧
(∀k. closed_in (subtopology euclidean s) k ⇒
closed_in (subtopology euclidean u) (IMAGE (g ∘ f) k)) ⇒
∀k. closed_in (subtopology euclidean t) k ⇒
closed_in (subtopology euclidean u) (IMAGE g k)
[CLOSED_MAP_IFF_UPPER_HEMICONTINUOUS_PREIMAGE] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ⇒
((∀u. closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean t) (IMAGE f u)) ⇔
∀u. open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean t)
{y | y ∈ t ∧ {x | x ∈ s ∧ f x = y} ⊆ u})
[CLOSED_MAP_IMP_OPEN_MAP] Theorem
⊢ ∀f s t.
IMAGE f s = t ∧
(∀u. closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean t) (IMAGE f u)) ∧
(∀u. open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean s)
{x | x ∈ s ∧ f x ∈ IMAGE f u}) ⇒
∀u. open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean t) (IMAGE f u)
[CLOSED_MAP_IMP_QUOTIENT_MAP] Theorem
⊢ ∀f s.
f continuous_on s ∧
(∀t. closed_in (subtopology euclidean s) t ⇒
closed_in (subtopology euclidean (IMAGE f s)) (IMAGE f t)) ⇒
∀t. t ⊆ IMAGE f s ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t} ⇔
open_in (subtopology euclidean (IMAGE f s)) t)
[CLOSED_MAP_OPEN_SUPERSET_PREIMAGE] Theorem
⊢ ∀f s t u w.
(∀k. closed_in (subtopology euclidean s) k ⇒
closed_in (subtopology euclidean t) (IMAGE f k)) ∧
open_in (subtopology euclidean s) u ∧ w ⊆ t ∧
{x | x ∈ s ∧ f x ∈ w} ⊆ u ⇒
∃v. open_in (subtopology euclidean t) v ∧ w ⊆ v ∧
{x | x ∈ s ∧ f x ∈ v} ⊆ u
[CLOSED_MAP_OPEN_SUPERSET_PREIMAGE_EQ] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ⇒
((∀k. closed_in (subtopology euclidean s) k ⇒
closed_in (subtopology euclidean t) (IMAGE f k)) ⇔
∀u w.
open_in (subtopology euclidean s) u ∧ w ⊆ t ∧
{x | x ∈ s ∧ f x ∈ w} ⊆ u ⇒
∃v. open_in (subtopology euclidean t) v ∧ w ⊆ v ∧
{x | x ∈ s ∧ f x ∈ v} ⊆ u)
[CLOSED_MAP_OPEN_SUPERSET_PREIMAGE_POINT] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ⇒
((∀k. closed_in (subtopology euclidean s) k ⇒
closed_in (subtopology euclidean t) (IMAGE f k)) ⇔
∀u y.
open_in (subtopology euclidean s) u ∧ y ∈ t ∧
{x | x ∈ s ∧ f x = y} ⊆ u ⇒
∃v. open_in (subtopology euclidean t) v ∧ y ∈ v ∧
{x | x ∈ s ∧ f x ∈ v} ⊆ u)
[CLOSED_MAP_RESTRICT] Theorem
⊢ ∀f s t t'.
(∀u. closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean t) (IMAGE f u)) ∧ t' ⊆ t ⇒
∀u. closed_in (subtopology euclidean {x | x ∈ s ∧ f x ∈ t'}) u ⇒
closed_in (subtopology euclidean t') (IMAGE f u)
[CLOSED_NEGATIONS] Theorem
⊢ ∀s. closed s ⇒ closed (IMAGE (λx. -x) s)
[CLOSED_OPEN_INTERVAL] Theorem
⊢ ∀a b. a ≤ b ⇒ interval [(a,b)] = interval (a,b) ∪ {a; b}
[CLOSED_POSITIVE_ORTHANT] Theorem
⊢ closed {x | 0 ≤ x}
[CLOSED_SCALING] Theorem
⊢ ∀s c. closed s ⇒ closed (IMAGE (λx. c * x) s)
[CLOSED_SEGMENT_LINEAR_IMAGE] Theorem
⊢ ∀f a b. linear f ⇒ segment [(f a,f b)] = IMAGE f (segment [(a,b)])
[CLOSED_SEQUENTIAL_LIMITS] Theorem
⊢ ∀s. closed s ⇔ ∀x l. (∀n. x n ∈ s) ∧ (x ⟶ l) sequentially ⇒ l ∈ s
[CLOSED_SING] Theorem
⊢ ∀a. closed {a}
[CLOSED_SPHERE] Theorem
⊢ ∀a r. closed (sphere (a,r))
[CLOSED_STANDARD_HYPERPLANE] Theorem
⊢ ∀a. closed {x | x = a}
[CLOSED_SUBSET] Theorem
⊢ ∀s t. s ⊆ t ∧ closed s ⇒ closed_in (subtopology euclidean t) s
[CLOSED_SUBSET_EQ] Theorem
⊢ ∀u s. closed s ⇒ (closed_in (subtopology euclidean u) s ⇔ s ⊆ u)
[CLOSED_SUBSTANDARD] Theorem
⊢ closed {x | x = 0}
[CLOSED_UNION] Theorem
⊢ ∀s t. closed s ∧ closed t ⇒ closed (s ∪ t)
[CLOSED_UNION_COMPACT_SUBSETS] Theorem
⊢ ∀s. closed s ⇒
∃f. (∀n. compact (f n)) ∧ (∀n. f n ⊆ s) ∧
(∀n. f n ⊆ f (n + 1)) ∧ BIGUNION {f n | n ∈ 𝕌(:num)} = s ∧
∀k. compact k ∧ k ⊆ s ⇒ ∃N. ∀n. n ≥ N ⇒ k ⊆ f n
[CLOSED_UNIV] Theorem
⊢ closed 𝕌(:real)
[CLOSEST_POINT_EXISTS] Theorem
⊢ ∀s a.
closed s ∧ s ≠ ∅ ⇒
closest_point s a ∈ s ∧
∀y. y ∈ s ⇒ dist (a,closest_point s a) ≤ dist (a,y)
[CLOSEST_POINT_IN_FRONTIER] Theorem
⊢ ∀s x.
closed s ∧ s ≠ ∅ ∧ x ∉ interior s ⇒
closest_point s x ∈ frontier s
[CLOSEST_POINT_IN_INTERIOR] Theorem
⊢ ∀s x.
closed s ∧ s ≠ ∅ ⇒
(closest_point s x ∈ interior s ⇔ x ∈ interior s)
[CLOSEST_POINT_IN_SET] Theorem
⊢ ∀s a. closed s ∧ s ≠ ∅ ⇒ closest_point s a ∈ s
[CLOSEST_POINT_LE] Theorem
⊢ ∀s a x. closed s ∧ x ∈ s ⇒ dist (a,closest_point s a) ≤ dist (a,x)
[CLOSEST_POINT_REFL] Theorem
⊢ ∀s x. closed s ∧ s ≠ ∅ ⇒ (closest_point s x = x ⇔ x ∈ s)
[CLOSEST_POINT_SELF] Theorem
⊢ ∀s x. x ∈ s ⇒ closest_point s x = x
[CLOSURE_APPROACHABLE] Theorem
⊢ ∀x s. x ∈ closure s ⇔ ∀e. 0 < e ⇒ ∃y. y ∈ s ∧ dist (y,x) < e
[CLOSURE_BALL] Theorem
⊢ ∀x e. 0 < e ⇒ closure (ball (x,e)) = cball (x,e)
[CLOSURE_BIGINTER_SUBSET] Theorem
⊢ ∀f. closure (BIGINTER f) ⊆ BIGINTER (IMAGE closure f)
[CLOSURE_BIGUNION] Theorem
⊢ ∀f. FINITE f ⇒ closure (BIGUNION f) = BIGUNION {closure s | s ∈ f}
[CLOSURE_BOUNDED_LINEAR_IMAGE] Theorem
⊢ ∀f s.
linear f ∧ bounded s ⇒ closure (IMAGE f s) = IMAGE f (closure s)
[CLOSURE_CLOSED] Theorem
⊢ ∀s. closed s ⇒ closure s = s
[CLOSURE_CLOSURE] Theorem
⊢ ∀s. closure (closure s) = closure s
[CLOSURE_COMPLEMENT] Theorem
⊢ ∀s. closure (𝕌(:real) DIFF s) = 𝕌(:real) DIFF interior s
[CLOSURE_EMPTY] Theorem
⊢ closure ∅ = ∅
[CLOSURE_EQ] Theorem
⊢ ∀s. closure s = s ⇔ closed s
[CLOSURE_EQ_EMPTY] Theorem
⊢ ∀s. closure s = ∅ ⇔ s = ∅
[CLOSURE_HALFSPACE_COMPONENT_GT] Theorem
⊢ ∀a. closure {x | x > a} = {x | x ≥ a}
[CLOSURE_HALFSPACE_COMPONENT_LT] Theorem
⊢ ∀a. closure {x | x < a} = {x | x ≤ a}
[CLOSURE_HALFSPACE_GT] Theorem
⊢ ∀a b. a ≠ 0 ⇒ closure {x | a * x > b} = {x | a * x ≥ b}
[CLOSURE_HALFSPACE_LT] Theorem
⊢ ∀a b. a ≠ 0 ⇒ closure {x | a * x < b} = {x | a * x ≤ b}
[CLOSURE_HULL] Theorem
⊢ ∀s. closure s = closed hull s
[CLOSURE_HYPERPLANE] Theorem
⊢ ∀a b. closure {x | a * x = b} = {x | a * x = b}
[CLOSURE_IMAGE_BOUNDED] Theorem
⊢ ∀f s.
f continuous_on closure s ∧ bounded s ⇒
closure (IMAGE f s) = IMAGE f (closure s)
[CLOSURE_IMAGE_CLOSURE] Theorem
⊢ ∀f s.
f continuous_on closure s ⇒
closure (IMAGE f (closure s)) = closure (IMAGE f s)
[CLOSURE_INJECTIVE_LINEAR_IMAGE] Theorem
⊢ ∀f s.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
closure (IMAGE f s) = IMAGE f (closure s)
[CLOSURE_INTERIOR] Theorem
⊢ ∀s. closure s = 𝕌(:real) DIFF interior (𝕌(:real) DIFF s)
[CLOSURE_INTERIOR_IDEMP] Theorem
⊢ ∀s. closure (interior (closure (interior s))) =
closure (interior s)
[CLOSURE_INTERIOR_UNION_CLOSED] Theorem
⊢ ∀s t.
closed s ∧ closed t ⇒
closure (interior (s ∪ t)) =
closure (interior s) ∪ closure (interior t)
[CLOSURE_INTERVAL] Theorem
⊢ (∀a b. closure (interval [(a,b)]) = interval [(a,b)]) ∧
∀a b.
closure (interval (a,b)) =
if interval (a,b) = ∅ then ∅ else interval [(a,b)]
[CLOSURE_INTER_SUBSET] Theorem
⊢ ∀s t. closure (s ∩ t) ⊆ closure s ∩ closure t
[CLOSURE_LINEAR_IMAGE_SUBSET] Theorem
⊢ ∀f s. linear f ⇒ IMAGE f (closure s) ⊆ closure (IMAGE f s)
[CLOSURE_MINIMAL] Theorem
⊢ ∀s t. s ⊆ t ∧ closed t ⇒ closure s ⊆ t
[CLOSURE_MINIMAL_EQ] Theorem
⊢ ∀s t. closed t ⇒ (closure s ⊆ t ⇔ s ⊆ t)
[CLOSURE_NEGATIONS] Theorem
⊢ ∀s. closure (IMAGE (λx. -x) s) = IMAGE (λx. -x) (closure s)
[CLOSURE_NONEMPTY_OPEN_INTER] Theorem
⊢ ∀s x. x ∈ closure s ⇔ ∀t. x ∈ t ∧ open t ⇒ s ∩ t ≠ ∅
[CLOSURE_OPEN_INTERVAL] Theorem
⊢ ∀a b.
interval (a,b) ≠ ∅ ⇒ closure (interval (a,b)) = interval [(a,b)]
[CLOSURE_OPEN_INTER_CLOSURE] Theorem
⊢ ∀s t. open s ⇒ closure (s ∩ closure t) = closure (s ∩ t)
[CLOSURE_OPEN_INTER_SUPERSET] Theorem
⊢ ∀s t. open s ∧ s ⊆ closure t ⇒ closure (s ∩ t) = closure s
[CLOSURE_OPEN_IN_INTER_CLOSURE] Theorem
⊢ ∀s t u.
open_in (subtopology euclidean u) s ∧ t ⊆ u ⇒
closure (s ∩ closure t) = closure (s ∩ t)
[CLOSURE_SEQUENTIAL] Theorem
⊢ ∀s l. l ∈ closure s ⇔ ∃x. (∀n. x n ∈ s) ∧ (x ⟶ l) sequentially
[CLOSURE_SING] Theorem
⊢ ∀x. closure {x} = {x}
[CLOSURE_SUBSET] Theorem
⊢ ∀s. s ⊆ closure s
[CLOSURE_SUBSET_EQ] Theorem
⊢ ∀s. closure s ⊆ s ⇔ closed s
[CLOSURE_SUMS] Theorem
⊢ ∀s t.
bounded s ∨ bounded t ⇒
closure {x + y | x ∈ s ∧ y ∈ t} =
{x + y | x ∈ closure s ∧ y ∈ closure t}
[CLOSURE_UNION] Theorem
⊢ ∀s t. closure (s ∪ t) = closure s ∪ closure t
[CLOSURE_UNION_FRONTIER] Theorem
⊢ ∀s. closure s = s ∪ frontier s
[CLOSURE_UNIQUE] Theorem
⊢ ∀s t.
s ⊆ t ∧ closed t ∧ (∀t'. s ⊆ t' ∧ closed t' ⇒ t ⊆ t') ⇒
closure s = t
[CLOSURE_UNIV] Theorem
⊢ closure 𝕌(:real) = 𝕌(:real)
[COBOUNDED_IMP_UNBOUNDED] Theorem
⊢ ∀s. bounded (𝕌(:real) DIFF s) ⇒ ¬bounded s
[COBOUNDED_INTER_UNBOUNDED] Theorem
⊢ ∀s t. bounded (𝕌(:real) DIFF s) ∧ ¬bounded t ⇒ s ∩ t ≠ ∅
[COLLINEAR_1] Theorem
⊢ ∀s. collinear s
[COLLINEAR_2] Theorem
⊢ ∀x y. collinear {x; y}
[COLLINEAR_3] Theorem
⊢ ∀x y z. collinear {x; y; z} ⇔ collinear {0; x − y; z − y}
[COLLINEAR_3_EXPAND] Theorem
⊢ ∀a b c. collinear {a; b; c} ⇔ a = c ∨ ∃u. b = u * a + (1 − u) * c
[COLLINEAR_3_TRANS] Theorem
⊢ ∀a b c d.
collinear {a; b; c} ∧ collinear {b; c; d} ∧ b ≠ c ⇒
collinear {a; b; d}
[COLLINEAR_4_3] Theorem
⊢ ∀a b c d.
a ≠ b ⇒
(collinear {a; b; c; d} ⇔
collinear {a; b; c} ∧ collinear {a; b; d})
[COLLINEAR_BETWEEN_CASES] Theorem
⊢ ∀a b c.
collinear {a; b; c} ⇔
between a (b,c) ∨ between b (c,a) ∨ between c (a,b)
[COLLINEAR_DIST_BETWEEN] Theorem
⊢ ∀a b x.
collinear {x; a; b} ∧ dist (x,a) ≤ dist (a,b) ∧
dist (x,b) ≤ dist (a,b) ⇒
between x (a,b)
[COLLINEAR_DIST_IN_CLOSED_SEGMENT] Theorem
⊢ ∀a b x.
collinear {x; a; b} ∧ dist (x,a) ≤ dist (a,b) ∧
dist (x,b) ≤ dist (a,b) ⇒
x ∈ segment [(a,b)]
[COLLINEAR_DIST_IN_OPEN_SEGMENT] Theorem
⊢ ∀a b x.
collinear {x; a; b} ∧ dist (x,a) < dist (a,b) ∧
dist (x,b) < dist (a,b) ⇒
x ∈ segment (a,b)
[COLLINEAR_EMPTY] Theorem
⊢ collinear ∅
[COLLINEAR_LEMMA] Theorem
⊢ ∀x y. collinear {0; x; y} ⇔ x = 0 ∨ y = 0 ∨ ∃c. y = c * x
[COLLINEAR_LEMMA_ALT] Theorem
⊢ ∀x y. collinear {0; x; y} ⇔ x = 0 ∨ ∃c. y = c * x
[COLLINEAR_MIDPOINT] Theorem
⊢ ∀a b. collinear {a; midpoint (a,b); b}
[COLLINEAR_SING] Theorem
⊢ ∀x. collinear {x}
[COLLINEAR_SMALL] Theorem
⊢ ∀s. FINITE s ∧ CARD s ≤ 2 ⇒ collinear s
[COLLINEAR_SUBSET] Theorem
⊢ ∀s t. collinear t ∧ s ⊆ t ⇒ collinear s
[COLLINEAR_TRIPLES] Theorem
⊢ ∀s a b.
a ≠ b ⇒
(collinear (a INSERT b INSERT s) ⇔
∀x. x ∈ s ⇒ collinear {a; b; x})
[COMPACT_AFFINITY] Theorem
⊢ ∀s a c. compact s ⇒ compact (IMAGE (λx. a + c * x) s)
[COMPACT_ATTAINS_INF] Theorem
⊢ ∀s. compact s ∧ s ≠ ∅ ⇒ ∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ x ≤ y
[COMPACT_ATTAINS_SUP] Theorem
⊢ ∀s. compact s ∧ s ≠ ∅ ⇒ ∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ y ≤ x
[COMPACT_BIGINTER] Theorem
⊢ ∀f. (∀s. s ∈ f ⇒ compact s) ∧ f ≠ ∅ ⇒ compact (BIGINTER f)
[COMPACT_BIGUNION] Theorem
⊢ ∀s. FINITE s ∧ (∀t. t ∈ s ⇒ compact t) ⇒ compact (BIGUNION s)
[COMPACT_CBALL] Theorem
⊢ ∀x e. compact (cball (x,e))
[COMPACT_CHAIN] Theorem
⊢ ∀f. (∀s. s ∈ f ⇒ compact s ∧ s ≠ ∅) ∧
(∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
BIGINTER f ≠ ∅
[COMPACT_CLOSED_DIFFERENCES] Theorem
⊢ ∀s t. compact s ∧ closed t ⇒ closed {x − y | x ∈ s ∧ y ∈ t}
[COMPACT_CLOSED_SUMS] Theorem
⊢ ∀s t. compact s ∧ closed t ⇒ closed {x + y | x ∈ s ∧ y ∈ t}
[COMPACT_CLOSURE] Theorem
⊢ ∀s. compact (closure s) ⇔ bounded s
[COMPACT_COMPONENTS] Theorem
⊢ ∀s c. compact s ∧ c ∈ components s ⇒ compact c
[COMPACT_CONTINUOUS_IMAGE] Theorem
⊢ ∀f s. f continuous_on s ∧ compact s ⇒ compact (IMAGE f s)
[COMPACT_CONTINUOUS_IMAGE_EQ] Theorem
⊢ ∀f s.
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
(f continuous_on s ⇔ ∀t. compact t ∧ t ⊆ s ⇒ compact (IMAGE f t))
[COMPACT_DIFF] Theorem
⊢ ∀s t. compact s ∧ open t ⇒ compact (s DIFF t)
[COMPACT_EMPTY] Theorem
⊢ compact ∅
[COMPACT_EQ_BOLZANO_WEIERSTRASS] Theorem
⊢ ∀s. compact s ⇔
∀t. INFINITE t ∧ t ⊆ s ⇒ ∃x. x ∈ s ∧ x limit_point_of t
[COMPACT_EQ_BOUNDED_CLOSED] Theorem
⊢ ∀s. compact s ⇔ bounded s ∧ closed s
[COMPACT_EQ_HEINE_BOREL] Theorem
⊢ ∀s. compact s ⇔
∀f. (∀t. t ∈ f ⇒ open t) ∧ s ⊆ BIGUNION f ⇒
∃f'. f' ⊆ f ∧ FINITE f' ∧ s ⊆ BIGUNION f'
[COMPACT_EQ_HEINE_BOREL_SUBTOPOLOGY] Theorem
⊢ ∀s. compact s ⇔
∀f. (∀t. t ∈ f ⇒ open_in (subtopology euclidean s) t) ∧
s ⊆ BIGUNION f ⇒
∃f'. f' ⊆ f ∧ FINITE f' ∧ s ⊆ BIGUNION f'
[COMPACT_FIP] Theorem
⊢ ∀f. (∀t. t ∈ f ⇒ compact t) ∧
(∀f'. FINITE f' ∧ f' ⊆ f ⇒ BIGINTER f' ≠ ∅) ⇒
BIGINTER f ≠ ∅
[COMPACT_FRONTIER] Theorem
⊢ ∀s. compact s ⇒ compact (frontier s)
[COMPACT_FRONTIER_BOUNDED] Theorem
⊢ ∀s. bounded s ⇒ compact (frontier s)
[COMPACT_IMP_BOUNDED] Theorem
⊢ ∀s. compact s ⇒ bounded s
[COMPACT_IMP_CLOSED] Theorem
⊢ ∀s. compact s ⇒ closed s
[COMPACT_IMP_COMPLETE] Theorem
⊢ ∀s. compact s ⇒ complete s
[COMPACT_IMP_FIP] Theorem
⊢ ∀s f.
compact s ∧ (∀t. t ∈ f ⇒ closed t) ∧
(∀f'. FINITE f' ∧ f' ⊆ f ⇒ s ∩ BIGINTER f' ≠ ∅) ⇒
s ∩ BIGINTER f ≠ ∅
[COMPACT_IMP_HEINE_BOREL] Theorem
⊢ ∀s. compact s ⇒
∀f. (∀t. t ∈ f ⇒ open t) ∧ s ⊆ BIGUNION f ⇒
∃f'. f' ⊆ f ∧ FINITE f' ∧ s ⊆ BIGUNION f'
[COMPACT_IMP_TOTALLY_BOUNDED] Theorem
⊢ ∀s. compact s ⇒
∀e. 0 < e ⇒
∃k. FINITE k ∧ k ⊆ s ∧
s ⊆ BIGUNION (IMAGE (λx. ball (x,e)) k)
[COMPACT_INSERT] Theorem
⊢ ∀a s. compact s ⇒ compact (a INSERT s)
[COMPACT_INTER] Theorem
⊢ ∀s t. compact s ∧ compact t ⇒ compact (s ∩ t)
[COMPACT_INTERVAL] Theorem
⊢ ∀a b. compact (interval [(a,b)])
[COMPACT_INTERVAL_EQ] Theorem
⊢ (∀a b. compact (interval [(a,b)])) ∧
∀a b. compact (interval (a,b)) ⇔ interval (a,b) = ∅
[COMPACT_INTER_CLOSED] Theorem
⊢ ∀s t. compact s ∧ closed t ⇒ compact (s ∩ t)
[COMPACT_LEMMA] Theorem
⊢ ∀s. bounded s ∧ (∀n. x n ∈ s) ⇒
∃l r.
(∀m n. m < n ⇒ r m < r n) ∧
∀e. 0 < e ⇒ ∃N. ∀n i. N ≤ n ⇒ abs (x (r n) − l) < e
[COMPACT_LINEAR_IMAGE] Theorem
⊢ ∀f s. compact s ∧ linear f ⇒ compact (IMAGE f s)
[COMPACT_NEGATIONS] Theorem
⊢ ∀s. compact s ⇒ compact (IMAGE (λx. -x) s)
[COMPACT_NEST] Theorem
⊢ ∀s. (∀n. compact (s n) ∧ s n ≠ ∅) ∧ (∀m n. m ≤ n ⇒ s n ⊆ s m) ⇒
BIGINTER {s n | n ∈ 𝕌(:num)} ≠ ∅
[COMPACT_REAL_LEMMA] Theorem
⊢ ∀s b.
(∀n. abs (s n) ≤ b) ⇒
∃l r.
(∀m n. m < n ⇒ r m < r n) ∧
∀e. 0 < e ⇒ ∃N. ∀n. N ≤ n ⇒ abs (s (r n) − l) < e
[COMPACT_SCALING] Theorem
⊢ ∀s c. compact s ⇒ compact (IMAGE (λx. c * x) s)
[COMPACT_SEQUENCE_WITH_LIMIT] Theorem
⊢ ∀f l. (f ⟶ l) sequentially ⇒ compact (l INSERT IMAGE f 𝕌(:num))
[COMPACT_SING] Theorem
⊢ ∀a. compact {a}
[COMPACT_SPHERE] Theorem
⊢ ∀a r. compact (sphere (a,r))
[COMPACT_TRANSLATION] Theorem
⊢ ∀s a. compact s ⇒ compact (IMAGE (λx. a + x) s)
[COMPACT_TRANSLATION_EQ] Theorem
⊢ ∀a s. compact (IMAGE (λx. a + x) s) ⇔ compact s
[COMPACT_UNIFORMLY_CONTINUOUS] Theorem
⊢ ∀f s. f continuous_on s ∧ compact s ⇒ f uniformly_continuous_on s
[COMPACT_UNIFORMLY_EQUICONTINUOUS] Theorem
⊢ ∀fs s.
(∀x e.
x ∈ s ∧ 0 < e ⇒
∃d. 0 < d ∧
∀f x'.
f ∈ fs ∧ x' ∈ s ∧ dist (x',x) < d ⇒ dist (f x',f x) < e) ∧
compact s ⇒
∀e. 0 < e ⇒
∃d. 0 < d ∧
∀f x x'.
f ∈ fs ∧ x ∈ s ∧ x' ∈ s ∧ dist (x',x) < d ⇒
dist (f x',f x) < e
[COMPACT_UNION] Theorem
⊢ ∀s t. compact s ∧ compact t ⇒ compact (s ∪ t)
[COMPLEMENT_CONNECTED_COMPONENT_BIGUNION] Theorem
⊢ ∀s x.
s DIFF connected_component s x =
BIGUNION
({connected_component s y | y | y ∈ s} DELETE
connected_component s x)
[COMPLETE_EQ_CLOSED] Theorem
⊢ ∀s. complete s ⇔ closed s
[COMPLETE_INJECTIVE_LINEAR_IMAGE] Theorem
⊢ ∀f. linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
∀s. complete s ⇒ complete (IMAGE f s)
[COMPLETE_INJECTIVE_LINEAR_IMAGE_EQ] Theorem
⊢ ∀f s.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
(complete (IMAGE f s) ⇔ complete s)
[COMPLETE_ISOMETRIC_IMAGE] Theorem
⊢ ∀f s e.
0 < e ∧ subspace s ∧ linear f ∧
(∀x. x ∈ s ⇒ abs (f x) ≥ e * abs x) ∧ complete s ⇒
complete (IMAGE f s)
[COMPLETE_UNIV] Theorem
⊢ complete 𝕌(:real)
[COMPONENTS_EMPTY] Theorem
⊢ components ∅ = ∅
[COMPONENTS_EQ] Theorem
⊢ ∀s c c'.
c ∈ components s ∧ c' ∈ components s ⇒ (c = c' ⇔ c ∩ c' ≠ ∅)
[COMPONENTS_EQ_EMPTY] Theorem
⊢ ∀s. components s = ∅ ⇔ s = ∅
[COMPONENTS_EQ_SING] Theorem
⊢ ∀s. components s = {s} ⇔ connected s ∧ s ≠ ∅
[COMPONENTS_EQ_SING_EXISTS] Theorem
⊢ ∀s. (∃a. components s = {a}) ⇔ connected s ∧ s ≠ ∅
[COMPONENTS_EQ_SING_N_EXISTS] Theorem
⊢ (∀s. components s = {s} ⇔ connected s ∧ s ≠ ∅) ∧
∀s. (∃a. components s = {a}) ⇔ connected s ∧ s ≠ ∅
[COMPONENTS_INTERMEDIATE_SUBSET] Theorem
⊢ ∀s t u. s ∈ components u ∧ s ⊆ t ∧ t ⊆ u ⇒ s ∈ components t
[COMPONENTS_MAXIMAL] Theorem
⊢ ∀s t c. c ∈ components s ∧ connected t ∧ t ⊆ s ∧ c ∩ t ≠ ∅ ⇒ t ⊆ c
[COMPONENTS_NONOVERLAP] Theorem
⊢ ∀s c c'.
c ∈ components s ∧ c' ∈ components s ⇒ (c ∩ c' = ∅ ⇔ c ≠ c')
[COMPONENTS_UNIQUE] Theorem
⊢ ∀s k.
BIGUNION k = s ∧
(∀c. c ∈ k ⇒
connected c ∧ c ≠ ∅ ∧
∀c'. connected c' ∧ c ⊆ c' ∧ c' ⊆ s ⇒ c' = c) ⇒
components s = k
[COMPONENTS_UNIQUE_EQ] Theorem
⊢ ∀s k.
components s = k ⇔
BIGUNION k = s ∧
∀c. c ∈ k ⇒
connected c ∧ c ≠ ∅ ∧
∀c'. connected c' ∧ c ⊆ c' ∧ c' ⊆ s ⇒ c' = c
[COMPONENTS_UNIV] Theorem
⊢ components 𝕌(:real) = {𝕌(:real)}
[CONDENSATION_POINT_IMP_LIMPT] Theorem
⊢ ∀x s. x condensation_point_of s ⇒ x limit_point_of s
[CONDENSATION_POINT_INFINITE_BALL] Theorem
⊢ ∀s x.
x condensation_point_of s ⇔
∀e. 0 < e ⇒ uncountable (s ∩ ball (x,e))
[CONDENSATION_POINT_INFINITE_BALL_CBALL] Theorem
⊢ (∀s x.
x condensation_point_of s ⇔
∀e. 0 < e ⇒ uncountable (s ∩ ball (x,e))) ∧
∀s x.
x condensation_point_of s ⇔
∀e. 0 < e ⇒ uncountable (s ∩ cball (x,e))
[CONDENSATION_POINT_INFINITE_CBALL] Theorem
⊢ ∀s x.
x condensation_point_of s ⇔
∀e. 0 < e ⇒ uncountable (s ∩ cball (x,e))
[CONDENSATION_POINT_OF_SUBSET] Theorem
⊢ ∀x s t.
x condensation_point_of s ∧ s ⊆ t ⇒ x condensation_point_of t
[CONNECTED_BIGUNION] Theorem
⊢ ∀P. (∀s. s ∈ P ⇒ connected s) ∧ BIGINTER P ≠ ∅ ⇒
connected (BIGUNION P)
[CONNECTED_CHAIN] Theorem
⊢ ∀f. (∀s. s ∈ f ⇒ compact s ∧ connected s) ∧
(∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
connected (BIGINTER f)
[CONNECTED_CHAIN_GEN] Theorem
⊢ ∀f. (∀s. s ∈ f ⇒ closed s ∧ connected s) ∧
(∃s. s ∈ f ∧ compact s) ∧ (∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
connected (BIGINTER f)
[CONNECTED_CLOPEN] Theorem
⊢ ∀s. connected s ⇔
∀t. open_in (subtopology euclidean s) t ∧
closed_in (subtopology euclidean s) t ⇒
t = ∅ ∨ t = s
[CONNECTED_CLOSED] Theorem
⊢ ∀s. connected s ⇔
¬∃e1 e2.
closed e1 ∧ closed e2 ∧ s ⊆ e1 ∪ e2 ∧ e1 ∩ e2 ∩ s = ∅ ∧
e1 ∩ s ≠ ∅ ∧ e2 ∩ s ≠ ∅
[CONNECTED_CLOSED_IN] Theorem
⊢ ∀s. connected s ⇔
¬∃e1 e2.
closed_in (subtopology euclidean s) e1 ∧
closed_in (subtopology euclidean s) e2 ∧ s ⊆ e1 ∪ e2 ∧
e1 ∩ e2 = ∅ ∧ e1 ≠ ∅ ∧ e2 ≠ ∅
[CONNECTED_CLOSED_IN_EQ] Theorem
⊢ ∀s. connected s ⇔
¬∃e1 e2.
closed_in (subtopology euclidean s) e1 ∧
closed_in (subtopology euclidean s) e2 ∧ e1 ∪ e2 = s ∧
e1 ∩ e2 = ∅ ∧ e1 ≠ ∅ ∧ e2 ≠ ∅
[CONNECTED_CLOSED_MONOTONE_PREIMAGE] Theorem
⊢ ∀f s t.
f continuous_on s ∧ IMAGE f s = t ∧
(∀c. closed_in (subtopology euclidean s) c ⇒
closed_in (subtopology euclidean t) (IMAGE f c)) ∧
(∀y. y ∈ t ⇒ connected {x | x ∈ s ∧ f x = y}) ⇒
∀c. connected c ∧ c ⊆ t ⇒ connected {x | x ∈ s ∧ f x ∈ c}
[CONNECTED_CLOSED_SET] Theorem
⊢ ∀s. closed s ⇒
(connected s ⇔
¬∃e1 e2.
closed e1 ∧ closed e2 ∧ e1 ≠ ∅ ∧ e2 ≠ ∅ ∧ e1 ∪ e2 = s ∧
e1 ∩ e2 = ∅)
[CONNECTED_CLOSURE] Theorem
⊢ ∀s. connected s ⇒ connected (closure s)
[CONNECTED_COMPONENT_BIGUNION] Theorem
⊢ ∀s x.
connected_component s x =
BIGUNION {t | connected t ∧ x ∈ t ∧ t ⊆ s}
[CONNECTED_COMPONENT_DISJOINT] Theorem
⊢ ∀s a b.
DISJOINT (connected_component s a) (connected_component s b) ⇔
a ∉ connected_component s b
[CONNECTED_COMPONENT_EMPTY] Theorem
⊢ ∀x. connected_component ∅ x = ∅
[CONNECTED_COMPONENT_EQ] Theorem
⊢ ∀s x y.
y ∈ connected_component s x ⇒
connected_component s y = connected_component s x
[CONNECTED_COMPONENT_EQUIVALENCE_RELATION] Theorem
⊢ ∀R s.
(∀x y. R x y ⇒ R y x) ∧ (∀x y z. R x y ∧ R y z ⇒ R x z) ∧
(∀a. a ∈ s ⇒
∃t. open_in (subtopology euclidean s) t ∧ a ∈ t ∧
∀x. x ∈ t ⇒ R a x) ⇒
∀a b. connected_component s a b ⇒ R a b
[CONNECTED_COMPONENT_EQ_EMPTY] Theorem
⊢ ∀s x. connected_component s x = ∅ ⇔ x ∉ s
[CONNECTED_COMPONENT_EQ_EQ] Theorem
⊢ ∀s x y.
connected_component s x = connected_component s y ⇔
x ∉ s ∧ y ∉ s ∨ x ∈ s ∧ y ∈ s ∧ connected_component s x y
[CONNECTED_COMPONENT_EQ_SELF] Theorem
⊢ ∀s x. connected s ∧ x ∈ s ⇒ connected_component s x = s
[CONNECTED_COMPONENT_EQ_UNIV] Theorem
⊢ ∀s x. connected_component s x = 𝕌(:real) ⇔ s = 𝕌(:real)
[CONNECTED_COMPONENT_IDEMP] Theorem
⊢ ∀s x.
connected_component (connected_component s x) x =
connected_component s x
[CONNECTED_COMPONENT_IN] Theorem
⊢ ∀s x y. connected_component s x y ⇒ x ∈ s ∧ y ∈ s
[CONNECTED_COMPONENT_INTERMEDIATE_SUBSET] Theorem
⊢ ∀t u a.
connected_component u a ⊆ t ∧ t ⊆ u ⇒
connected_component t a = connected_component u a
[CONNECTED_COMPONENT_MAXIMAL] Theorem
⊢ ∀s t x. x ∈ t ∧ connected t ∧ t ⊆ s ⇒ t ⊆ connected_component s x
[CONNECTED_COMPONENT_MONO] Theorem
⊢ ∀s t x. s ⊆ t ⇒ connected_component s x ⊆ connected_component t x
[CONNECTED_COMPONENT_NONOVERLAP] Theorem
⊢ ∀s a b.
connected_component s a ∩ connected_component s b = ∅ ⇔
a ∉ s ∨ b ∉ s ∨ connected_component s a ≠ connected_component s b
[CONNECTED_COMPONENT_OF_SUBSET] Theorem
⊢ ∀s t x.
s ⊆ t ∧ connected_component s x y ⇒ connected_component t x y
[CONNECTED_COMPONENT_OVERLAP] Theorem
⊢ ∀s a b.
connected_component s a ∩ connected_component s b ≠ ∅ ⇔
a ∈ s ∧ b ∈ s ∧ connected_component s a = connected_component s b
[CONNECTED_COMPONENT_REFL] Theorem
⊢ ∀s x. x ∈ s ⇒ connected_component s x x
[CONNECTED_COMPONENT_REFL_EQ] Theorem
⊢ ∀s x. connected_component s x x ⇔ x ∈ s
[CONNECTED_COMPONENT_SET] Theorem
⊢ ∀s x.
connected_component s x =
{y | ∃t. connected t ∧ t ⊆ s ∧ x ∈ t ∧ y ∈ t}
[CONNECTED_COMPONENT_SUBSET] Theorem
⊢ ∀s x. connected_component s x ⊆ s
[CONNECTED_COMPONENT_SYM] Theorem
⊢ ∀s x y. connected_component s x y ⇒ connected_component s y x
[CONNECTED_COMPONENT_SYM_EQ] Theorem
⊢ ∀s x y. connected_component s x y ⇔ connected_component s y x
[CONNECTED_COMPONENT_TRANS] Theorem
⊢ ∀s x y.
connected_component s x y ∧ connected_component s y z ⇒
connected_component s x z
[CONNECTED_COMPONENT_UNIQUE] Theorem
⊢ ∀s c x.
x ∈ c ∧ c ⊆ s ∧ connected c ∧
(∀c'. x ∈ c' ∧ c' ⊆ s ∧ connected c' ⇒ c' ⊆ c) ⇒
connected_component s x = c
[CONNECTED_COMPONENT_UNIV] Theorem
⊢ ∀x. connected_component 𝕌(:real) x = 𝕌(:real)
[CONNECTED_CONNECTED_COMPONENT] Theorem
⊢ ∀s x. connected (connected_component s x)
[CONNECTED_CONNECTED_COMPONENT_SET] Theorem
⊢ ∀s. connected s ⇔ ∀x. x ∈ s ⇒ connected_component s x = s
[CONNECTED_CONTINUOUS_IMAGE] Theorem
⊢ ∀f s. f continuous_on s ∧ connected s ⇒ connected (IMAGE f s)
[CONNECTED_DIFF_OPEN_FROM_CLOSED] Theorem
⊢ ∀s t u.
s ⊆ t ∧ t ⊆ u ∧ open s ∧ closed t ∧ connected u ∧
connected (t DIFF s) ⇒
connected (u DIFF s)
[CONNECTED_DISJOINT_BIGUNION_OPEN_UNIQUE] Theorem
⊢ ∀f f'.
pairwiseD DISJOINT f ∧ pairwiseD DISJOINT f' ∧
(∀s. s ∈ f ⇒ open s ∧ connected s ∧ s ≠ ∅) ∧
(∀s. s ∈ f' ⇒ open s ∧ connected s ∧ s ≠ ∅) ∧
BIGUNION f = BIGUNION f' ⇒
f = f'
[CONNECTED_EMPTY] Theorem
⊢ connected ∅
[CONNECTED_EQUIVALENCE_RELATION] Theorem
⊢ ∀R s.
connected s ∧ (∀x y. R x y ⇒ R y x) ∧
(∀x y z. R x y ∧ R y z ⇒ R x z) ∧
(∀a. a ∈ s ⇒
∃t. open_in (subtopology euclidean s) t ∧ a ∈ t ∧
∀x. x ∈ t ⇒ R a x) ⇒
∀a b. a ∈ s ∧ b ∈ s ⇒ R a b
[CONNECTED_EQUIVALENCE_RELATION_GEN] Theorem
⊢ ∀P R s.
connected s ∧ (∀x y. R x y ⇒ R y x) ∧
(∀x y z. R x y ∧ R y z ⇒ R x z) ∧
(∀t a.
open_in (subtopology euclidean s) t ∧ a ∈ t ⇒ ∃z. z ∈ t ∧ P z) ∧
(∀a. a ∈ s ⇒
∃t. open_in (subtopology euclidean s) t ∧ a ∈ t ∧
∀x y. x ∈ t ∧ y ∈ t ∧ P x ∧ P y ⇒ R x y) ⇒
∀a b. a ∈ s ∧ b ∈ s ∧ P a ∧ P b ⇒ R a b
[CONNECTED_EQ_COMPONENTS_SUBSET_SING] Theorem
⊢ ∀s. connected s ⇔ components s ⊆ {s}
[CONNECTED_EQ_COMPONENTS_SUBSET_SING_EXISTS] Theorem
⊢ ∀s. connected s ⇔ ∃a. components s ⊆ {a}
[CONNECTED_EQ_CONNECTED_COMPONENTS_EQ] Theorem
⊢ ∀s. connected s ⇔
∀c c'. c ∈ components s ∧ c' ∈ components s ⇒ c = c'
[CONNECTED_EQ_CONNECTED_COMPONENT_EQ] Theorem
⊢ ∀s. connected s ⇔
∀x y.
x ∈ s ∧ y ∈ s ⇒
connected_component s x = connected_component s y
[CONNECTED_FROM_CLOSED_UNION_AND_INTER] Theorem
⊢ ∀s t.
closed s ∧ closed t ∧ connected (s ∪ t) ∧ connected (s ∩ t) ⇒
connected s ∧ connected t
[CONNECTED_FROM_OPEN_UNION_AND_INTER] Theorem
⊢ ∀s t.
open s ∧ open t ∧ connected (s ∪ t) ∧ connected (s ∩ t) ⇒
connected s ∧ connected t
[CONNECTED_IFF_CONNECTABLE_POINTS] Theorem
⊢ ∀s. connected s ⇔
∀a b. a ∈ s ∧ b ∈ s ⇒ ∃t. connected t ∧ t ⊆ s ∧ a ∈ t ∧ b ∈ t
[CONNECTED_IFF_CONNECTED_COMPONENT] Theorem
⊢ ∀s. connected s ⇔ ∀x y. x ∈ s ∧ y ∈ s ⇒ connected_component s x y
[CONNECTED_IMP_PERFECT] Theorem
⊢ ∀s x. connected s ∧ ¬(∃a. s = {a}) ∧ x ∈ s ⇒ x limit_point_of s
[CONNECTED_IMP_PERFECT_CLOSED] Theorem
⊢ ∀s x.
connected s ∧ closed s ∧ ¬(∃a. s = {a}) ⇒
(x limit_point_of s ⇔ x ∈ s)
[CONNECTED_INDUCTION] Theorem
⊢ ∀P Q s.
connected s ∧
(∀t a.
open_in (subtopology euclidean s) t ∧ a ∈ t ⇒ ∃z. z ∈ t ∧ P z) ∧
(∀a. a ∈ s ⇒
∃t. open_in (subtopology euclidean s) t ∧ a ∈ t ∧
∀x y. x ∈ t ∧ y ∈ t ∧ P x ∧ P y ∧ Q x ⇒ Q y) ⇒
∀a b. a ∈ s ∧ b ∈ s ∧ P a ∧ P b ∧ Q a ⇒ Q b
[CONNECTED_INDUCTION_SIMPLE] Theorem
⊢ ∀P s.
connected s ∧
(∀a. a ∈ s ⇒
∃t. open_in (subtopology euclidean s) t ∧ a ∈ t ∧
∀x y. x ∈ t ∧ y ∈ t ∧ P x ⇒ P y) ⇒
∀a b. a ∈ s ∧ b ∈ s ∧ P a ⇒ P b
[CONNECTED_INTERMEDIATE_CLOSURE] Theorem
⊢ ∀s t. connected s ∧ s ⊆ t ∧ t ⊆ closure s ⇒ connected t
[CONNECTED_INTER_FRONTIER] Theorem
⊢ ∀s t. connected s ∧ s ∩ t ≠ ∅ ∧ s DIFF t ≠ ∅ ⇒ s ∩ frontier t ≠ ∅
[CONNECTED_IVT_COMPONENT] Theorem
⊢ ∀s x y a.
connected s ∧ x ∈ s ∧ y ∈ s ∧ x ≤ a ∧ a ≤ y ⇒ ∃z. z ∈ s ∧ z = a
[CONNECTED_IVT_HYPERPLANE] Theorem
⊢ ∀s x y a b.
connected s ∧ x ∈ s ∧ y ∈ s ∧ a * x ≤ b ∧ b ≤ a * y ⇒
∃z. z ∈ s ∧ a * z = b
[CONNECTED_LINEAR_IMAGE] Theorem
⊢ ∀f s. connected s ∧ linear f ⇒ connected (IMAGE f s)
[CONNECTED_MONOTONE_QUOTIENT_PREIMAGE] Theorem
⊢ ∀f s t.
f continuous_on s ∧ IMAGE f s = t ∧
(∀u. u ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
open_in (subtopology euclidean t) u)) ∧
(∀y. y ∈ t ⇒ connected {x | x ∈ s ∧ f x = y}) ∧ connected t ⇒
connected s
[CONNECTED_MONOTONE_QUOTIENT_PREIMAGE_GEN] Theorem
⊢ ∀f s t c.
IMAGE f s = t ∧
(∀u. u ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
open_in (subtopology euclidean t) u)) ∧
(∀y. y ∈ t ⇒ connected {x | x ∈ s ∧ f x = y}) ∧
(open_in (subtopology euclidean t) c ∨
closed_in (subtopology euclidean t) c) ∧ connected c ⇒
connected {x | x ∈ s ∧ f x ∈ c}
[CONNECTED_NEGATIONS] Theorem
⊢ ∀s. connected s ⇒ connected (IMAGE (λx. -x) s)
[CONNECTED_NEST] Theorem
⊢ ∀s. (∀n. compact (s n) ∧ connected (s n)) ∧
(∀m n. m ≤ n ⇒ s n ⊆ s m) ⇒
connected (BIGINTER {s n | n ∈ 𝕌(:num)})
[CONNECTED_NEST_GEN] Theorem
⊢ ∀s. (∀n. closed (s n) ∧ connected (s n)) ∧ (∃n. compact (s n)) ∧
(∀m n. m ≤ n ⇒ s n ⊆ s m) ⇒
connected (BIGINTER {s n | n ∈ 𝕌(:num)})
[CONNECTED_OPEN_IN] Theorem
⊢ ∀s. connected s ⇔
¬∃e1 e2.
open_in (subtopology euclidean s) e1 ∧
open_in (subtopology euclidean s) e2 ∧ s ⊆ e1 ∪ e2 ∧
e1 ∩ e2 = ∅ ∧ e1 ≠ ∅ ∧ e2 ≠ ∅
[CONNECTED_OPEN_IN_EQ] Theorem
⊢ ∀s. connected s ⇔
¬∃e1 e2.
open_in (subtopology euclidean s) e1 ∧
open_in (subtopology euclidean s) e2 ∧ e1 ∪ e2 = s ∧
e1 ∩ e2 = ∅ ∧ e1 ≠ ∅ ∧ e2 ≠ ∅
[CONNECTED_OPEN_MONOTONE_PREIMAGE] Theorem
⊢ ∀f s t.
f continuous_on s ∧ IMAGE f s = t ∧
(∀c. open_in (subtopology euclidean s) c ⇒
open_in (subtopology euclidean t) (IMAGE f c)) ∧
(∀y. y ∈ t ⇒ connected {x | x ∈ s ∧ f x = y}) ⇒
∀c. connected c ∧ c ⊆ t ⇒ connected {x | x ∈ s ∧ f x ∈ c}
[CONNECTED_OPEN_SET] Theorem
⊢ ∀s. open s ⇒
(connected s ⇔
¬∃e1 e2.
open e1 ∧ open e2 ∧ e1 ≠ ∅ ∧ e2 ≠ ∅ ∧ e1 ∪ e2 = s ∧
e1 ∩ e2 = ∅)
[CONNECTED_REAL_LEMMA] Theorem
⊢ ∀f a b e1 e2.
a ≤ b ∧ f a ∈ e1 ∧ f b ∈ e2 ∧
(∀e x.
a ≤ x ∧ x ≤ b ∧ 0 < e ⇒
∃d. 0 < d ∧ ∀y. abs (y − x) < d ⇒ dist (f y,f x) < e) ∧
(∀y. y ∈ e1 ⇒ ∃e. 0 < e ∧ ∀y'. dist (y',y) < e ⇒ y' ∈ e1) ∧
(∀y. y ∈ e2 ⇒ ∃e. 0 < e ∧ ∀y'. dist (y',y) < e ⇒ y' ∈ e2) ∧
¬(∃x. a ≤ x ∧ x ≤ b ∧ f x ∈ e1 ∧ f x ∈ e2) ⇒
∃x. a ≤ x ∧ x ≤ b ∧ f x ∉ e1 ∧ f x ∉ e2
[CONNECTED_SCALING] Theorem
⊢ ∀s c. connected s ⇒ connected (IMAGE (λx. c * x) s)
[CONNECTED_SEGMENT] Theorem
⊢ (∀a b. connected (segment [(a,b)])) ∧
∀a b. connected (segment (a,b))
[CONNECTED_SING] Theorem
⊢ ∀a. connected {a}
[CONNECTED_SUBSET_CLOPEN] Theorem
⊢ ∀u s c.
closed_in (subtopology euclidean u) s ∧
open_in (subtopology euclidean u) s ∧ connected c ∧ c ⊆ u ∧
c ∩ s ≠ ∅ ⇒
c ⊆ s
[CONNECTED_TRANSLATION] Theorem
⊢ ∀a s. connected s ⇒ connected (IMAGE (λx. a + x) s)
[CONNECTED_TRANSLATION_EQ] Theorem
⊢ ∀a s. connected (IMAGE (λx. a + x) s) ⇔ connected s
[CONNECTED_UNION] Theorem
⊢ ∀s t. connected s ∧ connected t ∧ s ∩ t ≠ ∅ ⇒ connected (s ∪ t)
[CONNECTED_UNION_STRONG] Theorem
⊢ ∀s t.
connected s ∧ connected t ∧ closure s ∩ t ≠ ∅ ⇒ connected (s ∪ t)
[CONNECTED_UNIV] Theorem
⊢ connected 𝕌(:real)
[CONTENT_0_SUBSET] Theorem
⊢ ∀s a b.
s ⊆ interval [(a,b)] ∧ content (interval [(a,b)]) = 0 ⇒
content s = 0
[CONTENT_0_SUBSET_GEN] Theorem
⊢ ∀s t. s ⊆ t ∧ bounded t ∧ content t = 0 ⇒ content s = 0
[CONTENT_CLOSED_INTERVAL] Theorem
⊢ ∀a b. a ≤ b ⇒ content (interval [(a,b)]) = b − a
[CONTENT_CLOSED_INTERVAL_CASES] Theorem
⊢ ∀a b. content (interval [(a,b)]) = if a ≤ b then b − a else 0
[CONTENT_EMPTY] Theorem
⊢ content ∅ = 0
[CONTENT_EQ_0] Theorem
⊢ ∀a b. content (interval [(a,b)]) = 0 ⇔ b ≤ a
[CONTENT_EQ_0_1] Theorem
⊢ ∀a b. content (interval [(a,b)]) = 0 ⇔ b ≤ a
[CONTENT_EQ_0_GEN] Theorem
⊢ ∀s. bounded s ⇒ (content s = 0 ⇔ ∃a. ∀x. x ∈ s ⇒ x = a)
[CONTENT_EQ_0_INTERIOR] Theorem
⊢ ∀a b.
content (interval [(a,b)]) = 0 ⇔ interior (interval [(a,b)]) = ∅
[CONTENT_LT_NZ] Theorem
⊢ ∀a b.
0 < content (interval [(a,b)]) ⇔ content (interval [(a,b)]) ≠ 0
[CONTENT_POS_LE] Theorem
⊢ ∀a b. 0 ≤ content (interval [(a,b)])
[CONTENT_POS_LT] Theorem
⊢ ∀a b. a < b ⇒ 0 < content (interval [(a,b)])
[CONTENT_POS_LT_EQ] Theorem
⊢ ∀a b. 0 < content (interval [(a,b)]) ⇔ a < b
[CONTENT_SUBSET] Theorem
⊢ ∀a b c d.
interval [(a,b)] ⊆ interval [(c,d)] ⇒
content (interval [(a,b)]) ≤ content (interval [(c,d)])
[CONTENT_UNIT] Theorem
⊢ content (interval [(0,1)]) = 1
[CONTINUOUS_ABS] Theorem
⊢ ∀f net. f continuous net ⇒ (λx. abs (f x)) continuous net
[CONTINUOUS_ABS_COMPOSE] Theorem
⊢ ∀net f. f continuous net ⇒ (λx. abs (f x)) continuous net
[CONTINUOUS_ADD] Theorem
⊢ ∀f g net.
f continuous net ∧ g continuous net ⇒
(λx. f x + g x) continuous net
[CONTINUOUS_AGREE_ON_CLOSURE] Theorem
⊢ ∀g h.
g continuous_on closure s ∧ h continuous_on closure s ∧
(∀x. x ∈ s ⇒ g x = h x) ⇒
∀x. x ∈ closure s ⇒ g x = h x
[CONTINUOUS_AT] Theorem
⊢ ∀f x. f continuous at x ⇔ (f ⟶ f x) (at x)
[CONTINUOUS_ATTAINS_INF] Theorem
⊢ ∀f s.
compact s ∧ s ≠ ∅ ∧ f continuous_on s ⇒
∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ f x ≤ f y
[CONTINUOUS_ATTAINS_SUP] Theorem
⊢ ∀f s.
compact s ∧ s ≠ ∅ ∧ f continuous_on s ⇒
∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ f y ≤ f x
[CONTINUOUS_AT_ABS] Theorem
⊢ ∀x. abs continuous at x
[CONTINUOUS_AT_AVOID] Theorem
⊢ ∀f x a.
f continuous at x ∧ f x ≠ a ⇒
∃e. 0 < e ∧ ∀y. dist (x,y) < e ⇒ f y ≠ a
[CONTINUOUS_AT_BALL] Theorem
⊢ ∀f x.
f continuous at x ⇔
∀e. 0 < e ⇒ ∃d. 0 < d ∧ IMAGE f (ball (x,d)) ⊆ ball (f x,e)
[CONTINUOUS_AT_COMPOSE] Theorem
⊢ ∀f g x.
f continuous at x ∧ g continuous at (f x) ⇒ g ∘ f continuous at x
[CONTINUOUS_AT_COMPOSE_EQ] Theorem
⊢ ∀f g h.
g continuous at x ∧ h continuous at (g x) ∧ (∀y. g (h y) = y) ∧
h (g x) = x ⇒
(f continuous at (g x) ⇔ (λx. f (g x)) continuous at x)
[CONTINUOUS_AT_DIST] Theorem
⊢ ∀a x. (λx. dist (a,x)) continuous at x
[CONTINUOUS_AT_DIST_CLOSEST_POINT] Theorem
⊢ ∀s x.
closed s ∧ s ≠ ∅ ⇒
(λx. dist (x,closest_point s x)) continuous at x
[CONTINUOUS_AT_ID] Theorem
⊢ ∀a. (λx. x) continuous at a
[CONTINUOUS_AT_IMP_CONTINUOUS_ON] Theorem
⊢ ∀f s. (∀x. x ∈ s ⇒ f continuous at x) ⇒ f continuous_on s
[CONTINUOUS_AT_INV] Theorem
⊢ ∀f a. f continuous at a ∧ f a ≠ 0 ⇒ realinv ∘ f continuous at a
[CONTINUOUS_AT_LIFT_DOT] Theorem
⊢ ∀a x. (λy. a * y) continuous at x
[CONTINUOUS_AT_OPEN] Theorem
⊢ ∀f x.
f continuous at x ⇔
∀t. open t ∧ f x ∈ t ⇒
∃s. open s ∧ x ∈ s ∧ ∀x'. x' ∈ s ⇒ f x' ∈ t
[CONTINUOUS_AT_RANGE] Theorem
⊢ ∀f x.
f continuous at x ⇔
∀e. 0 < e ⇒
∃d. 0 < d ∧ ∀x'. abs (x' − x) < d ⇒ abs (f x' − f x) < e
[CONTINUOUS_AT_SEQUENTIALLY] Theorem
⊢ ∀f a.
f continuous at a ⇔
∀x. (x ⟶ a) sequentially ⇒ (f ∘ x ⟶ f a) sequentially
[CONTINUOUS_AT_SETDIST] Theorem
⊢ ∀s x. (λy. setdist ({y},s)) continuous at x
[CONTINUOUS_AT_TRANSLATION] Theorem
⊢ ∀a z f. f continuous at (a + z) ⇔ (λx. f (a + x)) continuous at z
[CONTINUOUS_AT_WITHIN] Theorem
⊢ ∀f x s. f continuous at x ⇒ f continuous (at x within s)
[CONTINUOUS_AT_WITHIN_INV] Theorem
⊢ ∀f s a.
f continuous (at a within s) ∧ f a ≠ 0 ⇒
realinv ∘ f continuous (at a within s)
[CONTINUOUS_CLOSED_IMP_CAUCHY_CONTINUOUS] Theorem
⊢ ∀f s.
f continuous_on s ∧ closed s ⇒
∀x. cauchy x ∧ (∀n. x n ∈ s) ⇒ cauchy (f ∘ x)
[CONTINUOUS_CLOSED_IN_PREIMAGE] Theorem
⊢ ∀f s t.
f continuous_on s ∧ closed t ⇒
closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}
[CONTINUOUS_CLOSED_IN_PREIMAGE_CONSTANT] Theorem
⊢ ∀f s a.
f continuous_on s ⇒
closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x = a}
[CONTINUOUS_CLOSED_IN_PREIMAGE_EQ] Theorem
⊢ ∀f s.
f continuous_on s ⇔
∀t. closed t ⇒
closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}
[CONTINUOUS_CLOSED_IN_PREIMAGE_GEN] Theorem
⊢ ∀f s t u.
f continuous_on s ∧ IMAGE f s ⊆ t ∧
closed_in (subtopology euclidean t) u ⇒
closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u}
[CONTINUOUS_CLOSED_PREIMAGE] Theorem
⊢ ∀f s t.
f continuous_on s ∧ closed s ∧ closed t ⇒
closed {x | x ∈ s ∧ f x ∈ t}
[CONTINUOUS_CLOSED_PREIMAGE_CONSTANT] Theorem
⊢ ∀f s. f continuous_on s ∧ closed s ⇒ closed {x | x ∈ s ∧ f x = a}
[CONTINUOUS_CLOSED_PREIMAGE_UNIV] Theorem
⊢ ∀f s. (∀x. f continuous at x) ∧ closed s ⇒ closed {x | f x ∈ s}
[CONTINUOUS_CMUL] Theorem
⊢ ∀f c net. f continuous net ⇒ (λx. c * f x) continuous net
[CONTINUOUS_COMPONENT_COMPOSE] Theorem
⊢ ∀net f i. f continuous net ⇒ (λx. f x) continuous net
[CONTINUOUS_CONST] Theorem
⊢ ∀net c. (λx. c) continuous net
[CONTINUOUS_CONSTANT_ON_CLOSURE] Theorem
⊢ ∀f s a.
f continuous_on closure s ∧ (∀x. x ∈ s ⇒ f x = a) ⇒
∀x. x ∈ closure s ⇒ f x = a
[CONTINUOUS_DIAMETER] Theorem
⊢ ∀s e.
bounded s ∧ s ≠ ∅ ∧ 0 < e ⇒
∃d. 0 < d ∧
∀t. bounded t ∧ t ≠ ∅ ∧ hausdist (s,t) < d ⇒
abs (diameter s − diameter t) < e
[CONTINUOUS_DISCONNECTED_DISCRETE_FINITE_RANGE_CONSTANT_EQ] Theorem
⊢ (∀s. connected s ⇔
∀f t.
f continuous_on s ∧ IMAGE f s ⊆ t ∧
(∀y. y ∈ t ⇒ connected_component t y = {y}) ⇒
∃a. ∀x. x ∈ s ⇒ f x = a) ∧
(∀s. connected s ⇔
∀f. f continuous_on s ∧
(∀x. x ∈ s ⇒
∃e. 0 < e ∧
∀y. y ∈ s ∧ f y ≠ f x ⇒ e ≤ abs (f y − f x)) ⇒
∃a. ∀x. x ∈ s ⇒ f x = a) ∧
∀s. connected s ⇔
∀f. f continuous_on s ∧ FINITE (IMAGE f s) ⇒
∃a. ∀x. x ∈ s ⇒ f x = a
[CONTINUOUS_DISCONNECTED_RANGE_CONSTANT] Theorem
⊢ ∀f s.
connected s ∧ f continuous_on s ∧ IMAGE f s ⊆ t ∧
(∀y. y ∈ t ⇒ connected_component t y = {y}) ⇒
∃a. ∀x. x ∈ s ⇒ f x = a
[CONTINUOUS_DISCONNECTED_RANGE_CONSTANT_EQ] Theorem
⊢ ∀s. connected s ⇔
∀f t.
f continuous_on s ∧ IMAGE f s ⊆ t ∧
(∀y. y ∈ t ⇒ connected_component t y = {y}) ⇒
∃a. ∀x. x ∈ s ⇒ f x = a
[CONTINUOUS_DISCRETE_RANGE_CONSTANT] Theorem
⊢ ∀f s.
connected s ∧ f continuous_on s ∧
(∀x. x ∈ s ⇒
∃e. 0 < e ∧ ∀y. y ∈ s ∧ f y ≠ f x ⇒ e ≤ abs (f y − f x)) ⇒
∃a. ∀x. x ∈ s ⇒ f x = a
[CONTINUOUS_DISCRETE_RANGE_CONSTANT_EQ] Theorem
⊢ ∀s. connected s ⇔
∀f. f continuous_on s ∧
(∀x. x ∈ s ⇒
∃e. 0 < e ∧
∀y. y ∈ s ∧ f y ≠ f x ⇒ e ≤ abs (f y − f x)) ⇒
∃a. ∀x. x ∈ s ⇒ f x = a
[CONTINUOUS_DOT2] Theorem
⊢ ∀net f g.
f continuous net ∧ g continuous net ⇒
(λx. f x * g x) continuous net
[CONTINUOUS_FINITE_RANGE_CONSTANT] Theorem
⊢ ∀f s.
connected s ∧ f continuous_on s ∧ FINITE (IMAGE f s) ⇒
∃a. ∀x. x ∈ s ⇒ f x = a
[CONTINUOUS_FINITE_RANGE_CONSTANT_EQ] Theorem
⊢ ∀s. connected s ⇔
∀f. f continuous_on s ∧ FINITE (IMAGE f s) ⇒
∃a. ∀x. x ∈ s ⇒ f x = a
[CONTINUOUS_GE_ON_CLOSURE] Theorem
⊢ ∀f s a.
f continuous_on closure s ∧ (∀x. x ∈ s ⇒ a ≤ f x) ⇒
∀x. x ∈ closure s ⇒ a ≤ f x
[CONTINUOUS_IMP_CLOSED_MAP] Theorem
⊢ ∀f s t.
f continuous_on s ∧ IMAGE f s = t ∧ compact s ⇒
∀u. closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean t) (IMAGE f u)
[CONTINUOUS_IMP_QUOTIENT_MAP] Theorem
⊢ ∀f s t.
f continuous_on s ∧ IMAGE f s = t ∧ compact s ⇒
∀u. u ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
open_in (subtopology euclidean t) u)
[CONTINUOUS_INV] Theorem
⊢ ∀net f.
f continuous net ∧ f (netlimit net) ≠ 0 ⇒
realinv ∘ f continuous net
[CONTINUOUS_LEFT_INVERSE_IMP_QUOTIENT_MAP] Theorem
⊢ ∀f g s.
f continuous_on s ∧ g continuous_on IMAGE f s ∧
(∀x. x ∈ s ⇒ g (f x) = x) ⇒
∀u. u ⊆ IMAGE f s ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
open_in (subtopology euclidean (IMAGE f s)) u)
[CONTINUOUS_LEVELSET_OPEN] Theorem
⊢ ∀f s a.
connected s ∧ f continuous_on s ∧ open {x | x ∈ s ∧ f x = a} ∧
(∃x. x ∈ s ∧ f x = a) ⇒
∀x. x ∈ s ⇒ f x = a
[CONTINUOUS_LEVELSET_OPEN_IN] Theorem
⊢ ∀f s a.
connected s ∧ f continuous_on s ∧
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x = a} ∧
(∃x. x ∈ s ∧ f x = a) ⇒
∀x. x ∈ s ⇒ f x = a
[CONTINUOUS_LEVELSET_OPEN_IN_CASES] Theorem
⊢ ∀f s a.
connected s ∧ f continuous_on s ∧
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x = a} ⇒
(∀x. x ∈ s ⇒ f x ≠ a) ∨ ∀x. x ∈ s ⇒ f x = a
[CONTINUOUS_LE_ON_CLOSURE] Theorem
⊢ ∀f s a.
f continuous_on closure s ∧ (∀x. x ∈ s ⇒ f x ≤ a) ⇒
∀x. x ∈ closure s ⇒ f x ≤ a
[CONTINUOUS_MAP_CLOSURES] Theorem
⊢ ∀f. f continuous_on 𝕌(:real) ⇔
∀s. IMAGE f (closure s) ⊆ closure (IMAGE f s)
[CONTINUOUS_MAX] Theorem
⊢ ∀f g net.
f continuous net ∧ g continuous net ⇒
(λx. max (f x) (g x)) continuous net
[CONTINUOUS_MIN] Theorem
⊢ ∀f g net.
f continuous net ∧ g continuous net ⇒
(λx. min (f x) (g x)) continuous net
[CONTINUOUS_MUL] Theorem
⊢ ∀net f c.
c continuous net ∧ f continuous net ⇒
(λx. c x * f x) continuous net
[CONTINUOUS_NEG] Theorem
⊢ ∀f net. f continuous net ⇒ (λx. -f x) continuous net
[CONTINUOUS_ON] Theorem
⊢ ∀f s. f continuous_on s ⇔ ∀x. x ∈ s ⇒ (f ⟶ f x) (at x within s)
[CONTINUOUS_ON_ABS] Theorem
⊢ ∀f s. f continuous_on s ⇒ (λx. abs (f x)) continuous_on s
[CONTINUOUS_ON_ABS_COMPOSE] Theorem
⊢ ∀f s. f continuous_on s ⇒ (λx. abs (f x)) continuous_on s
[CONTINUOUS_ON_ADD] Theorem
⊢ ∀f g s.
f continuous_on s ∧ g continuous_on s ⇒
(λx. f x + g x) continuous_on s
[CONTINUOUS_ON_AVOID] Theorem
⊢ ∀f x s a.
f continuous_on s ∧ x ∈ s ∧ f x ≠ a ⇒
∃e. 0 < e ∧ ∀y. y ∈ s ∧ dist (x,y) < e ⇒ f y ≠ a
[CONTINUOUS_ON_CASES] Theorem
⊢ ∀P f g s t.
closed s ∧ closed t ∧ f continuous_on s ∧ g continuous_on t ∧
(∀x. x ∈ s ∧ ¬P x ∨ x ∈ t ∧ P x ⇒ f x = g x) ⇒
(λx. if P x then f x else g x) continuous_on s ∪ t
[CONTINUOUS_ON_CASES_1] Theorem
⊢ ∀f g s a.
f continuous_on {t | t ∈ s ∧ t ≤ a} ∧
g continuous_on equiv_class $<= s a ∧ (a ∈ s ⇒ f a = g a) ⇒
(λt. if t ≤ a then f t else g t) continuous_on s
[CONTINUOUS_ON_CASES_LE] Theorem
⊢ ∀f g h s a.
f continuous_on {t | t ∈ s ∧ h t ≤ a} ∧
g continuous_on {t | t ∈ s ∧ a ≤ h t} ∧ h continuous_on s ∧
(∀t. t ∈ s ∧ h t = a ⇒ f t = g t) ⇒
(λt. if h t ≤ a then f t else g t) continuous_on s
[CONTINUOUS_ON_CASES_LOCAL] Theorem
⊢ ∀P f g s t.
closed_in (subtopology euclidean (s ∪ t)) s ∧
closed_in (subtopology euclidean (s ∪ t)) t ∧ f continuous_on s ∧
g continuous_on t ∧ (∀x. x ∈ s ∧ ¬P x ∨ x ∈ t ∧ P x ⇒ f x = g x) ⇒
(λx. if P x then f x else g x) continuous_on s ∪ t
[CONTINUOUS_ON_CASES_LOCAL_OPEN] Theorem
⊢ ∀P f g s t.
open_in (subtopology euclidean (s ∪ t)) s ∧
open_in (subtopology euclidean (s ∪ t)) t ∧ f continuous_on s ∧
g continuous_on t ∧ (∀x. x ∈ s ∧ ¬P x ∨ x ∈ t ∧ P x ⇒ f x = g x) ⇒
(λx. if P x then f x else g x) continuous_on s ∪ t
[CONTINUOUS_ON_CASES_OPEN] Theorem
⊢ ∀P f g s t.
open s ∧ open t ∧ f continuous_on s ∧ g continuous_on t ∧
(∀x. x ∈ s ∧ ¬P x ∨ x ∈ t ∧ P x ⇒ f x = g x) ⇒
(λx. if P x then f x else g x) continuous_on s ∪ t
[CONTINUOUS_ON_CLOSED] Theorem
⊢ ∀f s.
f continuous_on s ⇔
∀t. closed_in (subtopology euclidean (IMAGE f s)) t ⇒
closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}
[CONTINUOUS_ON_CLOSED_GEN] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ⇒
(f continuous_on s ⇔
∀u. closed_in (subtopology euclidean t) u ⇒
closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u})
[CONTINUOUS_ON_CLOSURE] Theorem
⊢ ∀f s.
f continuous_on closure s ⇔
∀x e.
x ∈ closure s ∧ 0 < e ⇒
∃d. 0 < d ∧ ∀y. y ∈ s ∧ dist (y,x) < d ⇒ dist (f y,f x) < e
[CONTINUOUS_ON_CLOSURE_ABS_LE] Theorem
⊢ ∀f s x b.
f continuous_on closure s ∧ (∀y. y ∈ s ⇒ abs (f y) ≤ b) ∧
x ∈ closure s ⇒
abs (f x) ≤ b
[CONTINUOUS_ON_CLOSURE_COMPONENT_GE] Theorem
⊢ ∀f s x b.
f continuous_on closure s ∧ (∀y. y ∈ s ⇒ b ≤ f y) ∧ x ∈ closure s ⇒
b ≤ f x
[CONTINUOUS_ON_CLOSURE_COMPONENT_LE] Theorem
⊢ ∀f s x b.
f continuous_on closure s ∧ (∀y. y ∈ s ⇒ f y ≤ b) ∧ x ∈ closure s ⇒
f x ≤ b
[CONTINUOUS_ON_CLOSURE_SEQUENTIALLY] Theorem
⊢ ∀f s.
f continuous_on closure s ⇔
∀x a.
a ∈ closure s ∧ (∀n. x n ∈ s) ∧ (x ⟶ a) sequentially ⇒
(f ∘ x ⟶ f a) sequentially
[CONTINUOUS_ON_CMUL] Theorem
⊢ ∀f c s. f continuous_on s ⇒ (λx. c * f x) continuous_on s
[CONTINUOUS_ON_COMPONENTS_FINITE] Theorem
⊢ ∀f s.
FINITE (components s) ∧
(∀c. c ∈ components s ⇒ f continuous_on c) ⇒
f continuous_on s
[CONTINUOUS_ON_COMPONENTS_GEN] Theorem
⊢ ∀f s.
(∀c. c ∈ components s ⇒
open_in (subtopology euclidean s) c ∧ f continuous_on c) ⇒
f continuous_on s
[CONTINUOUS_ON_COMPONENT_COMPOSE] Theorem
⊢ ∀f s. f continuous_on s ⇒ (λx. f x) continuous_on s
[CONTINUOUS_ON_COMPOSE] Theorem
⊢ ∀f g s.
f continuous_on s ∧ g continuous_on IMAGE f s ⇒
g ∘ f continuous_on s
[CONTINUOUS_ON_COMPOSE_QUOTIENT] Theorem
⊢ ∀f g s t u.
IMAGE f s ⊆ t ∧ IMAGE g t ⊆ u ∧
(∀v. v ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ v} ⇔
open_in (subtopology euclidean t) v)) ∧
g ∘ f continuous_on s ⇒
g continuous_on t
[CONTINUOUS_ON_CONST] Theorem
⊢ ∀s c. (λx. c) continuous_on s
[CONTINUOUS_ON_DIST] Theorem
⊢ ∀a s. (λx. dist (a,x)) continuous_on s
[CONTINUOUS_ON_DIST_CLOSEST_POINT] Theorem
⊢ ∀s t.
closed s ∧ s ≠ ∅ ⇒
(λx. dist (x,closest_point s x)) continuous_on t
[CONTINUOUS_ON_DOT2] Theorem
⊢ ∀f g s.
f continuous_on s ∧ g continuous_on s ⇒
(λx. f x * g x) continuous_on s
[CONTINUOUS_ON_EMPTY] Theorem
⊢ ∀f. f continuous_on ∅
[CONTINUOUS_ON_EQ] Theorem
⊢ ∀f g s.
(∀x. x ∈ s ⇒ f x = g x) ∧ f continuous_on s ⇒ g continuous_on s
[CONTINUOUS_ON_EQ_CONTINUOUS_AT] Theorem
⊢ ∀f s. open s ⇒ (f continuous_on s ⇔ ∀x. x ∈ s ⇒ f continuous at x)
[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] Theorem
⊢ ∀f s. f continuous_on s ⇔ ∀x. x ∈ s ⇒ f continuous (at x within s)
[CONTINUOUS_ON_FINITE] Theorem
⊢ ∀f s. FINITE s ⇒ f continuous_on s
[CONTINUOUS_ON_ID] Theorem
⊢ ∀s. (λx. x) continuous_on s
[CONTINUOUS_ON_IMP_CLOSED_IN] Theorem
⊢ ∀f s t.
f continuous_on s ∧
closed_in (subtopology euclidean (IMAGE f s)) t ⇒
closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}
[CONTINUOUS_ON_IMP_OPEN_IN] Theorem
⊢ ∀f s t.
f continuous_on s ∧ open_in (subtopology euclidean (IMAGE f s)) t ⇒
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}
[CONTINUOUS_ON_INTERIOR] Theorem
⊢ ∀f s x. f continuous_on s ∧ x ∈ interior s ⇒ f continuous at x
[CONTINUOUS_ON_INV] Theorem
⊢ ∀f s.
f continuous_on s ∧ (∀x. x ∈ s ⇒ f x ≠ 0) ⇒
realinv ∘ f continuous_on s
[CONTINUOUS_ON_INVERSE] Theorem
⊢ ∀f g s.
f continuous_on s ∧ compact s ∧ (∀x. x ∈ s ⇒ g (f x) = x) ⇒
g continuous_on IMAGE f s
[CONTINUOUS_ON_INVERSE_CLOSED_MAP] Theorem
⊢ ∀f g s t.
f continuous_on s ∧ IMAGE f s = t ∧ (∀x. x ∈ s ⇒ g (f x) = x) ∧
(∀u. closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean t) (IMAGE f u)) ⇒
g continuous_on t
[CONTINUOUS_ON_INVERSE_OPEN_MAP] Theorem
⊢ ∀f g s t.
f continuous_on s ∧ IMAGE f s = t ∧ (∀x. x ∈ s ⇒ g (f x) = x) ∧
(∀u. open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean t) (IMAGE f u)) ⇒
g continuous_on t
[CONTINUOUS_ON_IVT] Theorem
⊢ ∀f a b y.
a ≤ b ∧ f a ≤ y ∧ y ≤ f b ∧ f continuous_on interval [(a,b)] ⇒
∃x. x ∈ interval [(a,b)] ∧ f x = y
[CONTINUOUS_ON_LIFT_DOT] Theorem
⊢ ∀s. (λy. a * y) continuous_on s
[CONTINUOUS_ON_MAX] Theorem
⊢ ∀f g s.
f continuous_on s ∧ g continuous_on s ⇒
(λx. max (f x) (g x)) continuous_on s
[CONTINUOUS_ON_MIN] Theorem
⊢ ∀f g s.
f continuous_on s ∧ g continuous_on s ⇒
(λx. min (f x) (g x)) continuous_on s
[CONTINUOUS_ON_MUL] Theorem
⊢ ∀s c f.
c continuous_on s ∧ f continuous_on s ⇒
(λx. c x * f x) continuous_on s
[CONTINUOUS_ON_NEG] Theorem
⊢ ∀f s. f continuous_on s ⇒ (λx. -f x) continuous_on s
[CONTINUOUS_ON_NO_LIMPT] Theorem
⊢ ∀f s. ¬(∃x. x limit_point_of s) ⇒ f continuous_on s
[CONTINUOUS_ON_OPEN] Theorem
⊢ ∀f s.
f continuous_on s ⇔
∀t. open_in (subtopology euclidean (IMAGE f s)) t ⇒
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}
[CONTINUOUS_ON_OPEN_AVOID] Theorem
⊢ ∀f x s a.
f continuous_on s ∧ open s ∧ x ∈ s ∧ f x ≠ a ⇒
∃e. 0 < e ∧ ∀y. dist (x,y) < e ⇒ f y ≠ a
[CONTINUOUS_ON_OPEN_GEN] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ⇒
(f continuous_on s ⇔
∀u. open_in (subtopology euclidean t) u ⇒
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u})
[CONTINUOUS_ON_POW] Theorem
⊢ ∀f s n. (λx. f x) continuous_on s ⇒ (λx. f x pow n) continuous_on s
[CONTINUOUS_ON_PRODUCT] Theorem
⊢ ∀f s t.
FINITE t ∧ (∀i. i ∈ t ⇒ (λx. f x i) continuous_on s) ⇒
(λx. product t (f x)) continuous_on s
[CONTINUOUS_ON_RANGE] Theorem
⊢ ∀f s.
f continuous_on s ⇔
∀x. x ∈ s ⇒
∀e. 0 < e ⇒
∃d. 0 < d ∧
∀x'. x' ∈ s ∧ abs (x' − x) < d ⇒ abs (f x' − f x) < e
[CONTINUOUS_ON_SEQUENTIALLY] Theorem
⊢ ∀f s.
f continuous_on s ⇔
∀x a.
a ∈ s ∧ (∀n. x n ∈ s) ∧ (x ⟶ a) sequentially ⇒
(f ∘ x ⟶ f a) sequentially
[CONTINUOUS_ON_SETDIST] Theorem
⊢ ∀s t. (λy. setdist ({y},s)) continuous_on t
[CONTINUOUS_ON_SING] Theorem
⊢ ∀f a. f continuous_on {a}
[CONTINUOUS_ON_SUB] Theorem
⊢ ∀f g s.
f continuous_on s ∧ g continuous_on s ⇒
(λx. f x − g x) continuous_on s
[CONTINUOUS_ON_SUBSET] Theorem
⊢ ∀f s t. f continuous_on s ∧ t ⊆ s ⇒ f continuous_on t
[CONTINUOUS_ON_SUM] Theorem
⊢ ∀t f s.
FINITE s ∧ (∀a. a ∈ s ⇒ f a continuous_on t) ⇒
(λx. sum s (λa. f a x)) continuous_on t
[CONTINUOUS_ON_UNION] Theorem
⊢ ∀f s t.
closed s ∧ closed t ∧ f continuous_on s ∧ f continuous_on t ⇒
f continuous_on s ∪ t
[CONTINUOUS_ON_UNION_LOCAL] Theorem
⊢ ∀f s.
closed_in (subtopology euclidean (s ∪ t)) s ∧
closed_in (subtopology euclidean (s ∪ t)) t ∧ f continuous_on s ∧
f continuous_on t ⇒
f continuous_on s ∪ t
[CONTINUOUS_ON_UNION_LOCAL_OPEN] Theorem
⊢ ∀f s.
open_in (subtopology euclidean (s ∪ t)) s ∧
open_in (subtopology euclidean (s ∪ t)) t ∧ f continuous_on s ∧
f continuous_on t ⇒
f continuous_on s ∪ t
[CONTINUOUS_ON_UNION_OPEN] Theorem
⊢ ∀f s t.
open s ∧ open t ∧ f continuous_on s ∧ f continuous_on t ⇒
f continuous_on s ∪ t
[CONTINUOUS_ON_VMUL] Theorem
⊢ ∀s c v. c continuous_on s ⇒ (λx. c x * v) continuous_on s
[CONTINUOUS_OPEN_IN_PREIMAGE] Theorem
⊢ ∀f s t.
f continuous_on s ∧ open t ⇒
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}
[CONTINUOUS_OPEN_IN_PREIMAGE_EQ] Theorem
⊢ ∀f s.
f continuous_on s ⇔
∀t. open t ⇒
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}
[CONTINUOUS_OPEN_IN_PREIMAGE_GEN] Theorem
⊢ ∀f s t u.
f continuous_on s ∧ IMAGE f s ⊆ t ∧
open_in (subtopology euclidean t) u ⇒
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u}
[CONTINUOUS_OPEN_PREIMAGE] Theorem
⊢ ∀f s t.
f continuous_on s ∧ open s ∧ open t ⇒ open {x | x ∈ s ∧ f x ∈ t}
[CONTINUOUS_OPEN_PREIMAGE_UNIV] Theorem
⊢ ∀f s. (∀x. f continuous at x) ∧ open s ⇒ open {x | f x ∈ s}
[CONTINUOUS_POW] Theorem
⊢ ∀net f n. (λx. f x) continuous net ⇒ (λx. f x pow n) continuous net
[CONTINUOUS_PRODUCT] Theorem
⊢ ∀net f t.
FINITE t ∧ (∀i. i ∈ t ⇒ (λx. f x i) continuous net) ⇒
(λx. product t (f x)) continuous net
[CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP] Theorem
⊢ ∀f g s t.
f continuous_on s ∧ IMAGE f s ⊆ t ∧ g continuous_on t ∧
IMAGE g t ⊆ s ∧ (∀y. y ∈ t ⇒ f (g y) = y) ⇒
∀u. u ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
open_in (subtopology euclidean t) u)
[CONTINUOUS_SUB] Theorem
⊢ ∀f g net.
f continuous net ∧ g continuous net ⇒
(λx. f x − g x) continuous net
[CONTINUOUS_SUM] Theorem
⊢ ∀net f s.
FINITE s ∧ (∀a. a ∈ s ⇒ f a continuous net) ⇒
(λx. sum s (λa. f a x)) continuous net
[CONTINUOUS_TRANSFORM_AT] Theorem
⊢ ∀f g x d.
0 < d ∧ (∀x'. dist (x',x) < d ⇒ f x' = g x') ∧ f continuous at x ⇒
g continuous at x
[CONTINUOUS_TRANSFORM_WITHIN] Theorem
⊢ ∀f g s x d.
0 < d ∧ x ∈ s ∧ (∀x'. x' ∈ s ∧ dist (x',x) < d ⇒ f x' = g x') ∧
f continuous (at x within s) ⇒
g continuous (at x within s)
[CONTINUOUS_TRANSFORM_WITHIN_OPEN] Theorem
⊢ ∀f g s a.
open s ∧ a ∈ s ∧ (∀x. x ∈ s ⇒ f x = g x) ∧ f continuous at a ⇒
g continuous at a
[CONTINUOUS_TRANSFORM_WITHIN_OPEN_IN] Theorem
⊢ ∀f g s t a.
open_in (subtopology euclidean t) s ∧ a ∈ s ∧
(∀x. x ∈ s ⇒ f x = g x) ∧ f continuous (at a within t) ⇒
g continuous (at a within t)
[CONTINUOUS_TRANSFORM_WITHIN_SET_IMP] Theorem
⊢ ∀f a s t.
eventually (λx. x ∈ t ⇒ x ∈ s) (at a) ∧
f continuous (at a within s) ⇒
f continuous (at a within t)
[CONTINUOUS_TRIVIAL_LIMIT] Theorem
⊢ ∀f net. trivial_limit net ⇒ f continuous net
[CONTINUOUS_UNIFORM_LIMIT] Theorem
⊢ ∀net f g s.
¬trivial_limit net ∧ eventually (λn. f n continuous_on s) net ∧
(∀e. 0 < e ⇒
eventually (λn. ∀x. x ∈ s ⇒ abs (f n x − g x) < e) net) ⇒
g continuous_on s
[CONTINUOUS_VMUL] Theorem
⊢ ∀net c v. c continuous net ⇒ (λx. c x * v) continuous net
[CONTINUOUS_WITHIN] Theorem
⊢ ∀f x. f continuous (at x within s) ⇔ (f ⟶ f x) (at x within s)
[CONTINUOUS_WITHIN_AVOID] Theorem
⊢ ∀f x s a.
f continuous (at x within s) ∧ x ∈ s ∧ f x ≠ a ⇒
∃e. 0 < e ∧ ∀y. y ∈ s ∧ dist (x,y) < e ⇒ f y ≠ a
[CONTINUOUS_WITHIN_BALL] Theorem
⊢ ∀f s x.
f continuous (at x within s) ⇔
∀e. 0 < e ⇒ ∃d. 0 < d ∧ IMAGE f (ball (x,d) ∩ s) ⊆ ball (f x,e)
[CONTINUOUS_WITHIN_CLOSED_NONTRIVIAL] Theorem
⊢ ∀a s. closed s ∧ a ∉ s ⇒ f continuous (at a within s)
[CONTINUOUS_WITHIN_COMPARISON] Theorem
⊢ ∀f g s a.
g continuous (at a within s) ∧
(∀x. x ∈ s ⇒ dist (f a,f x) ≤ dist (g a,g x)) ⇒
f continuous (at a within s)
[CONTINUOUS_WITHIN_COMPOSE] Theorem
⊢ ∀f g x s.
f continuous (at x within s) ∧
g continuous (at (f x) within IMAGE f s) ⇒
g ∘ f continuous (at x within s)
[CONTINUOUS_WITHIN_ID] Theorem
⊢ ∀a s. (λx. x) continuous (at a within s)
[CONTINUOUS_WITHIN_OPEN] Theorem
⊢ ∀f x u.
f continuous (at x within u) ⇔
∀t. open t ∧ f x ∈ t ⇒
∃s. open s ∧ x ∈ s ∧ ∀x'. x' ∈ s ∧ x' ∈ u ⇒ f x' ∈ t
[CONTINUOUS_WITHIN_SEQUENTIALLY] Theorem
⊢ ∀f s a.
f continuous (at a within s) ⇔
∀x. (∀n. x n ∈ s) ∧ (x ⟶ a) sequentially ⇒
(f ∘ x ⟶ f a) sequentially
[CONTINUOUS_WITHIN_SUBSET] Theorem
⊢ ∀f s t x.
f continuous (at x within s) ∧ t ⊆ s ⇒
f continuous (at x within t)
[CONTRACTION_IMP_CONTINUOUS_ON] Theorem
⊢ ∀f. (∀x y. x ∈ s ∧ y ∈ s ⇒ dist (f x,f y) ≤ dist (x,y)) ⇒
f continuous_on s
[CONVERGENT_BOUNDED_INCREASING] Theorem
⊢ ∀s b.
(∀m n. m ≤ n ⇒ s m ≤ s n) ∧ (∀n. abs (s n) ≤ b) ⇒
∃l. ∀e. 0 < e ⇒ ∃N. ∀n. N ≤ n ⇒ abs (s n − l) < e
[CONVERGENT_BOUNDED_MONOTONE] Theorem
⊢ ∀s b.
(∀n. abs (s n) ≤ b) ∧
((∀m n. m ≤ n ⇒ s m ≤ s n) ∨ ∀m n. m ≤ n ⇒ s n ≤ s m) ⇒
∃l. ∀e. 0 < e ⇒ ∃N. ∀n. N ≤ n ⇒ abs (s n − l) < e
[CONVERGENT_EQ_CAUCHY] Theorem
⊢ ∀s. (∃l. (s ⟶ l) sequentially) ⇔ cauchy s
[CONVERGENT_IMP_BOUNDED] Theorem
⊢ ∀s l. (s ⟶ l) sequentially ⇒ bounded (IMAGE s 𝕌(:num))
[CONVERGENT_IMP_CAUCHY] Theorem
⊢ ∀s l. (s ⟶ l) sequentially ⇒ cauchy s
[COUNTABLE_OPEN_INTERVAL] Theorem
⊢ ∀a b. countable (interval (a,b)) ⇔ interval (a,b) = ∅
[DECREASING_CLOSED_NEST] Theorem
⊢ ∀s. (∀n. closed (s n)) ∧ (∀n. s n ≠ ∅) ∧
(∀m n. m ≤ n ⇒ s n ⊆ s m) ∧
(∀e. 0 < e ⇒ ∃n. ∀x y. x ∈ s n ∧ y ∈ s n ⇒ dist (x,y) < e) ⇒
∃a. ∀n. a ∈ s n
[DECREASING_CLOSED_NEST_SING] Theorem
⊢ ∀s. (∀n. closed (s n)) ∧ (∀n. s n ≠ ∅) ∧
(∀m n. m ≤ n ⇒ s n ⊆ s m) ∧
(∀e. 0 < e ⇒ ∃n. ∀x y. x ∈ s n ∧ y ∈ s n ⇒ dist (x,y) < e) ⇒
∃a. BIGINTER {t | (∃n. t = s n)} = {a}
[DENSE_IMP_PERFECT] Theorem
⊢ ∀s. closure s = 𝕌(:real) ⇒ ∀x. x ∈ s ⇒ x limit_point_of s
[DENSE_LIMIT_POINTS] Theorem
⊢ ∀x. {x | x limit_point_of s} = 𝕌(:real) ⇔ closure s = 𝕌(:real)
[DENSE_OPEN_INTER] Theorem
⊢ ∀s t u.
open_in (subtopology euclidean u) s ∧ t ⊆ u ∨
open_in (subtopology euclidean u) t ∧ s ⊆ u ⇒
(u ⊆ closure (s ∩ t) ⇔ u ⊆ closure s ∧ u ⊆ closure t)
[DEPENDENT_CHOICE] Theorem
⊢ ∀P R.
(∃a. P 0 a) ∧ (∀n x. P n x ⇒ ∃y. P (SUC n) y ∧ R n x y) ⇒
∃f. (∀n. P n (f n)) ∧ ∀n. R n (f n) (f (SUC n))
[DEPENDENT_CHOICE_FIXED] Theorem
⊢ ∀P R a.
P 0 a ∧ (∀n x. P n x ⇒ ∃y. P (SUC n) y ∧ R n x y) ⇒
∃f. f 0 = a ∧ (∀n. P n (f n)) ∧ ∀n. R n (f n) (f (SUC n))
[DEPENDENT_EXPLICIT] Theorem
⊢ ∀p. dependent p ⇔
∃s u.
FINITE s ∧ s ⊆ p ∧ (∃v. v ∈ s ∧ u v ≠ 0) ∧
sum s (λv. u v * v) = 0
[DEPENDENT_MONO] Theorem
⊢ ∀s t. dependent s ∧ s ⊆ t ⇒ dependent t
[DIAMETER_BALL] Theorem
⊢ ∀a r. diameter (ball (a,r)) = if r < 0 then 0 else 2 * r
[DIAMETER_BOUNDED] Theorem
⊢ ∀s. bounded s ⇒
(∀x y. x ∈ s ∧ y ∈ s ⇒ abs (x − y) ≤ diameter s) ∧
∀d. 0 ≤ d ∧ d < diameter s ⇒
∃x y. x ∈ s ∧ y ∈ s ∧ abs (x − y) > d
[DIAMETER_BOUNDED_BOUND] Theorem
⊢ ∀s x y. bounded s ∧ x ∈ s ∧ y ∈ s ⇒ abs (x − y) ≤ diameter s
[DIAMETER_CBALL] Theorem
⊢ ∀a r. diameter (cball (a,r)) = if r < 0 then 0 else 2 * r
[DIAMETER_CLOSURE] Theorem
⊢ ∀s. bounded s ⇒ diameter (closure s) = diameter s
[DIAMETER_EMPTY] Theorem
⊢ diameter ∅ = 0
[DIAMETER_EQ_0] Theorem
⊢ ∀s. bounded s ⇒ (diameter s = 0 ⇔ s = ∅ ∨ ∃a. s = {a})
[DIAMETER_INTERVAL] Theorem
⊢ (∀a b.
diameter (interval [(a,b)]) =
if interval [(a,b)] = ∅ then 0 else abs (b − a)) ∧
∀a b.
diameter (interval (a,b)) =
if interval (a,b) = ∅ then 0 else abs (b − a)
[DIAMETER_LE] Theorem
⊢ ∀s d.
(s ≠ ∅ ∨ 0 ≤ d) ∧ (∀x y. x ∈ s ∧ y ∈ s ⇒ abs (x − y) ≤ d) ⇒
diameter s ≤ d
[DIAMETER_LINEAR_IMAGE] Theorem
⊢ ∀f s.
linear f ∧ (∀x. abs (f x) = abs x) ⇒
diameter (IMAGE f s) = diameter s
[DIAMETER_POS_LE] Theorem
⊢ ∀s. bounded s ⇒ 0 ≤ diameter s
[DIAMETER_SING] Theorem
⊢ ∀a. diameter {a} = 0
[DIAMETER_SUBSET] Theorem
⊢ ∀s t. s ⊆ t ∧ bounded t ⇒ diameter s ≤ diameter t
[DIAMETER_SUBSET_CBALL] Theorem
⊢ ∀s. bounded s ⇒ ∃z. s ⊆ cball (z,diameter s)
[DIAMETER_SUBSET_CBALL_NONEMPTY] Theorem
⊢ ∀s. bounded s ∧ s ≠ ∅ ⇒ ∃z. z ∈ s ∧ s ⊆ cball (z,diameter s)
[DIAMETER_SUMS] Theorem
⊢ ∀s t.
bounded s ∧ bounded t ⇒
diameter {x + y | x ∈ s ∧ y ∈ t} ≤ diameter s + diameter t
[DIFF_CLOSURE_SUBSET] Theorem
⊢ ∀s t. closure s DIFF closure t ⊆ closure (s DIFF t)
[DIM_LE_CARD] Theorem
⊢ ∀s. FINITE s ⇒ dim s ≤ CARD s
[DIM_SUBSET] Theorem
⊢ ∀s t. s ⊆ t ⇒ dim s ≤ dim t
[DIM_SUBSET_UNIV] Theorem
⊢ ∀s. dim s ≤ 1
[DIM_SUBSTANDARD] Theorem
⊢ dim {x | x = 0} = 0
[DIM_UNIQUE] Theorem
⊢ ∀v b. b ⊆ v ∧ v ⊆ span b ∧ independent b ∧ b HAS_SIZE n ⇒ dim v = n
[DIM_UNIV] Theorem
⊢ dim 𝕌(:real) = 1
[DINI] Theorem
⊢ ∀f g s.
compact s ∧ (∀n. f n continuous_on s) ∧ g continuous_on s ∧
(∀x. x ∈ s ⇒ ((λn. f n x) ⟶ g x) sequentially) ∧
(∀n x. x ∈ s ⇒ f n x ≤ f (n + 1) x) ⇒
∀e. 0 < e ⇒
eventually (λn. ∀x. x ∈ s ⇒ abs (f n x − g x) < e)
sequentially
[DISCRETE_BOUNDED_IMP_FINITE] Theorem
⊢ ∀s e.
0 < e ∧ (∀x y. x ∈ s ∧ y ∈ s ∧ abs (y − x) < e ⇒ y = x) ∧
bounded s ⇒
FINITE s
[DISCRETE_IMP_CLOSED] Theorem
⊢ ∀s e.
0 < e ∧ (∀x y. x ∈ s ∧ y ∈ s ∧ abs (y − x) < e ⇒ y = x) ⇒
closed s
[DISJOINT_INTERVAL] Theorem
⊢ ∀a b c d.
(interval [(a,b)] ∩ interval [(c,d)] = ∅ ⇔
b < a ∨ d < c ∨ b < c ∨ d < a) ∧
(interval [(a,b)] ∩ interval (c,d) = ∅ ⇔
b < a ∨ d ≤ c ∨ b ≤ c ∨ d ≤ a) ∧
(interval (a,b) ∩ interval [(c,d)] = ∅ ⇔
b ≤ a ∨ d < c ∨ b ≤ c ∨ d ≤ a) ∧
(interval (a,b) ∩ interval (c,d) = ∅ ⇔
b ≤ a ∨ d ≤ c ∨ b ≤ c ∨ d ≤ a)
[DISTANCE_ATTAINS_INF] Theorem
⊢ ∀s a.
closed s ∧ s ≠ ∅ ⇒
∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ dist (a,x) ≤ dist (a,y)
[DISTANCE_ATTAINS_SUP] Theorem
⊢ ∀s a.
compact s ∧ s ≠ ∅ ⇒
∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ dist (a,y) ≤ dist (a,x)
[DIST_0] Theorem
⊢ ∀x. dist (x,0) = abs x ∧ dist (0,x) = abs x
[DIST_CLOSEST_POINT_LIPSCHITZ] Theorem
⊢ ∀s x y.
closed s ∧ s ≠ ∅ ⇒
abs (dist (x,closest_point s x) − dist (y,closest_point s y)) ≤
dist (x,y)
[DIST_EQ] Theorem
⊢ ∀w x y z. dist (w,x) = dist (y,z) ⇔ (dist (w,x))² = (dist (y,z))²
[DIST_EQ_0] Theorem
⊢ ∀x y. dist (x,y) = 0 ⇔ x = y
[DIST_IN_CLOSED_SEGMENT] Theorem
⊢ ∀a b x.
x ∈ segment [(a,b)] ⇒
dist (x,a) ≤ dist (a,b) ∧ dist (x,b) ≤ dist (a,b)
[DIST_IN_OPEN_CLOSED_SEGMENT] Theorem
⊢ (∀a b x.
x ∈ segment [(a,b)] ⇒
dist (x,a) ≤ dist (a,b) ∧ dist (x,b) ≤ dist (a,b)) ∧
∀a b x.
x ∈ segment (a,b) ⇒
dist (x,a) < dist (a,b) ∧ dist (x,b) < dist (a,b)
[DIST_IN_OPEN_SEGMENT] Theorem
⊢ ∀a b x.
x ∈ segment (a,b) ⇒
dist (x,a) < dist (a,b) ∧ dist (x,b) < dist (a,b)
[DIST_LE_0] Theorem
⊢ ∀x y. dist (x,y) ≤ 0 ⇔ x = y
[DIST_MIDPOINT] Theorem
⊢ ∀a b.
dist (a,midpoint (a,b)) = dist (a,b) / 2 ∧
dist (b,midpoint (a,b)) = dist (a,b) / 2 ∧
dist (midpoint (a,b),a) = dist (a,b) / 2 ∧
dist (midpoint (a,b),b) = dist (a,b) / 2
[DIST_MUL] Theorem
⊢ ∀x y c. dist (c * x,c * y) = abs c * dist (x,y)
[DIST_NZ] Theorem
⊢ ∀x y. x ≠ y ⇔ 0 < dist (x,y)
[DIST_POS_LE] Theorem
⊢ ∀x y. 0 ≤ dist (x,y)
[DIST_POS_LT] Theorem
⊢ ∀x y. x ≠ y ⇒ 0 < dist (x,y)
[DIST_REFL] Theorem
⊢ ∀x. dist (x,x) = 0
[DIST_SYM] Theorem
⊢ ∀x y. dist (x,y) = dist (y,x)
[DIST_TRIANGLE] Theorem
⊢ ∀x y z. dist (x,z) ≤ dist (x,y) + dist (y,z)
[DIST_TRIANGLE_ADD] Theorem
⊢ ∀x x' y y'. dist (x + y,x' + y') ≤ dist (x,x') + dist (y,y')
[DIST_TRIANGLE_ADD_HALF] Theorem
⊢ ∀x x' y y'.
dist (x,x') < e / 2 ∧ dist (y,y') < e / 2 ⇒
dist (x + y,x' + y') < e
[DIST_TRIANGLE_ALT] Theorem
⊢ ∀x y z. dist (y,z) ≤ dist (x,y) + dist (x,z)
[DIST_TRIANGLE_EQ] Theorem
⊢ ∀x y z.
dist (x,z) = dist (x,y) + dist (y,z) ⇔
abs (x − y) * (y − z) = abs (y − z) * (x − y)
[DIST_TRIANGLE_HALF_L] Theorem
⊢ ∀x1 x2 y.
dist (x1,y) < e / 2 ∧ dist (x2,y) < e / 2 ⇒ dist (x1,x2) < e
[DIST_TRIANGLE_HALF_R] Theorem
⊢ ∀x1 x2 y.
dist (y,x1) < e / 2 ∧ dist (y,x2) < e / 2 ⇒ dist (x1,x2) < e
[DIST_TRIANGLE_LE] Theorem
⊢ ∀x y z e. dist (x,z) + dist (y,z) ≤ e ⇒ dist (x,y) ≤ e
[DIST_TRIANGLE_LT] Theorem
⊢ ∀x y z e. dist (x,z) + dist (y,z) < e ⇒ dist (x,y) < e
[DORDER_NET] Theorem
⊢ ∀n. dorder (netord n)
[EMPTY_AS_INTERVAL] Theorem
⊢ ∅ = interval [(1,0)]
[EMPTY_INTERIOR_FINITE] Theorem
⊢ ∀s. FINITE s ⇒ interior s = ∅
[ENDS_IN_INTERVAL] Theorem
⊢ (∀a b. a ∈ interval [(a,b)] ⇔ interval [(a,b)] ≠ ∅) ∧
(∀a b. b ∈ interval [(a,b)] ⇔ interval [(a,b)] ≠ ∅) ∧
(∀a b. a ∉ interval (a,b)) ∧ ∀a b. b ∉ interval (a,b)
[ENDS_IN_SEGMENT] Theorem
⊢ ∀a b. a ∈ segment [(a,b)] ∧ b ∈ segment [(a,b)]
[ENDS_IN_UNIT_INTERVAL] Theorem
⊢ 0 ∈ interval [(0,1)] ∧ 1 ∈ interval [(0,1)] ∧ 0 ∉ interval (0,1) ∧
1 ∉ interval (0,1)
[ENDS_NOT_IN_SEGMENT] Theorem
⊢ ∀a b. a ∉ segment (a,b) ∧ b ∉ segment (a,b)
[EQ_BALLS] Theorem
⊢ (∀a a' r r'.
ball (a,r) = ball (a',r') ⇔ a = a' ∧ r = r' ∨ r ≤ 0 ∧ r' ≤ 0) ∧
(∀a a' r r'. ball (a,r) = cball (a',r') ⇔ r ≤ 0 ∧ r' < 0) ∧
(∀a a' r r'. cball (a,r) = ball (a',r') ⇔ r < 0 ∧ r' ≤ 0) ∧
∀a a' r r'.
cball (a,r) = cball (a',r') ⇔ a = a' ∧ r = r' ∨ r < 0 ∧ r' < 0
[EQ_INTERVAL] Theorem
⊢ (∀a b c d.
interval [(a,b)] = interval [(c,d)] ⇔
interval [(a,b)] = ∅ ∧ interval [(c,d)] = ∅ ∨ a = c ∧ b = d) ∧
(∀a b c d.
interval [(a,b)] = interval (c,d) ⇔
interval [(a,b)] = ∅ ∧ interval (c,d) = ∅) ∧
(∀a b c d.
interval (a,b) = interval [(c,d)] ⇔
interval (a,b) = ∅ ∧ interval [(c,d)] = ∅) ∧
∀a b c d.
interval (a,b) = interval (c,d) ⇔
interval (a,b) = ∅ ∧ interval (c,d) = ∅ ∨ a = c ∧ b = d
[EVENTUALLY_AND] Theorem
⊢ ∀net p q.
eventually (λx. p x ∧ q x) net ⇔
eventually p net ∧ eventually q net
[EVENTUALLY_AT] Theorem
⊢ ∀a p.
eventually p (at a) ⇔
∃d. 0 < d ∧ ∀x. 0 < dist (x,a) ∧ dist (x,a) < d ⇒ p x
[EVENTUALLY_AT_INFINITY] Theorem
⊢ ∀p. eventually p at_infinity ⇔ ∃b. ∀x. abs x ≥ b ⇒ p x
[EVENTUALLY_AT_INFINITY_POS] Theorem
⊢ ∀p. eventually p at_infinity ⇔ ∃b. 0 < b ∧ ∀x. abs x ≥ b ⇒ p x
[EVENTUALLY_AT_NEGINFINITY] Theorem
⊢ ∀p. eventually p at_neginfinity ⇔ ∃b. ∀x. x ≤ b ⇒ p x
[EVENTUALLY_AT_POSINFINITY] Theorem
⊢ ∀p. eventually p at_posinfinity ⇔ ∃b. ∀x. x ≥ b ⇒ p x
[EVENTUALLY_FALSE] Theorem
⊢ ∀net. eventually (λx. F) net ⇔ trivial_limit net
[EVENTUALLY_FORALL] Theorem
⊢ ∀net p s.
FINITE s ∧ s ≠ ∅ ⇒
(eventually (λx. ∀a. a ∈ s ⇒ p a x) net ⇔
∀a. a ∈ s ⇒ eventually (p a) net)
[EVENTUALLY_HAPPENS] Theorem
⊢ ∀net p. eventually p net ⇒ trivial_limit net ∨ ∃x. p x
[EVENTUALLY_MONO] Theorem
⊢ ∀net p q. (∀x. p x ⇒ q x) ∧ eventually p net ⇒ eventually q net
[EVENTUALLY_MP] Theorem
⊢ ∀net p q.
eventually (λx. p x ⇒ q x) net ∧ eventually p net ⇒
eventually q net
[EVENTUALLY_SEQUENTIALLY] Theorem
⊢ ∀p. eventually p sequentially ⇔ ∃N. ∀n. N ≤ n ⇒ p n
[EVENTUALLY_TRUE] Theorem
⊢ ∀net. eventually (λx. T) net ⇔ T
[EVENTUALLY_WITHIN] Theorem
⊢ ∀s a p.
eventually p (at a within s) ⇔
∃d. 0 < d ∧ ∀x. x ∈ s ∧ 0 < dist (x,a) ∧ dist (x,a) < d ⇒ p x
[EVENTUALLY_WITHIN_INTERIOR] Theorem
⊢ ∀p s x.
x ∈ interior s ⇒
(eventually p (at x within s) ⇔ eventually p (at x))
[EVENTUALLY_WITHIN_LE] Theorem
⊢ ∀s a p.
eventually p (at a within s) ⇔
∃d. 0 < d ∧ ∀x. x ∈ s ∧ 0 < dist (x,a) ∧ dist (x,a) ≤ d ⇒ p x
[EXCHANGE_LEMMA] Theorem
⊢ ∀s t.
FINITE t ∧ independent s ∧ s ⊆ span t ⇒
∃t'. t' HAS_SIZE CARD t ∧ s ⊆ t' ∧ t' ⊆ s ∪ t ∧ s ⊆ span t'
[EXISTS_COMPONENT_SUPERSET] Theorem
⊢ ∀s t. t ⊆ s ∧ s ≠ ∅ ∧ connected t ⇒ ∃c. c ∈ components s ∧ t ⊆ c
[EXISTS_DIFF] Theorem
⊢ (∃s. P (𝕌(:α) DIFF s)) ⇔ ∃s. P s
[EXISTS_IN_INSERT] Theorem
⊢ ∀P a s. (∃x. x ∈ a INSERT s ∧ P x) ⇔ P a ∨ ∃x. x ∈ s ∧ P x
[EXTENSION_FROM_CLOPEN] Theorem
⊢ ∀f s t u.
open_in (subtopology euclidean s) t ∧
closed_in (subtopology euclidean s) t ∧ f continuous_on t ∧
IMAGE f t ⊆ u ∧ (u = ∅ ⇒ s = ∅) ⇒
∃g. g continuous_on s ∧ IMAGE g s ⊆ u ∧ ∀x. x ∈ t ⇒ g x = f x
[FINITE_BALL] Theorem
⊢ ∀a r. FINITE (ball (a,r)) ⇔ r ≤ 0
[FINITE_CBALL] Theorem
⊢ ∀a r. FINITE (cball (a,r)) ⇔ r ≤ 0
[FINITE_IMP_BOUNDED] Theorem
⊢ ∀s. FINITE s ⇒ bounded s
[FINITE_IMP_CLOSED] Theorem
⊢ ∀s. FINITE s ⇒ closed s
[FINITE_IMP_CLOSED_IN] Theorem
⊢ ∀s t. FINITE s ∧ s ⊆ t ⇒ closed_in (subtopology euclidean t) s
[FINITE_IMP_COMPACT] Theorem
⊢ ∀s. FINITE s ⇒ compact s
[FINITE_IMP_NOT_OPEN] Theorem
⊢ ∀s. FINITE s ∧ s ≠ ∅ ⇒ ¬open s
[FINITE_INTERVAL] Theorem
⊢ (∀a b. FINITE (interval [(a,b)]) ⇔ b ≤ a) ∧
∀a b. FINITE (interval (a,b)) ⇔ b ≤ a
[FINITE_INTER_NUMSEG] Theorem
⊢ ∀s m n. FINITE (s ∩ {m .. n})
[FINITE_SET_AVOID] Theorem
⊢ ∀a s. FINITE s ⇒ ∃d. 0 < d ∧ ∀x. x ∈ s ∧ x ≠ a ⇒ d ≤ dist (a,x)
[FINITE_SPHERE] Theorem
⊢ ∀a r. FINITE (sphere (a,r))
[FORALL_EVENTUALLY] Theorem
⊢ ∀net p s.
FINITE s ∧ s ≠ ∅ ⇒
((∀a. a ∈ s ⇒ eventually (p a) net) ⇔
eventually (λx. ∀a. a ∈ s ⇒ p a x) net)
[FORALL_IN_CLOSURE] Theorem
⊢ ∀f s t.
closed t ∧ f continuous_on closure s ∧ (∀x. x ∈ s ⇒ f x ∈ t) ⇒
∀x. x ∈ closure s ⇒ f x ∈ t
[FORALL_IN_CLOSURE_EQ] Theorem
⊢ ∀f s t.
closed t ∧ f continuous_on closure s ⇒
((∀x. x ∈ closure s ⇒ f x ∈ t) ⇔ ∀x. x ∈ s ⇒ f x ∈ t)
[FORALL_POS_MONO_1] Theorem
⊢ ∀P. (∀d e. d < e ∧ P d ⇒ P e) ∧ (∀n. P (&n + 1)⁻¹) ⇒
∀e. 0 < e ⇒ P e
[FRONTIER_BALL] Theorem
⊢ ∀a e. 0 < e ⇒ frontier (ball (a,e)) = sphere (a,e)
[FRONTIER_CBALL] Theorem
⊢ ∀a e. frontier (cball (a,e)) = sphere (a,e)
[FRONTIER_CLOSED] Theorem
⊢ ∀s. closed (frontier s)
[FRONTIER_CLOSED_INTERVAL] Theorem
⊢ ∀a b.
frontier (interval [(a,b)]) =
interval [(a,b)] DIFF interval (a,b)
[FRONTIER_CLOSURES] Theorem
⊢ ∀s. frontier s = closure s ∩ closure (𝕌(:real) DIFF s)
[FRONTIER_CLOSURE_SUBSET] Theorem
⊢ ∀s. frontier (closure s) ⊆ frontier s
[FRONTIER_COMPLEMENT] Theorem
⊢ ∀s. frontier (𝕌(:real) DIFF s) = frontier s
[FRONTIER_DISJOINT_EQ] Theorem
⊢ ∀s. frontier s ∩ s = ∅ ⇔ open s
[FRONTIER_EMPTY] Theorem
⊢ frontier ∅ = ∅
[FRONTIER_FRONTIER] Theorem
⊢ ∀s. open s ∨ closed s ⇒ frontier (frontier s) = frontier s
[FRONTIER_FRONTIER_FRONTIER] Theorem
⊢ ∀s. frontier (frontier (frontier s)) = frontier (frontier s)
[FRONTIER_FRONTIER_SUBSET] Theorem
⊢ ∀s. frontier (frontier s) ⊆ frontier s
[FRONTIER_HALFSPACE_GE] Theorem
⊢ ∀a b. ¬(a = 0 ∧ b = 0) ⇒ frontier {x | a * x ≥ b} = {x | a * x = b}
[FRONTIER_HALFSPACE_GT] Theorem
⊢ ∀a b. ¬(a = 0 ∧ b = 0) ⇒ frontier {x | a * x > b} = {x | a * x = b}
[FRONTIER_HALFSPACE_LE] Theorem
⊢ ∀a b. ¬(a = 0 ∧ b = 0) ⇒ frontier {x | a * x ≤ b} = {x | a * x = b}
[FRONTIER_HALFSPACE_LT] Theorem
⊢ ∀a b. ¬(a = 0 ∧ b = 0) ⇒ frontier {x | a * x < b} = {x | a * x = b}
[FRONTIER_INTERIORS] Theorem
⊢ ∀s. frontier s =
𝕌(:real) DIFF interior s DIFF interior (𝕌(:real) DIFF s)
[FRONTIER_INTERIOR_SUBSET] Theorem
⊢ ∀s. frontier (interior s) ⊆ frontier s
[FRONTIER_INTER_SUBSET] Theorem
⊢ ∀s t. frontier (s ∩ t) ⊆ frontier s ∪ frontier t
[FRONTIER_INTER_SUBSET_INTER] Theorem
⊢ ∀s t.
frontier (s ∩ t) ⊆
closure s ∩ frontier t ∪ frontier s ∩ closure t
[FRONTIER_OPEN_INTERVAL] Theorem
⊢ ∀a b.
frontier (interval (a,b)) =
if interval (a,b) = ∅ then ∅
else interval [(a,b)] DIFF interval (a,b)
[FRONTIER_SING] Theorem
⊢ ∀a. frontier {a} = {a}
[FRONTIER_STRADDLE] Theorem
⊢ ∀a s.
a ∈ frontier s ⇔
∀e. 0 < e ⇒
(∃x. x ∈ s ∧ dist (a,x) < e) ∧ ∃x. x ∉ s ∧ dist (a,x) < e
[FRONTIER_SUBSET_CLOSED] Theorem
⊢ ∀s. closed s ⇒ frontier s ⊆ s
[FRONTIER_SUBSET_COMPACT] Theorem
⊢ ∀s. compact s ⇒ frontier s ⊆ s
[FRONTIER_SUBSET_EQ] Theorem
⊢ ∀s. frontier s ⊆ s ⇔ closed s
[FRONTIER_UNION] Theorem
⊢ ∀s t.
closure s ∩ closure t = ∅ ⇒
frontier (s ∪ t) = frontier s ∪ frontier t
[FRONTIER_UNION_SUBSET] Theorem
⊢ ∀s t. frontier (s ∪ t) ⊆ frontier s ∪ frontier t
[FRONTIER_UNIV] Theorem
⊢ frontier 𝕌(:real) = ∅
[FUNCTION_FACTORS_LEFT_GEN] Theorem
⊢ ∀P f g.
(∀x y. P x ∧ P y ∧ g x = g y ⇒ f x = f y) ⇔
∃h. ∀x. P x ⇒ f x = h (g x)
[GDELTA_COMPLEMENT] Theorem
⊢ ∀s. gdelta (𝕌(:real) DIFF s) ⇔ fsigma s
[GREATER_EQ_REFL] Theorem
⊢ ∀m. m ≥ m
[HAS_SIZE_STDBASIS] Theorem
⊢ {i | 1 ≤ i ∧ i ≤ 1} HAS_SIZE 1
[HAUSDIST_ALT] Theorem
⊢ ∀s t.
bounded s ∧ bounded t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
hausdist (s,t) =
sup {abs (setdist ({x},s) − setdist ({x},t)) | x ∈ 𝕌(:real)}
[HAUSDIST_BALLS] Theorem
⊢ (∀a b r s.
hausdist (ball (a,r),ball (b,s)) =
if r ≤ 0 ∨ s ≤ 0 then 0 else dist (a,b) + abs (r − s)) ∧
(∀a b r s.
hausdist (ball (a,r),cball (b,s)) =
if r ≤ 0 ∨ s < 0 then 0 else dist (a,b) + abs (r − s)) ∧
(∀a b r s.
hausdist (cball (a,r),ball (b,s)) =
if r < 0 ∨ s ≤ 0 then 0 else dist (a,b) + abs (r − s)) ∧
∀a b r s.
hausdist (cball (a,r),cball (b,s)) =
if r < 0 ∨ s < 0 then 0 else dist (a,b) + abs (r − s)
[HAUSDIST_CLOSURE] Theorem
⊢ (∀s t. hausdist (closure s,t) = hausdist (s,t)) ∧
∀s t. hausdist (s,closure t) = hausdist (s,t)
[HAUSDIST_COMPACT_EXISTS] Theorem
⊢ ∀s t.
bounded s ∧ compact t ∧ t ≠ ∅ ⇒
∀x. x ∈ s ⇒ ∃y. y ∈ t ∧ dist (x,y) ≤ hausdist (s,t)
[HAUSDIST_COMPACT_NONTRIVIAL] Theorem
⊢ ∀s t.
compact s ∧ compact t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
hausdist (s,t) =
inf
{e |
0 ≤ e ∧ s ⊆ {x + y | x ∈ t ∧ abs y ≤ e} ∧
t ⊆ {x + y | x ∈ s ∧ abs y ≤ e}}
[HAUSDIST_COMPACT_SUMS] Theorem
⊢ ∀s t.
bounded s ∧ compact t ∧ t ≠ ∅ ⇒
s ⊆ {y + z | y ∈ t ∧ z ∈ cball (0,hausdist (s,t))}
[HAUSDIST_EMPTY] Theorem
⊢ (∀t. hausdist (∅,t) = 0) ∧ ∀s. hausdist (s,∅) = 0
[HAUSDIST_EQ] Theorem
⊢ ∀s t s' t'.
(∀b. (∀x. x ∈ s ⇒ setdist ({x},t) ≤ b) ∧
(∀y. y ∈ t ⇒ setdist ({y},s) ≤ b) ⇔
(∀x. x ∈ s' ⇒ setdist ({x},t') ≤ b) ∧
∀y. y ∈ t' ⇒ setdist ({y},s') ≤ b) ⇒
hausdist (s,t) = hausdist (s',t')
[HAUSDIST_EQ_0] Theorem
⊢ ∀s t.
bounded s ∧ bounded t ⇒
(hausdist (s,t) = 0 ⇔ s = ∅ ∨ t = ∅ ∨ closure s = closure t)
[HAUSDIST_INSERT_LE] Theorem
⊢ ∀s t a.
bounded s ∧ bounded t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
hausdist (a INSERT s,a INSERT t) ≤ hausdist (s,t)
[HAUSDIST_LINEAR_IMAGE] Theorem
⊢ ∀f s t.
linear f ∧ (∀x. abs (f x) = abs x) ⇒
hausdist (IMAGE f s,IMAGE f t) = hausdist (s,t)
[HAUSDIST_NONTRIVIAL] Theorem
⊢ ∀s t.
bounded s ∧ bounded t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
hausdist (s,t) =
sup ({setdist ({x},t) | x ∈ s} ∪ {setdist ({y},s) | y ∈ t})
[HAUSDIST_NONTRIVIAL_ALT] Theorem
⊢ ∀s t.
bounded s ∧ bounded t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
hausdist (s,t) =
max (sup {setdist ({x},t) | x ∈ s})
(sup {setdist ({y},s) | y ∈ t})
[HAUSDIST_POS_LE] Theorem
⊢ ∀s t. 0 ≤ hausdist (s,t)
[HAUSDIST_REFL] Theorem
⊢ ∀s. hausdist (s,s) = 0
[HAUSDIST_SETDIST_TRIANGLE] Theorem
⊢ ∀s t u.
t ≠ ∅ ∧ bounded s ∧ bounded t ⇒
setdist (s,u) ≤ hausdist (s,t) + setdist (t,u)
[HAUSDIST_SINGS] Theorem
⊢ ∀x y. hausdist ({x},{y}) = dist (x,y)
[HAUSDIST_SYM] Theorem
⊢ ∀s t. hausdist (s,t) = hausdist (t,s)
[HAUSDIST_TRANS] Theorem
⊢ ∀s t u.
bounded s ∧ bounded t ∧ bounded u ∧ t ≠ ∅ ⇒
hausdist (s,u) ≤ hausdist (s,t) + hausdist (t,u)
[HAUSDIST_TRANSLATION] Theorem
⊢ ∀a s t.
hausdist (IMAGE (λx. a + x) s,IMAGE (λx. a + x) t) =
hausdist (s,t)
[HAUSDIST_TRIANGLE] Theorem
⊢ ∀s t u.
bounded s ∧ bounded t ∧ bounded u ∧ t ≠ ∅ ⇒
hausdist (s,u) ≤ hausdist (s,t) + hausdist (t,u)
[HAUSDIST_UNION_LE] Theorem
⊢ ∀s t u.
bounded s ∧ bounded t ∧ bounded u ∧ t ≠ ∅ ∧ u ≠ ∅ ⇒
hausdist (s ∪ t,s ∪ u) ≤ hausdist (t,u)
[HEINE_BOREL_IMP_BOLZANO_WEIERSTRASS] Theorem
⊢ ∀s. (∀f. (∀t. t ∈ f ⇒ open t) ∧ s ⊆ BIGUNION f ⇒
∃f'. f' ⊆ f ∧ FINITE f' ∧ s ⊆ BIGUNION f') ⇒
∀t. INFINITE t ∧ t ⊆ s ⇒ ∃x. x ∈ s ∧ x limit_point_of t
[HEINE_BOREL_LEMMA] Theorem
⊢ ∀s. compact s ⇒
∀t. s ⊆ BIGUNION t ∧ (∀b. b ∈ t ⇒ open b) ⇒
∃e. 0 < e ∧ ∀x. x ∈ s ⇒ ∃b. b ∈ t ∧ ball (x,e) ⊆ b
[HOMEOMORPHIC_AFFINITY] Theorem
⊢ ∀s a c. c ≠ 0 ⇒ s homeomorphic IMAGE (λx. a + c * x) s
[HOMEOMORPHIC_BALLS] Theorem
⊢ ∀a b d e. 0 < d ∧ 0 < e ⇒ ball (a,d) homeomorphic ball (b,e)
[HOMEOMORPHIC_BALLS_CBALL_SPHERE] Theorem
⊢ (∀a b d e. 0 < d ∧ 0 < e ⇒ ball (a,d) homeomorphic ball (b,e)) ∧
(∀a b d e. 0 < d ∧ 0 < e ⇒ cball (a,d) homeomorphic cball (b,e)) ∧
∀a b d e. 0 < d ∧ 0 < e ⇒ sphere (a,d) homeomorphic sphere (b,e)
[HOMEOMORPHIC_CBALL] Theorem
⊢ ∀a b d e. 0 < d ∧ 0 < e ⇒ cball (a,d) homeomorphic cball (b,e)
[HOMEOMORPHIC_COMPACT] Theorem
⊢ ∀s f t.
compact s ∧ f continuous_on s ∧ IMAGE f s = t ∧
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
s homeomorphic t
[HOMEOMORPHIC_COMPACTNESS] Theorem
⊢ ∀s t. s homeomorphic t ⇒ (compact s ⇔ compact t)
[HOMEOMORPHIC_CONNECTEDNESS] Theorem
⊢ ∀s t. s homeomorphic t ⇒ (connected s ⇔ connected t)
[HOMEOMORPHIC_EMPTY] Theorem
⊢ (∀s. s homeomorphic ∅ ⇔ s = ∅) ∧ ∀s. ∅ homeomorphic s ⇔ s = ∅
[HOMEOMORPHIC_FINITE] Theorem
⊢ ∀s t. FINITE s ∧ FINITE t ⇒ (s homeomorphic t ⇔ CARD s = CARD t)
[HOMEOMORPHIC_FINITENESS] Theorem
⊢ ∀s t. s homeomorphic t ⇒ (FINITE s ⇔ FINITE t)
[HOMEOMORPHIC_FINITE_STRONG] Theorem
⊢ ∀s t.
FINITE s ∨ FINITE t ⇒
(s homeomorphic t ⇔ FINITE s ∧ FINITE t ∧ CARD s = CARD t)
[HOMEOMORPHIC_HYPERPLANES] Theorem
⊢ ∀a b c d.
a ≠ 0 ∧ c ≠ 0 ⇒ {x | a * x = b} homeomorphic {x | c * x = d}
[HOMEOMORPHIC_HYPERPLANE_STANDARD_HYPERPLANE] Theorem
⊢ ∀a b c. a ≠ 0 ⇒ {x | a * x = b} homeomorphic {x | x = c}
[HOMEOMORPHIC_IMP_CARD_EQ] Theorem
⊢ ∀s t. s homeomorphic t ⇒ s ≈ t
[HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ] Theorem
⊢ ∀f s t.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
(IMAGE f s homeomorphic t ⇔ s homeomorphic t)
[HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ] Theorem
⊢ ∀f s t.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
(s homeomorphic IMAGE f t ⇔ s homeomorphic t)
[HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF] Theorem
⊢ ∀f s.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒ IMAGE f s homeomorphic s
[HOMEOMORPHIC_LOCALLY] Theorem
⊢ ∀P Q.
(∀s t. s homeomorphic t ⇒ (P s ⇔ Q t)) ⇒
∀s t. s homeomorphic t ⇒ (locally P s ⇔ locally Q t)
[HOMEOMORPHIC_LOCAL_COMPACTNESS] Theorem
⊢ ∀s t. s homeomorphic t ⇒ (locally compact s ⇔ locally compact t)
[HOMEOMORPHIC_MINIMAL] Theorem
⊢ ∀s t.
s homeomorphic t ⇔
∃f g.
(∀x. x ∈ s ⇒ f x ∈ t ∧ g (f x) = x) ∧
(∀y. y ∈ t ⇒ g y ∈ s ∧ f (g y) = y) ∧ f continuous_on s ∧
g continuous_on t
[HOMEOMORPHIC_ONE_POINT_COMPACTIFICATIONS] Theorem
⊢ ∀s t a b.
compact s ∧ compact t ∧ a ∈ s ∧ b ∈ t ∧
s DELETE a homeomorphic t DELETE b ⇒
s homeomorphic t
[HOMEOMORPHIC_OPEN_INTERVALS] Theorem
⊢ ∀a b c d.
a < b ∧ c < d ⇒ interval (a,b) homeomorphic interval (c,d)
[HOMEOMORPHIC_OPEN_INTERVAL_UNIV] Theorem
⊢ ∀a b. a < b ⇒ interval (a,b) homeomorphic 𝕌(:real)
[HOMEOMORPHIC_REFL] Theorem
⊢ ∀s. s homeomorphic s
[HOMEOMORPHIC_SCALING] Theorem
⊢ ∀s c. c ≠ 0 ⇒ s homeomorphic IMAGE (λx. c * x) s
[HOMEOMORPHIC_SCALING_LEFT] Theorem
⊢ ∀c. 0 < c ⇒
∀s t. IMAGE (λx. c * x) s homeomorphic t ⇔ s homeomorphic t
[HOMEOMORPHIC_SCALING_RIGHT] Theorem
⊢ ∀c. 0 < c ⇒
∀s t. s homeomorphic IMAGE (λx. c * x) t ⇔ s homeomorphic t
[HOMEOMORPHIC_SING] Theorem
⊢ ∀a b. {a} homeomorphic {b}
[HOMEOMORPHIC_SPHERE] Theorem
⊢ ∀a b d e. 0 < d ∧ 0 < e ⇒ sphere (a,d) homeomorphic sphere (b,e)
[HOMEOMORPHIC_STANDARD_HYPERPLANE_HYPERPLANE] Theorem
⊢ ∀a b c. a ≠ 0 ⇒ {x | x = c} homeomorphic {x | a * x = b}
[HOMEOMORPHIC_SYM] Theorem
⊢ ∀s t. s homeomorphic t ⇔ t homeomorphic s
[HOMEOMORPHIC_TRANS] Theorem
⊢ ∀s t u. s homeomorphic t ∧ t homeomorphic u ⇒ s homeomorphic u
[HOMEOMORPHIC_TRANSLATION] Theorem
⊢ ∀s a. s homeomorphic IMAGE (λx. a + x) s
[HOMEOMORPHIC_TRANSLATION_LEFT_EQ] Theorem
⊢ ∀a s t. IMAGE (λx. a + x) s homeomorphic t ⇔ s homeomorphic t
[HOMEOMORPHIC_TRANSLATION_RIGHT_EQ] Theorem
⊢ ∀a s t. s homeomorphic IMAGE (λx. a + x) t ⇔ s homeomorphic t
[HOMEOMORPHIC_TRANSLATION_SELF] Theorem
⊢ ∀a s. IMAGE (λx. a + x) s homeomorphic s
[HOMEOMORPHISM] Theorem
⊢ ∀s t f g.
homeomorphism (s,t) (f,g) ⇔
f continuous_on s ∧ IMAGE f s ⊆ t ∧ g continuous_on t ∧
IMAGE g t ⊆ s ∧ (∀x. x ∈ s ⇒ g (f x) = x) ∧
∀y. y ∈ t ⇒ f (g y) = y
[HOMEOMORPHISM_COMPACT] Theorem
⊢ ∀s f t.
compact s ∧ f continuous_on s ∧ IMAGE f s = t ∧
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
∃g. homeomorphism (s,t) (f,g)
[HOMEOMORPHISM_COMPOSE] Theorem
⊢ ∀f g h k s t u.
homeomorphism (s,t) (f,g) ∧ homeomorphism (t,u) (h,k) ⇒
homeomorphism (s,u) (h ∘ f,g ∘ k)
[HOMEOMORPHISM_FROM_COMPOSITION_INJECTIVE] Theorem
⊢ ∀f g s t u.
f continuous_on s ∧ IMAGE f s ⊆ t ∧ g continuous_on t ∧
IMAGE g t ⊆ u ∧ (∀x y. x ∈ t ∧ y ∈ t ∧ g x = g y ⇒ x = y) ∧
(∃h. homeomorphism (s,u) (g ∘ f,h)) ⇒
(∃f'. homeomorphism (s,t) (f,f')) ∧
∃g'. homeomorphism (t,u) (g,g')
[HOMEOMORPHISM_FROM_COMPOSITION_SURJECTIVE] Theorem
⊢ ∀f g s t u.
f continuous_on s ∧ IMAGE f s = t ∧ g continuous_on t ∧
IMAGE g t ⊆ u ∧ (∃h. homeomorphism (s,u) (g ∘ f,h)) ⇒
(∃f'. homeomorphism (s,t) (f,f')) ∧
∃g'. homeomorphism (t,u) (g,g')
[HOMEOMORPHISM_ID] Theorem
⊢ ∀s. homeomorphism (s,s) ((λx. x),(λx. x))
[HOMEOMORPHISM_IMP_CLOSED_MAP] Theorem
⊢ ∀f g s t u.
homeomorphism (s,t) (f,g) ∧ closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean t) (IMAGE f u)
[HOMEOMORPHISM_IMP_OPEN_MAP] Theorem
⊢ ∀f g s t u.
homeomorphism (s,t) (f,g) ∧ open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean t) (IMAGE f u)
[HOMEOMORPHISM_IMP_QUOTIENT_MAP] Theorem
⊢ ∀f g s t.
homeomorphism (s,t) (f,g) ⇒
∀u. u ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
open_in (subtopology euclidean t) u)
[HOMEOMORPHISM_INJECTIVE_CLOSED_MAP] Theorem
⊢ ∀f s t.
f continuous_on s ∧ IMAGE f s = t ∧
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧
(∀u. closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean t) (IMAGE f u)) ⇒
∃g. homeomorphism (s,t) (f,g)
[HOMEOMORPHISM_INJECTIVE_CLOSED_MAP_EQ] Theorem
⊢ ∀f s t.
f continuous_on s ∧ IMAGE f s = t ∧
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
((∃g. homeomorphism (s,t) (f,g)) ⇔
∀u. closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean t) (IMAGE f u))
[HOMEOMORPHISM_INJECTIVE_OPEN_MAP] Theorem
⊢ ∀f s t.
f continuous_on s ∧ IMAGE f s = t ∧
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧
(∀u. open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean t) (IMAGE f u)) ⇒
∃g. homeomorphism (s,t) (f,g)
[HOMEOMORPHISM_INJECTIVE_OPEN_MAP_EQ] Theorem
⊢ ∀f s t.
f continuous_on s ∧ IMAGE f s = t ∧
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
((∃g. homeomorphism (s,t) (f,g)) ⇔
∀u. open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean t) (IMAGE f u))
[HOMEOMORPHISM_LOCALLY] Theorem
⊢ ∀P Q f g.
(∀s t. homeomorphism (s,t) (f,g) ⇒ (P s ⇔ Q t)) ⇒
∀s t. homeomorphism (s,t) (f,g) ⇒ (locally P s ⇔ locally Q t)
[HOMEOMORPHISM_OF_SUBSETS] Theorem
⊢ ∀f g s t s' t'.
homeomorphism (s,t) (f,g) ∧ s' ⊆ s ∧ t' ⊆ t ∧ IMAGE f s' = t' ⇒
homeomorphism (s',t') (f,g)
[HOMEOMORPHISM_SYM] Theorem
⊢ ∀f g s t. homeomorphism (s,t) (f,g) ⇔ homeomorphism (t,s) (g,f)
[IMAGE_AFFINITY_INTERVAL] Theorem
⊢ ∀a b m c.
IMAGE (λx. m * x + c) (interval [(a,b)]) =
if interval [(a,b)] = ∅ then ∅
else if 0 ≤ m then interval [(m * a + c,m * b + c)]
else interval [(m * b + c,m * a + c)]
[IMAGE_CLOSURE_SUBSET] Theorem
⊢ ∀f s t.
f continuous_on closure s ∧ closed t ∧ IMAGE f s ⊆ t ⇒
IMAGE f (closure s) ⊆ t
[IMAGE_STRETCH_INTERVAL] Theorem
⊢ ∀a b m.
IMAGE (λx. @f. f = m 1 * x) (interval [(a,b)]) =
if interval [(a,b)] = ∅ then ∅
else
interval
[((@f. f = min (m 1 * a) (m 1 * b)),
@f. f = max (m 1 * a) (m 1 * b))]
[IMAGE_TWIZZLE_INTERVAL] Theorem
⊢ ∀p a b. IMAGE (λx. x) (interval [(a,b)]) = interval [(a,b)]
[INDEPENDENT_BOUND] Theorem
⊢ ∀s. independent s ⇒ FINITE s ∧ CARD s ≤ 1
[INDEPENDENT_CARD_LE_DIM] Theorem
⊢ ∀v b. b ⊆ v ∧ independent b ⇒ FINITE b ∧ CARD b ≤ dim v
[INDEPENDENT_EMPTY] Theorem
⊢ independent ∅
[INDEPENDENT_INJECTIVE_IMAGE] Theorem
⊢ ∀f s.
independent s ∧ linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
independent (IMAGE f s)
[INDEPENDENT_INJECTIVE_IMAGE_GEN] Theorem
⊢ ∀f s.
independent s ∧ linear f ∧
(∀x y. x ∈ span s ∧ y ∈ span s ∧ f x = f y ⇒ x = y) ⇒
independent (IMAGE f s)
[INDEPENDENT_INSERT] Theorem
⊢ ∀a s.
independent (a INSERT s) ⇔
if a ∈ s then independent s else independent s ∧ a ∉ span s
[INDEPENDENT_MONO] Theorem
⊢ ∀s t. independent t ∧ s ⊆ t ⇒ independent s
[INDEPENDENT_NONZERO] Theorem
⊢ ∀s. independent s ⇒ 0 ∉ s
[INDEPENDENT_SING] Theorem
⊢ ∀x. independent {x} ⇔ x ≠ 0
[INDEPENDENT_SPAN_BOUND] Theorem
⊢ ∀s t.
FINITE t ∧ independent s ∧ s ⊆ span t ⇒
FINITE s ∧ CARD s ≤ CARD t
[INDEPENDENT_STDBASIS] Theorem
⊢ independent {i | 1 ≤ i ∧ i ≤ 1}
[INFINITE_OPEN_IN] Theorem
⊢ ∀u s.
open_in (subtopology euclidean u) s ∧
(∃x. x ∈ s ∧ x limit_point_of u) ⇒
INFINITE s
[INFINITE_SUPERSET] Theorem
⊢ ∀s t. INFINITE s ∧ s ⊆ t ⇒ INFINITE t
[INFSUM_0] Theorem
⊢ suminf s (λi. 0) = 0
[INFSUM_ADD] Theorem
⊢ ∀x y s.
summable s x ∧ summable s y ⇒
suminf s (λi. x i + y i) = suminf s x + suminf s y
[INFSUM_CMUL] Theorem
⊢ ∀s x c. summable s x ⇒ suminf s (λn. c * x n) = c * suminf s x
[INFSUM_EQ] Theorem
⊢ ∀f g k.
summable k f ∧ summable k g ∧ (∀x. x ∈ k ⇒ f x = g x) ⇒
suminf k f = suminf k g
[INFSUM_LINEAR] Theorem
⊢ ∀f h s.
summable s f ∧ linear h ⇒ suminf s (λn. h (f n)) = h (suminf s f)
[INFSUM_NEG] Theorem
⊢ ∀s x. summable s x ⇒ suminf s (λn. -x n) = -suminf s x
[INFSUM_RESTRICT] Theorem
⊢ ∀k a. suminf 𝕌(:num) (λn. if n ∈ k then a n else 0) = suminf k a
[INFSUM_SUB] Theorem
⊢ ∀x y s.
summable s x ∧ summable s y ⇒
suminf s (λi. x i − y i) = suminf s x − suminf s y
[INFSUM_UNIQUE] Theorem
⊢ ∀f l s. (f sums l) s ⇒ suminf s f = l
[INF_INSERT] Theorem
⊢ ∀x s.
bounded s ⇒ inf (x INSERT s) = if s = ∅ then x else min x (inf s)
[INJECTIVE_IMP_ISOMETRIC] Theorem
⊢ ∀f s.
closed s ∧ subspace s ∧ linear f ∧ (∀x. x ∈ s ∧ f x = 0 ⇒ x = 0) ⇒
∃e. 0 < e ∧ ∀x. x ∈ s ⇒ abs (f x) ≥ e * abs x
[INJECTIVE_MAP_OPEN_IFF_CLOSED] Theorem
⊢ ∀f s t.
f continuous_on s ∧ IMAGE f s = t ∧
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
((∀u. open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean t) (IMAGE f u)) ⇔
∀u. closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean t) (IMAGE f u))
[INTERIOR_BALL] Theorem
⊢ ∀a r. interior (ball (a,r)) = ball (a,r)
[INTERIOR_BIGINTER_SUBSET] Theorem
⊢ ∀f. interior (BIGINTER f) ⊆ BIGINTER (IMAGE interior f)
[INTERIOR_BIJECTIVE_LINEAR_IMAGE] Theorem
⊢ ∀f s.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ∧ (∀y. ∃x. f x = y) ⇒
interior (IMAGE f s) = IMAGE f (interior s)
[INTERIOR_CBALL] Theorem
⊢ ∀x e. interior (cball (x,e)) = ball (x,e)
[INTERIOR_CLOSED_EQ_EMPTY_AS_FRONTIER] Theorem
⊢ ∀s. closed s ∧ interior s = ∅ ⇔ ∃t. open t ∧ s = frontier t
[INTERIOR_CLOSED_INTERVAL] Theorem
⊢ ∀a b. interior (interval [(a,b)]) = interval (a,b)
[INTERIOR_CLOSED_UNION_EMPTY_INTERIOR] Theorem
⊢ ∀s t. closed s ∧ interior t = ∅ ⇒ interior (s ∪ t) = interior s
[INTERIOR_CLOSURE] Theorem
⊢ ∀s. interior s = 𝕌(:real) DIFF closure (𝕌(:real) DIFF s)
[INTERIOR_CLOSURE_IDEMP] Theorem
⊢ ∀s. interior (closure (interior (closure s))) =
interior (closure s)
[INTERIOR_CLOSURE_INTER_OPEN] Theorem
⊢ ∀s t.
open s ∧ open t ⇒
interior (closure (s ∩ t)) =
interior (closure s) ∩ interior (closure t)
[INTERIOR_COMPLEMENT] Theorem
⊢ ∀s. interior (𝕌(:real) DIFF s) = 𝕌(:real) DIFF closure s
[INTERIOR_DIFF] Theorem
⊢ ∀s t. interior (s DIFF t) = interior s DIFF closure t
[INTERIOR_EMPTY] Theorem
⊢ interior ∅ = ∅
[INTERIOR_EQ] Theorem
⊢ ∀s. interior s = s ⇔ open s
[INTERIOR_EQ_EMPTY] Theorem
⊢ ∀s. interior s = ∅ ⇔ ∀t. open t ∧ t ⊆ s ⇒ t = ∅
[INTERIOR_EQ_EMPTY_ALT] Theorem
⊢ ∀s. interior s = ∅ ⇔ ∀t. open t ∧ t ≠ ∅ ⇒ t DIFF s ≠ ∅
[INTERIOR_FINITE_BIGINTER] Theorem
⊢ ∀s. FINITE s ⇒ interior (BIGINTER s) = BIGINTER (IMAGE interior s)
[INTERIOR_FRONTIER] Theorem
⊢ ∀s. interior (frontier s) =
interior (closure s) DIFF closure (interior s)
[INTERIOR_FRONTIER_EMPTY] Theorem
⊢ ∀s. open s ∨ closed s ⇒ interior (frontier s) = ∅
[INTERIOR_HALFSPACE_COMPONENT_GE] Theorem
⊢ ∀a. interior {x | x ≥ a} = {x | x > a}
[INTERIOR_HALFSPACE_COMPONENT_LE] Theorem
⊢ ∀a. interior {x | x ≤ a} = {x | x < a}
[INTERIOR_HALFSPACE_GE] Theorem
⊢ ∀a b. a ≠ 0 ⇒ interior {x | a * x ≥ b} = {x | a * x > b}
[INTERIOR_HALFSPACE_LE] Theorem
⊢ ∀a b. a ≠ 0 ⇒ interior {x | a * x ≤ b} = {x | a * x < b}
[INTERIOR_HYPERPLANE] Theorem
⊢ ∀a b. a ≠ 0 ⇒ interior {x | a * x = b} = ∅
[INTERIOR_IMAGE_SUBSET] Theorem
⊢ ∀f s.
(∀x. f continuous at x) ∧ (∀x y. f x = f y ⇒ x = y) ⇒
interior (IMAGE f s) ⊆ IMAGE f (interior s)
[INTERIOR_INJECTIVE_LINEAR_IMAGE] Theorem
⊢ ∀f s.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
interior (IMAGE f s) = IMAGE f (interior s)
[INTERIOR_INTER] Theorem
⊢ ∀s t. interior (s ∩ t) = interior s ∩ interior t
[INTERIOR_INTERIOR] Theorem
⊢ ∀s. interior (interior s) = interior s
[INTERIOR_INTERVAL] Theorem
⊢ (∀a b. interior (interval [(a,b)]) = interval (a,b)) ∧
∀a b. interior (interval (a,b)) = interval (a,b)
[INTERIOR_LIMIT_POINT] Theorem
⊢ ∀s x. x ∈ interior s ⇒ x limit_point_of s
[INTERIOR_MAXIMAL] Theorem
⊢ ∀s t. t ⊆ s ∧ open t ⇒ t ⊆ interior s
[INTERIOR_MAXIMAL_EQ] Theorem
⊢ ∀s t. open s ⇒ (s ⊆ interior t ⇔ s ⊆ t)
[INTERIOR_NEGATIONS] Theorem
⊢ ∀s. interior (IMAGE (λx. -x) s) = IMAGE (λx. -x) (interior s)
[INTERIOR_OPEN] Theorem
⊢ ∀s. open s ⇒ interior s = s
[INTERIOR_SING] Theorem
⊢ ∀a. interior {a} = ∅
[INTERIOR_STANDARD_HYPERPLANE] Theorem
⊢ ∀a. interior {x | x = a} = ∅
[INTERIOR_SUBSET] Theorem
⊢ ∀s. interior s ⊆ s
[INTERIOR_TRANSLATION] Theorem
⊢ ∀a s.
interior (IMAGE (λx. a + x) s) = IMAGE (λx. a + x) (interior s)
[INTERIOR_UNIONS_OPEN_SUBSETS] Theorem
⊢ ∀s. BIGUNION {t | open t ∧ t ⊆ s} = interior s
[INTERIOR_UNION_EQ_EMPTY] Theorem
⊢ ∀s t.
closed s ∨ closed t ⇒
(interior (s ∪ t) = ∅ ⇔ interior s = ∅ ∧ interior t = ∅)
[INTERIOR_UNIQUE] Theorem
⊢ ∀s t.
t ⊆ s ∧ open t ∧ (∀t'. t' ⊆ s ∧ open t' ⇒ t' ⊆ t) ⇒
interior s = t
[INTERIOR_UNIV] Theorem
⊢ interior 𝕌(:real) = 𝕌(:real)
[INTERVAL] Theorem
⊢ (∀a b.
interval [(a,b)] =
if a ≤ b then cball (midpoint (a,b),dist (a,b) / 2) else ∅) ∧
∀a b.
interval (a,b) =
if a < b then ball (midpoint (a,b),dist (a,b) / 2) else ∅
[INTERVAL_BOUNDS_EMPTY] Theorem
⊢ interval_upperbound ∅ = 0 ∧ interval_lowerbound ∅ = 0
[INTERVAL_BOUNDS_NULL] Theorem
⊢ ∀a b.
content (interval [(a,b)]) = 0 ⇒
interval_upperbound (interval [(a,b)]) =
interval_lowerbound (interval [(a,b)])
[INTERVAL_CASES] Theorem
⊢ ∀x. x ∈ interval [(a,b)] ⇒ x ∈ interval (a,b) ∨ x = a ∨ x = b
[INTERVAL_CONTAINS_COMPACT_NEIGHBOURHOOD] Theorem
⊢ ∀s x.
is_interval s ∧ x ∈ s ⇒
∃a b d.
0 < d ∧ x ∈ interval [(a,b)] ∧ interval [(a,b)] ⊆ s ∧
ball (x,d) ∩ s ⊆ interval [(a,b)]
[INTERVAL_EQ_EMPTY] Theorem
⊢ ∀a b. (b < a ⇔ interval [(a,b)] = ∅) ∧ (b ≤ a ⇔ interval (a,b) = ∅)
[INTERVAL_IMAGE_STRETCH_INTERVAL] Theorem
⊢ ∀a b m. ∃u v.
IMAGE (λx. @f. f = m 1 * x) (interval [(a,b)]) = interval [(u,v)]
[INTERVAL_LOWERBOUND] Theorem
⊢ ∀a b. a ≤ b ⇒ interval_lowerbound (interval [(a,b)]) = a
[INTERVAL_LOWERBOUND_NONEMPTY] Theorem
⊢ ∀a b.
interval [(a,b)] ≠ ∅ ⇒ interval_lowerbound (interval [(a,b)]) = a
[INTERVAL_NE_EMPTY] Theorem
⊢ (interval [(a,b)] ≠ ∅ ⇔ a ≤ b) ∧ (interval (a,b) ≠ ∅ ⇔ a < b)
[INTERVAL_OPEN_SUBSET_CLOSED] Theorem
⊢ ∀a b. interval (a,b) ⊆ interval [(a,b)]
[INTERVAL_SING] Theorem
⊢ interval [(a,a)] = {a} ∧ interval (a,a) = ∅
[INTERVAL_SUBSET_IS_INTERVAL] Theorem
⊢ ∀s a b.
is_interval s ⇒
(interval [(a,b)] ⊆ s ⇔ interval [(a,b)] = ∅ ∨ a ∈ s ∧ b ∈ s)
[INTERVAL_TRANSLATION] Theorem
⊢ (∀c a b.
interval [(c + a,c + b)] = IMAGE (λx. c + x) (interval [(a,b)])) ∧
∀c a b. interval (c + a,c + b) = IMAGE (λx. c + x) (interval (a,b))
[INTERVAL_UPPERBOUND] Theorem
⊢ ∀a b. a ≤ b ⇒ interval_upperbound (interval [(a,b)]) = b
[INTERVAL_UPPERBOUND_NONEMPTY] Theorem
⊢ ∀a b.
interval [(a,b)] ≠ ∅ ⇒ interval_upperbound (interval [(a,b)]) = b
[INTER_BALLS_EQ_EMPTY] Theorem
⊢ (∀a b r s.
ball (a,r) ∩ ball (b,s) = ∅ ⇔ r ≤ 0 ∨ s ≤ 0 ∨ r + s ≤ dist (a,b)) ∧
(∀a b r s.
ball (a,r) ∩ cball (b,s) = ∅ ⇔
r ≤ 0 ∨ s < 0 ∨ r + s ≤ dist (a,b)) ∧
(∀a b r s.
cball (a,r) ∩ ball (b,s) = ∅ ⇔
r < 0 ∨ s ≤ 0 ∨ r + s ≤ dist (a,b)) ∧
∀a b r s.
cball (a,r) ∩ cball (b,s) = ∅ ⇔
r < 0 ∨ s < 0 ∨ r + s < dist (a,b)
[INTER_INTERVAL] Theorem
⊢ interval [(a,b)] ∩ interval [(c,d)] = interval [(max a c,min b d)]
[INTER_INTERVAL_MIXED_EQ_EMPTY] Theorem
⊢ ∀a b c d.
interval (c,d) ≠ ∅ ⇒
(interval (a,b) ∩ interval [(c,d)] = ∅ ⇔
interval (a,b) ∩ interval (c,d) = ∅)
[IN_BALL] Theorem
⊢ ∀x y e. y ∈ ball (x,e) ⇔ dist (x,y) < e
[IN_BALL_0] Theorem
⊢ ∀x e. x ∈ ball (0,e) ⇔ abs x < e
[IN_CBALL] Theorem
⊢ ∀x y e. y ∈ cball (x,e) ⇔ dist (x,y) ≤ e
[IN_CBALL_0] Theorem
⊢ ∀x e. x ∈ cball (0,e) ⇔ abs x ≤ e
[IN_CLOSURE_DELETE] Theorem
⊢ ∀s x. x ∈ closure (s DELETE x) ⇔ x limit_point_of s
[IN_COMPONENTS] Theorem
⊢ ∀u s. s ∈ components u ⇔ ∃x. x ∈ u ∧ s = connected_component u x
[IN_COMPONENTS_BIGUNION_COMPLEMENT] Theorem
⊢ ∀s c.
c ∈ components s ⇒ s DIFF c = BIGUNION (components s DELETE c)
[IN_COMPONENTS_CONNECTED] Theorem
⊢ ∀s c. c ∈ components s ⇒ connected c
[IN_COMPONENTS_MAXIMAL] Theorem
⊢ ∀s c.
c ∈ components s ⇔
c ≠ ∅ ∧ c ⊆ s ∧ connected c ∧
∀c'. c' ≠ ∅ ∧ c ⊆ c' ∧ c' ⊆ s ∧ connected c' ⇒ c' = c
[IN_COMPONENTS_NONEMPTY] Theorem
⊢ ∀s c. c ∈ components s ⇒ c ≠ ∅
[IN_COMPONENTS_SELF] Theorem
⊢ ∀s. s ∈ components s ⇔ connected s ∧ s ≠ ∅
[IN_COMPONENTS_SUBSET] Theorem
⊢ ∀s c. c ∈ components s ⇒ c ⊆ s
[IN_DIRECTION] Theorem
⊢ ∀a v x y.
netord (a in_direction v) x y ⇔
0 < dist (x,a) ∧ dist (x,a) ≤ dist (y,a) ∧
∃c. 0 ≤ c ∧ x − a = c * v
[IN_INTERIOR] Theorem
⊢ ∀x s. x ∈ interior s ⇔ ∃e. 0 < e ∧ ball (x,e) ⊆ s
[IN_INTERIOR_CBALL] Theorem
⊢ ∀x s. x ∈ interior s ⇔ ∃e. 0 < e ∧ cball (x,e) ⊆ s
[IN_INTERIOR_LINEAR_IMAGE] Theorem
⊢ ∀f g s x.
linear f ∧ linear g ∧ f ∘ g = I ∧ x ∈ interior s ⇒
f x ∈ interior (IMAGE f s)
[IN_INTERVAL] Theorem
⊢ (x ∈ interval (a,b) ⇔ a < x ∧ x < b) ∧
(x ∈ interval [(a,b)] ⇔ a ≤ x ∧ x ≤ b)
[IN_INTERVAL_REFLECT] Theorem
⊢ (∀a b x. -x ∈ interval [(-b,-a)] ⇔ x ∈ interval [(a,b)]) ∧
∀a b x. -x ∈ interval (-b,-a) ⇔ x ∈ interval (a,b)
[IN_OPEN_SEGMENT] Theorem
⊢ ∀a b x. x ∈ segment (a,b) ⇔ x ∈ segment [(a,b)] ∧ x ≠ a ∧ x ≠ b
[IN_OPEN_SEGMENT_ALT] Theorem
⊢ ∀a b x.
x ∈ segment (a,b) ⇔ x ∈ segment [(a,b)] ∧ x ≠ a ∧ x ≠ b ∧ a ≠ b
[IN_SEGMENT] Theorem
⊢ ∀a b x.
(x ∈ segment [(a,b)] ⇔
∃u. 0 ≤ u ∧ u ≤ 1 ∧ x = (1 − u) * a + u * b) ∧
(x ∈ segment (a,b) ⇔
a ≠ b ∧ ∃u. 0 < u ∧ u < 1 ∧ x = (1 − u) * a + u * b)
[IN_SEGMENT_COMPONENT] Theorem
⊢ ∀a b x i. x ∈ segment [(a,b)] ⇒ min a b ≤ x ∧ x ≤ max a b
[IN_SPAN_DELETE] Theorem
⊢ ∀a b s.
a ∈ span s ∧ a ∉ span (s DELETE b) ⇒
b ∈ span (a INSERT s DELETE b)
[IN_SPAN_INSERT] Theorem
⊢ ∀a b s. a ∈ span (b INSERT s) ∧ a ∉ span s ⇒ b ∈ span (a INSERT s)
[IN_SPHERE] Theorem
⊢ ∀x y e. y ∈ sphere (x,e) ⇔ dist (x,y) = e
[IN_SPHERE_0] Theorem
⊢ ∀x e. x ∈ sphere (0,e) ⇔ abs x = e
[ISOMETRY_IMP_EMBEDDING] Theorem
⊢ ∀f s t.
IMAGE f s = t ∧
(∀x y. x ∈ s ∧ y ∈ s ⇒ dist (f x,f y) = dist (x,y)) ⇒
∃g. homeomorphism (s,t) (f,g)
[ISOMETRY_IMP_HOMEOMORPHISM_COMPACT] Theorem
⊢ ∀f s.
compact s ∧ IMAGE f s ⊆ s ∧
(∀x y. x ∈ s ∧ y ∈ s ⇒ dist (f x,f y) = dist (x,y)) ⇒
∃g. homeomorphism (s,s) (f,g)
[ISOMETRY_IMP_OPEN_MAP] Theorem
⊢ ∀f s t u.
IMAGE f s = t ∧
(∀x y. x ∈ s ∧ y ∈ s ⇒ dist (f x,f y) = dist (x,y)) ∧
open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean t) (IMAGE f u)
[ISOMETRY_ON_IMP_CONTINUOUS_ON] Theorem
⊢ ∀f. (∀x y. x ∈ s ∧ y ∈ s ⇒ dist (f x,f y) = dist (x,y)) ⇒
f continuous_on s
[IS_INTERVAL] Theorem
⊢ ∀s. is_interval s ⇔ ∀a b x. a ∈ s ∧ b ∈ s ∧ a ≤ x ∧ x ≤ b ⇒ x ∈ s
[IS_INTERVAL_CASES] Theorem
⊢ ∀s. is_interval s ⇔
s = ∅ ∨ s = 𝕌(:real) ∨ (∃a. s = {x | a < x}) ∨
(∃a. s = {x | a ≤ x}) ∨ (∃b. s = {x | x ≤ b}) ∨
(∃b. s = {x | x < b}) ∨ (∃a b. s = {x | a < x ∧ x < b}) ∨
(∃a b. s = {x | a < x ∧ x ≤ b}) ∨
(∃a b. s = {x | a ≤ x ∧ x < b}) ∨ ∃a b. s = {x | a ≤ x ∧ x ≤ b}
[IS_INTERVAL_COMPACT] Theorem
⊢ ∀s. is_interval s ∧ compact s ⇔ ∃a b. s = interval [(a,b)]
[IS_INTERVAL_EMPTY] Theorem
⊢ is_interval ∅
[IS_INTERVAL_IMP_LOCALLY_COMPACT] Theorem
⊢ ∀s. is_interval s ⇒ locally compact s
[IS_INTERVAL_INTER] Theorem
⊢ ∀s t. is_interval s ∧ is_interval t ⇒ is_interval (s ∩ t)
[IS_INTERVAL_INTERVAL] Theorem
⊢ ∀a b. is_interval (interval (a,b)) ∧ is_interval (interval [(a,b)])
[IS_INTERVAL_POINTWISE] Theorem
⊢ ∀s x. is_interval s ⇒ (∃a. a ∈ s ∧ a = x) ⇒ x ∈ s
[IS_INTERVAL_SCALING] Theorem
⊢ ∀s c. is_interval s ⇒ is_interval (IMAGE (λx. c * x) s)
[IS_INTERVAL_SCALING_EQ] Theorem
⊢ ∀s c. is_interval (IMAGE (λx. c * x) s) ⇔ c = 0 ∨ is_interval s
[IS_INTERVAL_SING] Theorem
⊢ ∀a. is_interval {a}
[IS_INTERVAL_SUMS] Theorem
⊢ ∀s t.
is_interval s ∧ is_interval t ⇒
is_interval {x + y | x ∈ s ∧ y ∈ t}
[IS_INTERVAL_UNIV] Theorem
⊢ is_interval 𝕌(:real)
[JOINABLE_COMPONENTS_EQ] Theorem
⊢ ∀s t c1 c2.
connected t ∧ t ⊆ s ∧ c1 ∈ components s ∧ c2 ∈ components s ∧
c1 ∩ t ≠ ∅ ∧ c2 ∩ t ≠ ∅ ⇒
c1 = c2
[JOINABLE_CONNECTED_COMPONENT_EQ] Theorem
⊢ ∀s t x y.
connected t ∧ t ⊆ s ∧ connected_component s x ∩ t ≠ ∅ ∧
connected_component s y ∩ t ≠ ∅ ⇒
connected_component s x = connected_component s y
[LEBESGUE_COVERING_LEMMA] Theorem
⊢ ∀s c.
compact s ∧ c ≠ ∅ ∧ s ⊆ BIGUNION c ∧ (∀b. b ∈ c ⇒ open b) ⇒
∃d. 0 < d ∧ ∀t. t ⊆ s ∧ diameter t ≤ d ⇒ ∃b. b ∈ c ∧ t ⊆ b
[LE_1] Theorem
⊢ (∀n. n ≠ 0 ⇒ 0 < n) ∧ (∀n. n ≠ 0 ⇒ 1 ≤ n) ∧ (∀n. 0 < n ⇒ n ≠ 0) ∧
(∀n. 0 < n ⇒ 1 ≤ n) ∧ (∀n. 1 ≤ n ⇒ 0 < n) ∧ ∀n. 1 ≤ n ⇒ n ≠ 0
[LIFT_TO_QUOTIENT_SPACE] Theorem
⊢ ∀f h s t u.
IMAGE f s = t ∧
(∀v. v ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ v} ⇔
open_in (subtopology euclidean t) v)) ∧ h continuous_on s ∧
IMAGE h s = u ∧ (∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ h x = h y) ⇒
∃g. g continuous_on t ∧ IMAGE g t = u ∧ ∀x. x ∈ s ⇒ h x = g (f x)
[LIFT_TO_QUOTIENT_SPACE_UNIQUE] Theorem
⊢ ∀f g s t u.
IMAGE f s = t ∧ IMAGE g s = u ∧
(∀v. v ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ v} ⇔
open_in (subtopology euclidean t) v)) ∧
(∀v. v ⊆ u ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ g x ∈ v} ⇔
open_in (subtopology euclidean u) v)) ∧
(∀x y. x ∈ s ∧ y ∈ s ⇒ (f x = f y ⇔ g x = g y)) ⇒
t homeomorphic u
[LIMIT_POINT_FINITE] Theorem
⊢ ∀s a. FINITE s ⇒ ¬(a limit_point_of s)
[LIMIT_POINT_UNION] Theorem
⊢ ∀s t x.
x limit_point_of s ∪ t ⇔ x limit_point_of s ∨ x limit_point_of t
[LIMPT_APPROACHABLE] Theorem
⊢ ∀x s.
x limit_point_of s ⇔
∀e. 0 < e ⇒ ∃x'. x' ∈ s ∧ x' ≠ x ∧ dist (x',x) < e
[LIMPT_APPROACHABLE_LE] Theorem
⊢ ∀x s.
x limit_point_of s ⇔
∀e. 0 < e ⇒ ∃x'. x' ∈ s ∧ x' ≠ x ∧ dist (x',x) ≤ e
[LIMPT_BALL] Theorem
⊢ ∀x y e. y limit_point_of ball (x,e) ⇔ 0 < e ∧ y ∈ cball (x,e)
[LIMPT_EMPTY] Theorem
⊢ ∀x. ¬(x limit_point_of ∅)
[LIMPT_INFINITE_BALL] Theorem
⊢ ∀s x. x limit_point_of s ⇔ ∀e. 0 < e ⇒ INFINITE (s ∩ ball (x,e))
[LIMPT_INFINITE_CBALL] Theorem
⊢ ∀s x. x limit_point_of s ⇔ ∀e. 0 < e ⇒ INFINITE (s ∩ cball (x,e))
[LIMPT_INFINITE_OPEN] Theorem
⊢ ∀s x. x limit_point_of s ⇔ ∀t. x ∈ t ∧ open t ⇒ INFINITE (s ∩ t)
[LIMPT_INFINITE_OPEN_BALL_CBALL] Theorem
⊢ (∀s x. x limit_point_of s ⇔ ∀t. x ∈ t ∧ open t ⇒ INFINITE (s ∩ t)) ∧
(∀s x. x limit_point_of s ⇔ ∀e. 0 < e ⇒ INFINITE (s ∩ ball (x,e))) ∧
∀s x. x limit_point_of s ⇔ ∀e. 0 < e ⇒ INFINITE (s ∩ cball (x,e))
[LIMPT_INJECTIVE_LINEAR_IMAGE_EQ] Theorem
⊢ ∀f s.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
(f x limit_point_of IMAGE f s ⇔ x limit_point_of s)
[LIMPT_INSERT] Theorem
⊢ ∀s x y. x limit_point_of y INSERT s ⇔ x limit_point_of s
[LIMPT_OF_CLOSURE] Theorem
⊢ ∀x s. x limit_point_of closure s ⇔ x limit_point_of s
[LIMPT_OF_LIMPTS] Theorem
⊢ ∀x s.
x limit_point_of {y | y limit_point_of s} ⇒ x limit_point_of s
[LIMPT_OF_OPEN] Theorem
⊢ ∀s x. open s ∧ x ∈ s ⇒ x limit_point_of s
[LIMPT_OF_OPEN_IN] Theorem
⊢ ∀s t x.
open_in (subtopology euclidean s) t ∧ x limit_point_of s ∧ x ∈ t ⇒
x limit_point_of t
[LIMPT_OF_SEQUENCE_SUBSEQUENCE] Theorem
⊢ ∀f l.
l limit_point_of IMAGE f 𝕌(:num) ⇒
∃r. (∀m n. m < n ⇒ r m < r n) ∧ (f ∘ r ⟶ l) sequentially
[LIMPT_OF_UNIV] Theorem
⊢ ∀x. x limit_point_of 𝕌(:real)
[LIMPT_SEQUENTIAL] Theorem
⊢ ∀x s.
x limit_point_of s ⇔
∃f. (∀n. f n ∈ s DELETE x) ∧ (f ⟶ x) sequentially
[LIMPT_SEQUENTIAL_INJ] Theorem
⊢ ∀x s.
x limit_point_of s ⇔
∃f. (∀n. f n ∈ s DELETE x) ∧ (∀m n. f m = f n ⇔ m = n) ∧
(f ⟶ x) sequentially
[LIMPT_SING] Theorem
⊢ ∀x y. ¬(x limit_point_of {y})
[LIMPT_SUBSET] Theorem
⊢ ∀x s t. x limit_point_of s ∧ s ⊆ t ⇒ x limit_point_of t
[LIMPT_UNIV] Theorem
⊢ ∀x. x limit_point_of 𝕌(:real)
[LIM_ABS] Theorem
⊢ ∀net f l. (f ⟶ l) net ⇒ ((λx. abs (f x)) ⟶ abs l) net
[LIM_ABS_LBOUND] Theorem
⊢ ∀net f l b.
¬trivial_limit net ∧ (f ⟶ l) net ∧
eventually (λx. b ≤ abs (f x)) net ⇒
b ≤ abs l
[LIM_ABS_UBOUND] Theorem
⊢ ∀net f l b.
¬trivial_limit net ∧ (f ⟶ l) net ∧
eventually (λx. abs (f x) ≤ b) net ⇒
abs l ≤ b
[LIM_ADD] Theorem
⊢ ∀net f g l m.
(f ⟶ l) net ∧ (g ⟶ m) net ⇒ ((λx. f x + g x) ⟶ (l + m)) net
[LIM_AT] Theorem
⊢ ∀f l a.
(f ⟶ l) (at a) ⇔
∀e. 0 < e ⇒
∃d. 0 < d ∧
∀x. 0 < dist (x,a) ∧ dist (x,a) < d ⇒ dist (f x,l) < e
[LIM_AT_ID] Theorem
⊢ ∀a. ((λx. x) ⟶ a) (at a)
[LIM_AT_INFINITY] Theorem
⊢ ∀f l.
(f ⟶ l) at_infinity ⇔
∀e. 0 < e ⇒ ∃b. ∀x. abs x ≥ b ⇒ dist (f x,l) < e
[LIM_AT_INFINITY_POS] Theorem
⊢ ∀f l.
(f ⟶ l) at_infinity ⇔
∀e. 0 < e ⇒ ∃b. 0 < b ∧ ∀x. abs x ≥ b ⇒ dist (f x,l) < e
[LIM_AT_LE] Theorem
⊢ ∀f l a.
(f ⟶ l) (at a) ⇔
∀e. 0 < e ⇒
∃d. 0 < d ∧
∀x. 0 < dist (x,a) ∧ dist (x,a) ≤ d ⇒ dist (f x,l) < e
[LIM_AT_NEGINFINITY] Theorem
⊢ ∀f l.
(f ⟶ l) at_neginfinity ⇔
∀e. 0 < e ⇒ ∃b. ∀x. x ≤ b ⇒ dist (f x,l) < e
[LIM_AT_POSINFINITY] Theorem
⊢ ∀f l.
(f ⟶ l) at_posinfinity ⇔
∀e. 0 < e ⇒ ∃b. ∀x. x ≥ b ⇒ dist (f x,l) < e
[LIM_AT_WITHIN] Theorem
⊢ ∀f l a s. (f ⟶ l) (at a) ⇒ (f ⟶ l) (at a within s)
[LIM_AT_ZERO] Theorem
⊢ ∀f l a. (f ⟶ l) (at a) ⇔ ((λx. f (a + x)) ⟶ l) (at 0)
[LIM_BILINEAR] Theorem
⊢ ∀net h f g l m.
(f ⟶ l) net ∧ (g ⟶ m) net ∧ bilinear h ⇒
((λx. h (f x) (g x)) ⟶ h l m) net
[LIM_CASES_COFINITE_SEQUENTIALLY] Theorem
⊢ ∀f g l.
FINITE {n | (¬P n)} ⇒
(((λn. if P n then f n else g n) ⟶ l) sequentially ⇔
(f ⟶ l) sequentially)
[LIM_CASES_FINITE_SEQUENTIALLY] Theorem
⊢ ∀f g l.
FINITE {n | P n} ⇒
(((λn. if P n then f n else g n) ⟶ l) sequentially ⇔
(g ⟶ l) sequentially)
[LIM_CASES_SEQUENTIALLY] Theorem
⊢ ∀f g l m.
(((λn. if m ≤ n then f n else g n) ⟶ l) sequentially ⇔
(f ⟶ l) sequentially) ∧
(((λn. if m < n then f n else g n) ⟶ l) sequentially ⇔
(f ⟶ l) sequentially) ∧
(((λn. if n ≤ m then f n else g n) ⟶ l) sequentially ⇔
(g ⟶ l) sequentially) ∧
(((λn. if n < m then f n else g n) ⟶ l) sequentially ⇔
(g ⟶ l) sequentially)
[LIM_CMUL] Theorem
⊢ ∀f l c. (f ⟶ l) net ⇒ ((λx. c * f x) ⟶ (c * l)) net
[LIM_CMUL_EQ] Theorem
⊢ ∀net f l c. c ≠ 0 ⇒ (((λx. c * f x) ⟶ (c * l)) net ⇔ (f ⟶ l) net)
[LIM_COMPONENT] Theorem
⊢ ∀net f i l. (f ⟶ l) net ⇒ ((λa. f a) ⟶ l) net
[LIM_COMPONENT_EQ] Theorem
⊢ ∀net f i l b.
(f ⟶ l) net ∧ ¬trivial_limit net ∧ eventually (λx. f x = b) net ⇒
l = b
[LIM_COMPONENT_LBOUND] Theorem
⊢ ∀net f l b.
¬trivial_limit net ∧ (f ⟶ l) net ∧ eventually (λx. b ≤ f x) net ⇒
b ≤ l
[LIM_COMPONENT_LE] Theorem
⊢ ∀net f g l m.
¬trivial_limit net ∧ (f ⟶ l) net ∧ (g ⟶ m) net ∧
eventually (λx. f x ≤ g x) net ⇒
l ≤ m
[LIM_COMPONENT_UBOUND] Theorem
⊢ ∀net f l b k.
¬trivial_limit net ∧ (f ⟶ l) net ∧ eventually (λx. f x ≤ b) net ⇒
l ≤ b
[LIM_COMPOSE_AT] Theorem
⊢ ∀net f g y z.
(f ⟶ y) net ∧ eventually (λw. f w = y ⇒ g y = z) net ∧
(g ⟶ z) (at y) ⇒
(g ∘ f ⟶ z) net
[LIM_COMPOSE_WITHIN] Theorem
⊢ ∀net f g s y z.
(f ⟶ y) net ∧
eventually (λw. f w ∈ s ∧ (f w = y ⇒ g y = z)) net ∧
(g ⟶ z) (at y within s) ⇒
(g ∘ f ⟶ z) net
[LIM_CONG_AT] Theorem
⊢ (∀x. x ≠ a ⇒ f x = g x) ⇒ (((λx. f x) ⟶ l) (at a) ⇔ (g ⟶ l) (at a))
[LIM_CONG_WITHIN] Theorem
⊢ (∀x. x ≠ a ⇒ f x = g x) ⇒
(((λx. f x) ⟶ l) (at a within s) ⇔ (g ⟶ l) (at a within s))
[LIM_CONST] Theorem
⊢ ∀net a. ((λx. a) ⟶ a) net
[LIM_CONST_EQ] Theorem
⊢ ∀net c d. ((λx. c) ⟶ d) net ⇔ trivial_limit net ∨ c = d
[LIM_CONTINUOUS_FUNCTION] Theorem
⊢ ∀f net g l.
f continuous at l ∧ (g ⟶ l) net ⇒ ((λx. f (g x)) ⟶ f l) net
[LIM_DEF] Theorem
⊢ ∀f l net.
(f ⟶ l) net ⇔
trivial_limit net ∨
∀e. 0 < e ⇒
∃y. (∃x. netord net x y) ∧
∀x. netord net x y ⇒ dist (f x,l) < e
[LIM_DROP_LBOUND] Theorem
⊢ ∀net f l b.
(f ⟶ l) net ∧ ¬trivial_limit net ∧ eventually (λx. b ≤ f x) net ⇒
b ≤ l
[LIM_DROP_LE] Theorem
⊢ ∀net f g l m.
¬trivial_limit net ∧ (f ⟶ l) net ∧ (g ⟶ m) net ∧
eventually (λx. f x ≤ g x) net ⇒
l ≤ m
[LIM_DROP_UBOUND] Theorem
⊢ ∀net f l b.
(f ⟶ l) net ∧ ¬trivial_limit net ∧ eventually (λx. f x ≤ b) net ⇒
l ≤ b
[LIM_EVENTUALLY] Theorem
⊢ ∀net f l. eventually (λx. f x = l) net ⇒ (f ⟶ l) net
[LIM_INFINITY_POSINFINITY] Theorem
⊢ ∀f l. (f ⟶ l) at_infinity ⇒ (f ⟶ l) at_posinfinity
[LIM_INV] Theorem
⊢ ∀net f l. (f ⟶ l) net ∧ l ≠ 0 ⇒ (realinv ∘ f ⟶ l⁻¹) net
[LIM_IN_CLOSED_SET] Theorem
⊢ ∀net f s l.
closed s ∧ eventually (λx. f x ∈ s) net ∧ ¬trivial_limit net ∧
(f ⟶ l) net ⇒
l ∈ s
[LIM_LIFT_DOT] Theorem
⊢ ∀f a. (f ⟶ l) net ⇒ ((λy. a * f y) ⟶ (a * l)) net
[LIM_LINEAR] Theorem
⊢ ∀net h f l. (f ⟶ l) net ∧ linear h ⇒ ((λx. h (f x)) ⟶ h l) net
[LIM_MAX] Theorem
⊢ ∀net f g l m.
(f ⟶ l) net ∧ (g ⟶ m) net ⇒ ((λx. max (f x) (g x)) ⟶ max l m) net
[LIM_MIN] Theorem
⊢ ∀net f g l m.
(f ⟶ l) net ∧ (g ⟶ m) net ⇒ ((λx. min (f x) (g x)) ⟶ min l m) net
[LIM_MUL] Theorem
⊢ ∀net f l c d.
(c ⟶ d) net ∧ (f ⟶ l) net ⇒ ((λx. c x * f x) ⟶ (d * l)) net
[LIM_NEG] Theorem
⊢ ∀net f l. (f ⟶ l) net ⇒ ((λx. -f x) ⟶ -l) net
[LIM_NEG_EQ] Theorem
⊢ ∀net f l. ((λx. -f x) ⟶ -l) net ⇔ (f ⟶ l) net
[LIM_NULL] Theorem
⊢ ∀net f l. (f ⟶ l) net ⇔ ((λx. f x − l) ⟶ 0) net
[LIM_NULL_ABS] Theorem
⊢ ∀net f. (f ⟶ 0) net ⇔ ((λx. abs (f x)) ⟶ 0) net
[LIM_NULL_ADD] Theorem
⊢ ∀net f g. (f ⟶ 0) net ∧ (g ⟶ 0) net ⇒ ((λx. f x + g x) ⟶ 0) net
[LIM_NULL_CMUL] Theorem
⊢ ∀net f c. (f ⟶ 0) net ⇒ ((λx. c * f x) ⟶ 0) net
[LIM_NULL_CMUL_BOUNDED] Theorem
⊢ ∀f g B.
eventually (λa. g a = 0 ∨ abs (f a) ≤ B) net ∧ (g ⟶ 0) net ⇒
((λn. f n * g n) ⟶ 0) net
[LIM_NULL_CMUL_EQ] Theorem
⊢ ∀net f c. c ≠ 0 ⇒ (((λx. c * f x) ⟶ 0) net ⇔ (f ⟶ 0) net)
[LIM_NULL_COMPARISON] Theorem
⊢ ∀net f g.
eventually (λx. abs (f x) ≤ g x) net ∧ ((λx. g x) ⟶ 0) net ⇒
(f ⟶ 0) net
[LIM_NULL_SUB] Theorem
⊢ ∀net f g. (f ⟶ 0) net ∧ (g ⟶ 0) net ⇒ ((λx. f x − g x) ⟶ 0) net
[LIM_NULL_SUM] Theorem
⊢ ∀net f s.
FINITE s ∧ (∀a. a ∈ s ⇒ ((λx. f x a) ⟶ 0) net) ⇒
((λx. sum s (f x)) ⟶ 0) net
[LIM_POSINFINITY_SEQUENTIALLY] Theorem
⊢ ∀f l. (f ⟶ l) at_posinfinity ⇒ ((λn. f (&n)) ⟶ l) sequentially
[LIM_SEQUENTIALLY] Theorem
⊢ ∀s l.
(s ⟶ l) sequentially ⇔
∀e. 0 < e ⇒ ∃N. ∀n. N ≤ n ⇒ dist (s n,l) < e
[LIM_SUB] Theorem
⊢ ∀net f g l m.
(f ⟶ l) net ∧ (g ⟶ m) net ⇒ ((λx. f x − g x) ⟶ (l − m)) net
[LIM_SUBSEQUENCE] Theorem
⊢ ∀s r l.
(∀m n. m < n ⇒ r m < r n) ∧ (s ⟶ l) sequentially ⇒
(s ∘ r ⟶ l) sequentially
[LIM_SUBSEQUENCE_WEAK] Theorem
⊢ ∀s r l.
(∀m n. m ≤ n ⇒ r m ≤ r n) ∧ (∀n. ∃m. n ≤ r m) ∧
(s ⟶ l) sequentially ⇒
(s ∘ r ⟶ l) sequentially
[LIM_SUM] Theorem
⊢ ∀net f l s.
FINITE s ∧ (∀i. i ∈ s ⇒ (f i ⟶ l i) net) ⇒
((λx. sum s (λi. f i x)) ⟶ sum s l) net
[LIM_TRANSFORM] Theorem
⊢ ∀net f g l. ((λx. f x − g x) ⟶ 0) net ∧ (f ⟶ l) net ⇒ (g ⟶ l) net
[LIM_TRANSFORM_AT] Theorem
⊢ ∀f g x d.
0 < d ∧ (∀x'. 0 < dist (x',x) ∧ dist (x',x) < d ⇒ f x' = g x') ∧
(f ⟶ l) (at x) ⇒
(g ⟶ l) (at x)
[LIM_TRANSFORM_AWAY_AT] Theorem
⊢ ∀f g a b.
a ≠ b ∧ (∀x. x ≠ a ∧ x ≠ b ⇒ f x = g x) ∧ (f ⟶ l) (at a) ⇒
(g ⟶ l) (at a)
[LIM_TRANSFORM_AWAY_WITHIN] Theorem
⊢ ∀f g a b s.
a ≠ b ∧ (∀x. x ∈ s ∧ x ≠ a ∧ x ≠ b ⇒ f x = g x) ∧
(f ⟶ l) (at a within s) ⇒
(g ⟶ l) (at a within s)
[LIM_TRANSFORM_BOUND] Theorem
⊢ ∀f g.
eventually (λn. abs (f n) ≤ abs (g n)) net ∧ (g ⟶ 0) net ⇒
(f ⟶ 0) net
[LIM_TRANSFORM_EQ] Theorem
⊢ ∀net f g l. ((λx. f x − g x) ⟶ 0) net ⇒ ((f ⟶ l) net ⇔ (g ⟶ l) net)
[LIM_TRANSFORM_EVENTUALLY] Theorem
⊢ ∀net f g l.
eventually (λx. f x = g x) net ∧ (f ⟶ l) net ⇒ (g ⟶ l) net
[LIM_TRANSFORM_WITHIN] Theorem
⊢ ∀f g x s d.
0 < d ∧
(∀x'. x' ∈ s ∧ 0 < dist (x',x) ∧ dist (x',x) < d ⇒ f x' = g x') ∧
(f ⟶ l) (at x within s) ⇒
(g ⟶ l) (at x within s)
[LIM_TRANSFORM_WITHIN_OPEN] Theorem
⊢ ∀f g s a l.
open s ∧ a ∈ s ∧ (∀x. x ∈ s ∧ x ≠ a ⇒ f x = g x) ∧ (f ⟶ l) (at a) ⇒
(g ⟶ l) (at a)
[LIM_TRANSFORM_WITHIN_OPEN_IN] Theorem
⊢ ∀f g s t a l.
open_in (subtopology euclidean t) s ∧ a ∈ s ∧
(∀x. x ∈ s ∧ x ≠ a ⇒ f x = g x) ∧ (f ⟶ l) (at a within t) ⇒
(g ⟶ l) (at a within t)
[LIM_TRANSFORM_WITHIN_SET] Theorem
⊢ ∀f a s t.
eventually (λx. x ∈ s ⇔ x ∈ t) (at a) ⇒
((f ⟶ l) (at a within s) ⇔ (f ⟶ l) (at a within t))
[LIM_TRANSFORM_WITHIN_SET_IMP] Theorem
⊢ ∀f l a s t.
eventually (λx. x ∈ t ⇒ x ∈ s) (at a) ∧ (f ⟶ l) (at a within s) ⇒
(f ⟶ l) (at a within t)
[LIM_UNION] Theorem
⊢ ∀f x l s t.
(f ⟶ l) (at x within s) ∧ (f ⟶ l) (at x within t) ⇒
(f ⟶ l) (at x within s ∪ t)
[LIM_UNION_UNIV] Theorem
⊢ ∀f x l s t.
(f ⟶ l) (at x within s) ∧ (f ⟶ l) (at x within t) ∧
s ∪ t = 𝕌(:real) ⇒
(f ⟶ l) (at x)
[LIM_UNIQUE] Theorem
⊢ ∀net f l l'.
¬trivial_limit net ∧ (f ⟶ l) net ∧ (f ⟶ l') net ⇒ l = l'
[LIM_VMUL] Theorem
⊢ ∀net c d v. (c ⟶ d) net ⇒ ((λx. c x * v) ⟶ (d * v)) net
[LIM_WITHIN] Theorem
⊢ ∀f l a s.
(f ⟶ l) (at a within s) ⇔
∀e. 0 < e ⇒
∃d. 0 < d ∧
∀x. x ∈ s ∧ 0 < dist (x,a) ∧ dist (x,a) < d ⇒
dist (f x,l) < e
[LIM_WITHIN_CLOSED_TRIVIAL] Theorem
⊢ ∀a s. closed s ∧ a ∉ s ⇒ trivial_limit (at a within s)
[LIM_WITHIN_EMPTY] Theorem
⊢ ∀f l x. (f ⟶ l) (at x within ∅)
[LIM_WITHIN_ID] Theorem
⊢ ∀a s. ((λx. x) ⟶ a) (at a within s)
[LIM_WITHIN_INTERIOR] Theorem
⊢ ∀f l s x.
x ∈ interior s ⇒ ((f ⟶ l) (at x within s) ⇔ (f ⟶ l) (at x))
[LIM_WITHIN_LE] Theorem
⊢ ∀f l a s.
(f ⟶ l) (at a within s) ⇔
∀e. 0 < e ⇒
∃d. 0 < d ∧
∀x. x ∈ s ∧ 0 < dist (x,a) ∧ dist (x,a) ≤ d ⇒
dist (f x,l) < e
[LIM_WITHIN_OPEN] Theorem
⊢ ∀f l a s.
a ∈ s ∧ open s ⇒ ((f ⟶ l) (at a within s) ⇔ (f ⟶ l) (at a))
[LIM_WITHIN_SUBSET] Theorem
⊢ ∀f l a s. (f ⟶ l) (at a within s) ∧ t ⊆ s ⇒ (f ⟶ l) (at a within t)
[LIM_WITHIN_UNION] Theorem
⊢ (f ⟶ l) (at x within s ∪ t) ⇔
(f ⟶ l) (at x within s) ∧ (f ⟶ l) (at x within t)
[LINEAR_0] Theorem
⊢ ∀f. linear f ⇒ f 0 = 0
[LINEAR_ADD] Theorem
⊢ ∀f x y. linear f ⇒ f (x + y) = f x + f y
[LINEAR_BOUNDED] Theorem
⊢ ∀f. linear f ⇒ ∃B. ∀x. abs (f x) ≤ B * abs x
[LINEAR_BOUNDED_POS] Theorem
⊢ ∀f. linear f ⇒ ∃B. 0 < B ∧ ∀x. abs (f x) ≤ B * abs x
[LINEAR_CMUL] Theorem
⊢ ∀f c x. linear f ⇒ f (c * x) = c * f x
[LINEAR_COMPOSE] Theorem
⊢ ∀f g. linear f ∧ linear g ⇒ linear (g ∘ f)
[LINEAR_COMPOSE_ADD] Theorem
⊢ ∀f g. linear f ∧ linear g ⇒ linear (λx. f x + g x)
[LINEAR_COMPOSE_CMUL] Theorem
⊢ ∀f c. linear f ⇒ linear (λx. c * f x)
[LINEAR_COMPOSE_NEG] Theorem
⊢ ∀f. linear f ⇒ linear (λx. -f x)
[LINEAR_COMPOSE_SUB] Theorem
⊢ ∀f g. linear f ∧ linear g ⇒ linear (λx. f x − g x)
[LINEAR_COMPOSE_SUM] Theorem
⊢ ∀f s.
FINITE s ∧ (∀a. a ∈ s ⇒ linear (f a)) ⇒
linear (λx. sum s (λa. f a x))
[LINEAR_CONTINUOUS_AT] Theorem
⊢ ∀f a. linear f ⇒ f continuous at a
[LINEAR_CONTINUOUS_COMPOSE] Theorem
⊢ ∀net f g.
f continuous net ∧ linear g ⇒ (λx. g (f x)) continuous net
[LINEAR_CONTINUOUS_ON] Theorem
⊢ ∀f s. linear f ⇒ f continuous_on s
[LINEAR_CONTINUOUS_ON_COMPOSE] Theorem
⊢ ∀f g s.
f continuous_on s ∧ linear g ⇒ (λx. g (f x)) continuous_on s
[LINEAR_CONTINUOUS_WITHIN] Theorem
⊢ ∀f s x. linear f ⇒ f continuous (at x within s)
[LINEAR_EQ] Theorem
⊢ ∀f g b s.
linear f ∧ linear g ∧ s ⊆ span b ∧ (∀x. x ∈ b ⇒ f x = g x) ⇒
∀x. x ∈ s ⇒ f x = g x
[LINEAR_EQ_0] Theorem
⊢ ∀f b s.
linear f ∧ s ⊆ span b ∧ (∀x. x ∈ b ⇒ f x = 0) ⇒
∀x. x ∈ s ⇒ f x = 0
[LINEAR_EQ_0_SPAN] Theorem
⊢ ∀f b. linear f ∧ (∀x. x ∈ b ⇒ f x = 0) ⇒ ∀x. x ∈ span b ⇒ f x = 0
[LINEAR_EQ_STDBASIS] Theorem
⊢ ∀f g. linear f ∧ linear g ∧ (∀i. 1 ≤ i ∧ i ≤ 1 ⇒ f i = g i) ⇒ f = g
[LINEAR_ID] Theorem
⊢ linear (λx. x)
[LINEAR_IMAGE_SUBSET_INTERIOR] Theorem
⊢ ∀f s.
linear f ∧ (∀y. ∃x. f x = y) ⇒
IMAGE f (interior s) ⊆ interior (IMAGE f s)
[LINEAR_INDEPENDENT_EXTEND] Theorem
⊢ ∀f b. independent b ⇒ ∃g. linear g ∧ ∀x. x ∈ b ⇒ g x = f x
[LINEAR_INDEPENDENT_EXTEND_LEMMA] Theorem
⊢ ∀f b.
FINITE b ⇒
independent b ⇒
∃g. (∀x y. x ∈ span b ∧ y ∈ span b ⇒ g (x + y) = g x + g y) ∧
(∀x c. x ∈ span b ⇒ g (c * x) = c * g x) ∧
∀x. x ∈ b ⇒ g x = f x
[LINEAR_INJECTIVE_0_SUBSPACE] Theorem
⊢ ∀f s.
linear f ∧ subspace s ⇒
((∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇔
∀x. x ∈ s ∧ f x = 0 ⇒ x = 0)
[LINEAR_INJECTIVE_BOUNDED_BELOW_POS] Theorem
⊢ ∀f. linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
∃B. 0 < B ∧ ∀x. abs x * B ≤ abs (f x)
[LINEAR_INJECTIVE_IMP_SURJECTIVE] Theorem
⊢ ∀f. linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒ ∀y. ∃x. f x = y
[LINEAR_INJECTIVE_LEFT_INVERSE] Theorem
⊢ ∀f. linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
∃g. linear g ∧ g ∘ f = (λx. x)
[LINEAR_INTERIOR_IMAGE_SUBSET] Theorem
⊢ ∀f s.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
interior (IMAGE f s) ⊆ IMAGE f (interior s)
[LINEAR_INVERTIBLE_BOUNDED_BELOW] Theorem
⊢ ∀f g.
linear f ∧ linear g ∧ g ∘ f = I ⇒ ∃B. ∀x. B * abs x ≤ abs (f x)
[LINEAR_INVERTIBLE_BOUNDED_BELOW_POS] Theorem
⊢ ∀f g.
linear f ∧ linear g ∧ g ∘ f = I ⇒
∃B. 0 < B ∧ ∀x. B * abs x ≤ abs (f x)
[LINEAR_LIM_0] Theorem
⊢ ∀f. linear f ⇒ (f ⟶ 0) (at 0)
[LINEAR_MUL_COMPONENT] Theorem
⊢ ∀f v. linear f ⇒ linear (λx. f x * v)
[LINEAR_NEG] Theorem
⊢ ∀f x. linear f ⇒ f (-x) = -f x
[LINEAR_NEGATION] Theorem
⊢ linear (λx. -x)
[LINEAR_OPEN_MAPPING] Theorem
⊢ ∀f g.
linear f ∧ linear g ∧ f ∘ g = I ⇒ ∀s. open s ⇒ open (IMAGE f s)
[LINEAR_SCALING] Theorem
⊢ ∀c. linear (λx. c * x)
[LINEAR_SUB] Theorem
⊢ ∀f x y. linear f ⇒ f (x − y) = f x − f y
[LINEAR_SUM] Theorem
⊢ ∀f g s. linear f ∧ FINITE s ⇒ f (sum s g) = sum s (f ∘ g)
[LINEAR_SUM_MUL] Theorem
⊢ ∀f s c v.
linear f ∧ FINITE s ⇒
f (sum s (λi. c i * v i)) = sum s (λi. c i * f (v i))
[LINEAR_UNIFORMLY_CONTINUOUS_ON] Theorem
⊢ ∀f s. linear f ⇒ f uniformly_continuous_on s
[LINEAR_ZERO] Theorem
⊢ linear (λx. 0)
[LOCALLY_CLOSED] Theorem
⊢ ∀s. locally closed s ⇔ locally compact s
[LOCALLY_COMPACT] Theorem
⊢ ∀s. locally compact s ⇔
∀x. x ∈ s ⇒
∃u v.
x ∈ u ∧ u ⊆ v ∧ v ⊆ s ∧
open_in (subtopology euclidean s) u ∧ compact v
[LOCALLY_COMPACT_ALT] Theorem
⊢ ∀s. locally compact s ⇔
∀x. x ∈ s ⇒
∃u. x ∈ u ∧ open_in (subtopology euclidean s) u ∧
compact (closure u) ∧ closure u ⊆ s
[LOCALLY_COMPACT_CLOSED_IN] Theorem
⊢ ∀s t.
closed_in (subtopology euclidean s) t ∧ locally compact s ⇒
locally compact t
[LOCALLY_COMPACT_CLOSED_INTER_OPEN] Theorem
⊢ ∀s. locally compact s ⇔ ∃t u. closed t ∧ open u ∧ s = t ∩ u
[LOCALLY_COMPACT_CLOSED_IN_OPEN] Theorem
⊢ ∀s. locally compact s ⇒
∃t. open t ∧ closed_in (subtopology euclidean t) s
[LOCALLY_COMPACT_CLOSED_UNION] Theorem
⊢ ∀s t.
locally compact s ∧ locally compact t ∧
closed_in (subtopology euclidean (s ∪ t)) s ∧
closed_in (subtopology euclidean (s ∪ t)) t ⇒
locally compact (s ∪ t)
[LOCALLY_COMPACT_COMPACT] Theorem
⊢ ∀s. locally compact s ⇔
∀k. k ⊆ s ∧ compact k ⇒
∃u v.
k ⊆ u ∧ u ⊆ v ∧ v ⊆ s ∧
open_in (subtopology euclidean s) u ∧ compact v
[LOCALLY_COMPACT_COMPACT_ALT] Theorem
⊢ ∀s. locally compact s ⇔
∀k. k ⊆ s ∧ compact k ⇒
∃u. k ⊆ u ∧ open_in (subtopology euclidean s) u ∧
compact (closure u) ∧ closure u ⊆ s
[LOCALLY_COMPACT_COMPACT_SUBOPEN] Theorem
⊢ ∀s. locally compact s ⇔
∀k t.
k ⊆ s ∧ compact k ∧ open t ∧ k ⊆ t ⇒
∃u v.
k ⊆ u ∧ u ⊆ v ∧ u ⊆ t ∧ v ⊆ s ∧
open_in (subtopology euclidean s) u ∧ compact v
[LOCALLY_COMPACT_DELETE] Theorem
⊢ ∀s a. locally compact s ⇒ locally compact (s DELETE a)
[LOCALLY_COMPACT_INTER] Theorem
⊢ ∀s t.
locally compact s ∧ locally compact t ⇒ locally compact (s ∩ t)
[LOCALLY_COMPACT_INTER_CBALL] Theorem
⊢ ∀s. locally compact s ⇔
∀x. x ∈ s ⇒ ∃e. 0 < e ∧ closed (cball (x,e) ∩ s)
[LOCALLY_COMPACT_INTER_CBALLS] Theorem
⊢ ∀s. locally compact s ⇔
∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ∀d. d ≤ e ⇒ closed (cball (x,d) ∩ s)
[LOCALLY_COMPACT_OPEN_IN] Theorem
⊢ ∀s t.
open_in (subtopology euclidean s) t ∧ locally compact s ⇒
locally compact t
[LOCALLY_COMPACT_OPEN_INTER_CLOSURE] Theorem
⊢ ∀s. locally compact s ⇒ ∃t. open t ∧ s = t ∩ closure s
[LOCALLY_COMPACT_OPEN_UNION] Theorem
⊢ ∀s t.
locally compact s ∧ locally compact t ∧
open_in (subtopology euclidean (s ∪ t)) s ∧
open_in (subtopology euclidean (s ∪ t)) t ⇒
locally compact (s ∪ t)
[LOCALLY_COMPACT_PROPER_IMAGE] Theorem
⊢ ∀f s.
f continuous_on s ∧
(∀k. k ⊆ IMAGE f s ∧ compact k ⇒ compact {x | x ∈ s ∧ f x ∈ k}) ∧
locally compact s ⇒
locally compact (IMAGE f s)
[LOCALLY_COMPACT_PROPER_IMAGE_EQ] Theorem
⊢ ∀f s.
f continuous_on s ∧
(∀k. k ⊆ IMAGE f s ∧ compact k ⇒ compact {x | x ∈ s ∧ f x ∈ k}) ⇒
(locally compact s ⇔ locally compact (IMAGE f s))
[LOCALLY_COMPACT_TRANSLATION_EQ] Theorem
⊢ ∀a s. locally compact (IMAGE (λx. a + x) s) ⇔ locally compact s
[LOCALLY_COMPACT_UNIV] Theorem
⊢ locally compact 𝕌(:real)
[LOCALLY_DIFF_CLOSED] Theorem
⊢ ∀P s t.
locally P s ∧ closed_in (subtopology euclidean s) t ⇒
locally P (s DIFF t)
[LOCALLY_EMPTY] Theorem
⊢ ∀P. locally P ∅
[LOCALLY_INJECTIVE_LINEAR_IMAGE] Theorem
⊢ ∀P Q.
(∀f s.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒ (P (IMAGE f s) ⇔ Q s)) ⇒
∀f s.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
(locally P (IMAGE f s) ⇔ locally Q s)
[LOCALLY_INTER] Theorem
⊢ ∀P. (∀s t. P s ∧ P t ⇒ P (s ∩ t)) ⇒
∀s t. locally P s ∧ locally P t ⇒ locally P (s ∩ t)
[LOCALLY_MONO] Theorem
⊢ ∀P Q s. (∀t. P t ⇒ Q t) ∧ locally P s ⇒ locally Q s
[LOCALLY_OPEN_MAP_IMAGE] Theorem
⊢ ∀P Q f s.
f continuous_on s ∧
(∀t. open_in (subtopology euclidean s) t ⇒
open_in (subtopology euclidean (IMAGE f s)) (IMAGE f t)) ∧
(∀t. t ⊆ s ∧ P t ⇒ Q (IMAGE f t)) ∧ locally P s ⇒
locally Q (IMAGE f s)
[LOCALLY_OPEN_SUBSET] Theorem
⊢ ∀P s t.
locally P s ∧ open_in (subtopology euclidean s) t ⇒ locally P t
[LOCALLY_SING] Theorem
⊢ ∀P a. locally P {a} ⇔ P {a}
[LOCALLY_TRANSLATION] Theorem
⊢ ∀P. (∀a s. P (IMAGE (λx. a + x) s) ⇔ P s) ⇒
∀a s. locally P (IMAGE (λx. a + x) s) ⇔ locally P s
[LOWER_HEMICONTINUOUS] Theorem
⊢ ∀f t s.
(∀x. x ∈ s ⇒ f x ⊆ t) ⇒
((∀u. closed_in (subtopology euclidean t) u ⇒
closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ⇔
∀u. open_in (subtopology euclidean t) u ⇒
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∩ u ≠ ∅})
[MAPPING_CONNECTED_ONTO_SEGMENT] Theorem
⊢ ∀s a b.
connected s ∧ ¬(∃a. s ⊆ {a}) ⇒
∃f. f continuous_on s ∧ IMAGE f s = segment [(a,b)]
[MAXIMAL_INDEPENDENT_SUBSET] Theorem
⊢ ∀v. ∃b. b ⊆ v ∧ independent b ∧ v ⊆ span b
[MAXIMAL_INDEPENDENT_SUBSET_EXTEND] Theorem
⊢ ∀s v.
s ⊆ v ∧ independent s ⇒
∃b. s ⊆ b ∧ b ⊆ v ∧ independent b ∧ v ⊆ span b
[METRIZABLE_SPACE_EUCLIDEAN] Theorem
⊢ metrizable_space euclidean
[MIDPOINT_COLLINEAR] Theorem
⊢ ∀a b c.
a ≠ c ⇒
(b = midpoint (a,c) ⇔
collinear {a; b; c} ∧ dist (a,b) = dist (b,c))
[MIDPOINT_EQ_ENDPOINT] Theorem
⊢ ∀a b.
(midpoint (a,b) = a ⇔ a = b) ∧ (midpoint (a,b) = b ⇔ a = b) ∧
(a = midpoint (a,b) ⇔ a = b) ∧ (b = midpoint (a,b) ⇔ a = b)
[MIDPOINT_IN_SEGMENT] Theorem
⊢ (∀a b. midpoint (a,b) ∈ segment [(a,b)]) ∧
∀a b. midpoint (a,b) ∈ segment (a,b) ⇔ a ≠ b
[MIDPOINT_LINEAR_IMAGE] Theorem
⊢ ∀f a b. linear f ⇒ midpoint (f a,f b) = f (midpoint (a,b))
[MIDPOINT_REFL] Theorem
⊢ ∀x. midpoint (x,x) = x
[MIDPOINT_SYM] Theorem
⊢ ∀a b. midpoint (a,b) = midpoint (b,a)
[MONOTONE_BIGGER] Theorem
⊢ ∀r. (∀m n. m < n ⇒ r m < r n) ⇒ ∀n. n ≤ r n
[MONOTONE_SUBSEQUENCE] Theorem
⊢ ∀s. ∃r.
(∀m n. m < n ⇒ r m < r n) ∧
((∀m n. m ≤ n ⇒ s (r m) ≤ s (r n)) ∨
∀m n. m ≤ n ⇒ s (r n) ≤ s (r m))
[MUL_CAUCHY_SCHWARZ_EQUAL] Theorem
⊢ ∀x y. (x * y)² = x * x * (y * y) ⇔ collinear {0; x; y}
[MUMFORD_LEMMA] Theorem
⊢ ∀f s t y.
f continuous_on s ∧ IMAGE f s ⊆ t ∧ locally compact s ∧ y ∈ t ∧
compact {x | x ∈ s ∧ f x = y} ⇒
∃u v.
open_in (subtopology euclidean s) u ∧
open_in (subtopology euclidean t) v ∧
{x | x ∈ s ∧ f x = y} ⊆ u ∧ y ∈ v ∧ IMAGE f u ⊆ v ∧
∀k. k ⊆ v ∧ compact k ⇒ compact {x | x ∈ u ∧ f x ∈ k}
[NEGATIONS_BALL] Theorem
⊢ ∀r. IMAGE (λx. -x) (ball (0,r)) = ball (0,r)
[NEGATIONS_CBALL] Theorem
⊢ ∀r. IMAGE (λx. -x) (cball (0,r)) = cball (0,r)
[NEGATIONS_SPHERE] Theorem
⊢ ∀r. IMAGE (λx. -x) (sphere (0,r)) = sphere (0,r)
[NET] Theorem
⊢ ∀n x y.
(∀z. netord n z x ⇒ netord n z y) ∨
∀z. netord n z y ⇒ netord n z x
[NETLIMIT_AT] Theorem
⊢ ∀a. netlimit (at a) = a
[NETLIMIT_WITHIN] Theorem
⊢ ∀a s. ¬trivial_limit (at a within s) ⇒ netlimit (at a within s) = a
[NETLIMIT_WITHIN_INTERIOR] Theorem
⊢ ∀s x. x ∈ interior s ⇒ netlimit (at x within s) = x
[NET_DILEMMA] Theorem
⊢ ∀net.
(∃a. (∃x. netord net x a) ∧ ∀x. netord net x a ⇒ P x) ∧
(∃b. (∃x. netord net x b) ∧ ∀x. netord net x b ⇒ Q x) ⇒
∃c. (∃x. netord net x c) ∧ ∀x. netord net x c ⇒ P x ∧ Q x
[NONTRIVIAL_LIMIT_WITHIN] Theorem
⊢ ∀net s. trivial_limit net ⇒ trivial_limit (net within s)
[NOT_BOUNDED_UNIV] Theorem
⊢ ¬bounded 𝕌(:real)
[NOT_EVENTUALLY] Theorem
⊢ ∀net p. (∀x. ¬p x) ∧ ¬trivial_limit net ⇒ ¬eventually p net
[NOT_INTERVAL_UNIV] Theorem
⊢ (∀a b. interval [(a,b)] ≠ 𝕌(:real)) ∧
∀a b. interval (a,b) ≠ 𝕌(:real)
[NOWHERE_DENSE] Theorem
⊢ ∀s. interior (closure s) = ∅ ⇔
∀t. open t ∧ t ≠ ∅ ⇒ ∃u. open u ∧ u ≠ ∅ ∧ u ⊆ t ∧ u ∩ s = ∅
[NOWHERE_DENSE_COUNTABLE_BIGUNION] Theorem
⊢ ∀g. countable g ∧ (∀s. s ∈ g ⇒ interior (closure s) = ∅) ⇒
interior (BIGUNION g) = ∅
[NOWHERE_DENSE_COUNTABLE_BIGUNION_CLOSED] Theorem
⊢ ∀g. countable g ∧ (∀s. s ∈ g ⇒ closed s ∧ interior s = ∅) ⇒
interior (BIGUNION g) = ∅
[NOWHERE_DENSE_UNION] Theorem
⊢ ∀s t.
interior (closure (s ∪ t)) = ∅ ⇔
interior (closure s) = ∅ ∧ interior (closure t) = ∅
[NO_LIMIT_POINT_IMP_CLOSED] Theorem
⊢ ∀s. ¬(∃x. x limit_point_of s) ⇒ closed s
[OLDNET] Theorem
⊢ ∀n x y.
netord n x x ∧ netord n y y ⇒
∃z. netord n z z ∧ ∀w. netord n w z ⇒ netord n w x ∧ netord n w y
[OPEN] Theorem
⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ∀x'. abs (x' − x) < e ⇒ x' ∈ s
[OPEN_AFFINITY] Theorem
⊢ ∀s a c. open s ∧ c ≠ 0 ⇒ open (IMAGE (λx. a + c * x) s)
[OPEN_BALL] Theorem
⊢ ∀x e. open (ball (x,e))
[OPEN_BIGINTER] Theorem
⊢ ∀s. FINITE s ∧ (∀t. t ∈ s ⇒ open t) ⇒ open (BIGINTER s)
[OPEN_BIGUNION] Theorem
⊢ ∀f. (∀s. s ∈ f ⇒ open s) ⇒ open (BIGUNION f)
[OPEN_BIJECTIVE_LINEAR_IMAGE_EQ] Theorem
⊢ ∀f s.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ∧ (∀y. ∃x. f x = y) ⇒
(open (IMAGE f s) ⇔ open s)
[OPEN_CLOSED] Theorem
⊢ ∀s. open s ⇔ closed (𝕌(:real) DIFF s)
[OPEN_CLOSED_INTERVAL] Theorem
⊢ ∀a b. interval (a,b) = interval [(a,b)] DIFF {a; b}
[OPEN_CLOSED_INTERVAL_CONVEX] Theorem
⊢ ∀a b x y e.
x ∈ interval (a,b) ∧ y ∈ interval [(a,b)] ∧ 0 < e ∧ e ≤ 1 ⇒
e * x + (1 − e) * y ∈ interval (a,b)
[OPEN_CONTAINS_BALL] Theorem
⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ball (x,e) ⊆ s
[OPEN_CONTAINS_BALL_EQ] Theorem
⊢ ∀s. open s ⇒ ∀x. x ∈ s ⇔ ∃e. 0 < e ∧ ball (x,e) ⊆ s
[OPEN_CONTAINS_CBALL] Theorem
⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ cball (x,e) ⊆ s
[OPEN_CONTAINS_CBALL_EQ] Theorem
⊢ ∀s. open s ⇒ ∀x. x ∈ s ⇔ ∃e. 0 < e ∧ cball (x,e) ⊆ s
[OPEN_CONTAINS_INTERVAL] Theorem
⊢ ∀s. open s ⇔
∀x. x ∈ s ⇒ ∃a b. x ∈ interval (a,b) ∧ interval [(a,b)] ⊆ s
[OPEN_CONTAINS_INTERVAL_OPEN_INTERVAL] Theorem
⊢ (∀s. open s ⇔
∀x. x ∈ s ⇒ ∃a b. x ∈ interval (a,b) ∧ interval [(a,b)] ⊆ s) ∧
∀s. open s ⇔
∀x. x ∈ s ⇒ ∃a b. x ∈ interval (a,b) ∧ interval (a,b) ⊆ s
[OPEN_CONTAINS_OPEN_INTERVAL] Theorem
⊢ ∀s. open s ⇔
∀x. x ∈ s ⇒ ∃a b. x ∈ interval (a,b) ∧ interval (a,b) ⊆ s
[OPEN_DELETE] Theorem
⊢ ∀s x. open s ⇒ open (s DELETE x)
[OPEN_DIFF] Theorem
⊢ ∀s t. open s ∧ closed t ⇒ open (s DIFF t)
[OPEN_EMPTY] Theorem
⊢ open ∅
[OPEN_EXISTS] Theorem
⊢ ∀Q. (∀a. open {x | Q a x}) ⇒ open {x | (∃a. Q a x)}
[OPEN_EXISTS_IN] Theorem
⊢ ∀P Q. (∀a. P a ⇒ open {x | Q a x}) ⇒ open {x | (∃a. P a ∧ Q a x)}
[OPEN_HALFSPACE_COMPONENT_GT] Theorem
⊢ ∀a. open {x | x > a}
[OPEN_HALFSPACE_COMPONENT_LT] Theorem
⊢ ∀a. open {x | x < a}
[OPEN_HALFSPACE_GT] Theorem
⊢ ∀a b. open {x | a * x > b}
[OPEN_HALFSPACE_LT] Theorem
⊢ ∀a b. open {x | a * x < b}
[OPEN_IMP_INFINITE] Theorem
⊢ ∀s. open s ⇒ s = ∅ ∨ INFINITE s
[OPEN_IMP_LOCALLY_COMPACT] Theorem
⊢ ∀s. open s ⇒ locally compact s
[OPEN_IN] Theorem
⊢ ∀s. open s ⇔ open_in euclidean s
[OPEN_INTER] Theorem
⊢ ∀s t. open s ∧ open t ⇒ open (s ∩ t)
[OPEN_INTERIOR] Theorem
⊢ ∀s. open (interior s)
[OPEN_INTERVAL] Theorem
⊢ ∀a b. open (interval (a,b))
[OPEN_INTERVAL_EQ] Theorem
⊢ (∀a b. open (interval [(a,b)]) ⇔ interval [(a,b)] = ∅) ∧
∀a b. open (interval (a,b))
[OPEN_INTERVAL_LEFT] Theorem
⊢ ∀b. open {x | x < b}
[OPEN_INTERVAL_LEMMA] Theorem
⊢ ∀a b x.
a < x ∧ x < b ⇒
∃d. 0 < d ∧ ∀x'. abs (x' − x) < d ⇒ a < x' ∧ x' < b
[OPEN_INTERVAL_LOWERBOUND] Theorem
⊢ ∀a b. a < b ⇒ interval_lowerbound (interval (a,b)) = a
[OPEN_INTERVAL_MIDPOINT] Theorem
⊢ ∀a b. interval (a,b) ≠ ∅ ⇒ 2⁻¹ * (a + b) ∈ interval (a,b)
[OPEN_INTERVAL_RIGHT] Theorem
⊢ ∀a. open {x | a < x}
[OPEN_INTERVAL_UPPERBOUND] Theorem
⊢ ∀a b. a < b ⇒ interval_upperbound (interval (a,b)) = b
[OPEN_INTER_CLOSURE_EQ_EMPTY] Theorem
⊢ ∀s t. open s ⇒ (s ∩ closure t = ∅ ⇔ s ∩ t = ∅)
[OPEN_INTER_CLOSURE_SUBSET] Theorem
⊢ ∀s t. open s ⇒ s ∩ closure t ⊆ closure (s ∩ t)
[OPEN_IN_CONNECTED_COMPONENT] Theorem
⊢ ∀s x.
FINITE {connected_component s x | x | x ∈ s} ⇒
open_in (subtopology euclidean s) (connected_component s x)
[OPEN_IN_CONTAINS_BALL] Theorem
⊢ ∀s t.
open_in (subtopology euclidean t) s ⇔
s ⊆ t ∧ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ball (x,e) ∩ t ⊆ s
[OPEN_IN_CONTAINS_CBALL] Theorem
⊢ ∀s t.
open_in (subtopology euclidean t) s ⇔
s ⊆ t ∧ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ cball (x,e) ∩ t ⊆ s
[OPEN_IN_DELETE] Theorem
⊢ ∀u s a.
open_in (subtopology euclidean u) s ⇒
open_in (subtopology euclidean u) (s DELETE a)
[OPEN_IN_INTER_OPEN] Theorem
⊢ ∀s t u.
open_in (subtopology euclidean u) s ∧ open t ⇒
open_in (subtopology euclidean u) (s ∩ t)
[OPEN_IN_LOCALLY_COMPACT] Theorem
⊢ ∀s t.
locally compact s ⇒
(open_in (subtopology euclidean s) t ⇔
t ⊆ s ∧
∀k. compact k ∧ k ⊆ s ⇒
open_in (subtopology euclidean k) (k ∩ t))
[OPEN_IN_OPEN] Theorem
⊢ ∀s u. open_in (subtopology euclidean u) s ⇔ ∃t. open t ∧ s = u ∩ t
[OPEN_IN_OPEN_EQ] Theorem
⊢ ∀s t.
open s ⇒ (open_in (subtopology euclidean s) t ⇔ open t ∧ t ⊆ s)
[OPEN_IN_OPEN_INTER] Theorem
⊢ ∀u s. open s ⇒ open_in (subtopology euclidean u) (u ∩ s)
[OPEN_IN_OPEN_TRANS] Theorem
⊢ ∀s t. open_in (subtopology euclidean t) s ∧ open t ⇒ open s
[OPEN_IN_REFL] Theorem
⊢ ∀s. open_in (subtopology euclidean s) s
[OPEN_IN_SING] Theorem
⊢ ∀s a.
open_in (subtopology euclidean s) {a} ⇔
a ∈ s ∧ ¬(a limit_point_of s)
[OPEN_IN_SUBSET_TRANS] Theorem
⊢ ∀s t u.
open_in (subtopology euclidean u) s ∧ s ⊆ t ∧ t ⊆ u ⇒
open_in (subtopology euclidean t) s
[OPEN_IN_SUBTOPOLOGY_INTER_SUBSET] Theorem
⊢ ∀s u v.
open_in (subtopology euclidean u) (u ∩ s) ∧ v ⊆ u ⇒
open_in (subtopology euclidean v) (v ∩ s)
[OPEN_IN_TRANS] Theorem
⊢ ∀s t u.
open_in (subtopology euclidean t) s ∧
open_in (subtopology euclidean u) t ⇒
open_in (subtopology euclidean u) s
[OPEN_IN_TRANS_EQ] Theorem
⊢ ∀s t.
(∀u. open_in (subtopology euclidean t) u ⇒
open_in (subtopology euclidean s) t) ⇔
open_in (subtopology euclidean s) t
[OPEN_MAP_CLOSED_SUPERSET_PREIMAGE] Theorem
⊢ ∀f s t u w.
(∀k. open_in (subtopology euclidean s) k ⇒
open_in (subtopology euclidean t) (IMAGE f k)) ∧
closed_in (subtopology euclidean s) u ∧ w ⊆ t ∧
{x | x ∈ s ∧ f x ∈ w} ⊆ u ⇒
∃v. closed_in (subtopology euclidean t) v ∧ w ⊆ v ∧
{x | x ∈ s ∧ f x ∈ v} ⊆ u
[OPEN_MAP_CLOSED_SUPERSET_PREIMAGE_EQ] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ⇒
((∀k. open_in (subtopology euclidean s) k ⇒
open_in (subtopology euclidean t) (IMAGE f k)) ⇔
∀u w.
closed_in (subtopology euclidean s) u ∧ w ⊆ t ∧
{x | x ∈ s ∧ f x ∈ w} ⊆ u ⇒
∃v. closed_in (subtopology euclidean t) v ∧ w ⊆ v ∧
{x | x ∈ s ∧ f x ∈ v} ⊆ u)
[OPEN_MAP_FROM_COMPOSITION_INJECTIVE] Theorem
⊢ ∀f g s t u.
IMAGE f s ⊆ t ∧ IMAGE g t ⊆ u ∧ g continuous_on t ∧
(∀x y. x ∈ t ∧ y ∈ t ∧ g x = g y ⇒ x = y) ∧
(∀k. open_in (subtopology euclidean s) k ⇒
open_in (subtopology euclidean u) (IMAGE (g ∘ f) k)) ⇒
∀k. open_in (subtopology euclidean s) k ⇒
open_in (subtopology euclidean t) (IMAGE f k)
[OPEN_MAP_FROM_COMPOSITION_SURJECTIVE] Theorem
⊢ ∀f g s t u.
f continuous_on s ∧ IMAGE f s = t ∧ IMAGE g t ⊆ u ∧
(∀k. open_in (subtopology euclidean s) k ⇒
open_in (subtopology euclidean u) (IMAGE (g ∘ f) k)) ⇒
∀k. open_in (subtopology euclidean t) k ⇒
open_in (subtopology euclidean u) (IMAGE g k)
[OPEN_MAP_IFF_LOWER_HEMICONTINUOUS_PREIMAGE] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ⇒
((∀u. open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean t) (IMAGE f u)) ⇔
∀u. closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean t)
{y | y ∈ t ∧ {x | x ∈ s ∧ f x = y} ⊆ u})
[OPEN_MAP_IMP_CLOSED_MAP] Theorem
⊢ ∀f s t.
IMAGE f s = t ∧
(∀u. open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean t) (IMAGE f u)) ∧
(∀u. closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean s)
{x | x ∈ s ∧ f x ∈ IMAGE f u}) ⇒
∀u. closed_in (subtopology euclidean s) u ⇒
closed_in (subtopology euclidean t) (IMAGE f u)
[OPEN_MAP_IMP_QUOTIENT_MAP] Theorem
⊢ ∀f s.
f continuous_on s ∧
(∀t. open_in (subtopology euclidean s) t ⇒
open_in (subtopology euclidean (IMAGE f s)) (IMAGE f t)) ⇒
∀t. t ⊆ IMAGE f s ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t} ⇔
open_in (subtopology euclidean (IMAGE f s)) t)
[OPEN_MAP_INTERIORS] Theorem
⊢ ∀f. (∀s. open s ⇒ open (IMAGE f s)) ⇔
∀s. IMAGE f (interior s) ⊆ interior (IMAGE f s)
[OPEN_MAP_RESTRICT] Theorem
⊢ ∀f s t t'.
(∀u. open_in (subtopology euclidean s) u ⇒
open_in (subtopology euclidean t) (IMAGE f u)) ∧ t' ⊆ t ⇒
∀u. open_in (subtopology euclidean {x | x ∈ s ∧ f x ∈ t'}) u ⇒
open_in (subtopology euclidean t') (IMAGE f u)
[OPEN_NEGATIONS] Theorem
⊢ ∀s. open s ⇒ open (IMAGE (λx. -x) s)
[OPEN_OPEN_IN_TRANS] Theorem
⊢ ∀s t. open s ∧ open t ∧ t ⊆ s ⇒ open_in (subtopology euclidean s) t
[OPEN_POSITIVE_MULTIPLES] Theorem
⊢ ∀s. open s ⇒ open {c * x | 0 < c ∧ x ∈ s}
[OPEN_POSITIVE_ORTHANT] Theorem
⊢ open {x | 0 < x}
[OPEN_SCALING] Theorem
⊢ ∀s c. c ≠ 0 ∧ open s ⇒ open (IMAGE (λx. c * x) s)
[OPEN_SEGMENT] Theorem
⊢ ∀a b. open (segment (a,b))
[OPEN_SEGMENT_ALT] Theorem
⊢ ∀a b. a ≠ b ⇒ segment (a,b) = {(1 − u) * a + u * b | 0 < u ∧ u < 1}
[OPEN_SEGMENT_LINEAR_IMAGE] Theorem
⊢ ∀f a b.
linear f ∧ (∀x y. f x = f y ⇒ x = y) ⇒
segment (f a,f b) = IMAGE f (segment (a,b))
[OPEN_SUBSET] Theorem
⊢ ∀s t. s ⊆ t ∧ open s ⇒ open_in (subtopology euclidean t) s
[OPEN_SUBSET_INTERIOR] Theorem
⊢ ∀s t. open s ⇒ (s ⊆ interior t ⇔ s ⊆ t)
[OPEN_SUB_OPEN] Theorem
⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃t. open t ∧ x ∈ t ∧ t ⊆ s
[OPEN_SUMS] Theorem
⊢ ∀s t. open s ∨ open t ⇒ open {x + y | x ∈ s ∧ y ∈ t}
[OPEN_SURJECTIVE_LINEAR_IMAGE] Theorem
⊢ ∀f. linear f ∧ (∀y. ∃x. f x = y) ⇒ ∀s. open s ⇒ open (IMAGE f s)
[OPEN_TRANSLATION] Theorem
⊢ ∀s a. open s ⇒ open (IMAGE (λx. a + x) s)
[OPEN_TRANSLATION_EQ] Theorem
⊢ ∀a s. open (IMAGE (λx. a + x) s) ⇔ open s
[OPEN_UNION] Theorem
⊢ ∀s t. open s ∧ open t ⇒ open (s ∪ t)
[OPEN_UNION_COMPACT_SUBSETS] Theorem
⊢ ∀s. open s ⇒
∃f. (∀n. compact (f n)) ∧ (∀n. f n ⊆ s) ∧
(∀n. f n ⊆ interior (f (n + 1))) ∧
BIGUNION {f n | n ∈ 𝕌(:num)} = s ∧
∀k. compact k ∧ k ⊆ s ⇒ ∃N. ∀n. n ≥ N ⇒ k ⊆ f n
[OPEN_UNIV] Theorem
⊢ open 𝕌(:real)
[PAIRWISE_DISJOINT_COMPONENTS] Theorem
⊢ ∀u. pairwiseD DISJOINT (components u)
[PARTIAL_SUMS_COMPONENT_LE_INFSUM] Theorem
⊢ ∀f s n.
(∀i. i ∈ s ⇒ 0 ≤ f i) ∧ summable s f ⇒
sum (s ∩ {0 .. n}) f ≤ suminf s f
[PARTIAL_SUMS_DROP_LE_INFSUM] Theorem
⊢ ∀f s n.
(∀i. i ∈ s ⇒ 0 ≤ f i) ∧ summable s f ⇒
sum (s ∩ {0 .. n}) f ≤ suminf s f
[PASTING_LEMMA] Theorem
⊢ ∀f g t s k.
(∀i. i ∈ k ⇒
open_in (subtopology euclidean s) (t i) ∧
f i continuous_on t i) ∧
(∀i j x. i ∈ k ∧ j ∈ k ∧ x ∈ s ∩ t i ∩ t j ⇒ f i x = f j x) ∧
(∀x. x ∈ s ⇒ ∃j. j ∈ k ∧ x ∈ t j ∧ g x = f j x) ⇒
g continuous_on s
[PASTING_LEMMA_CLOSED] Theorem
⊢ ∀f g t s k.
FINITE k ∧
(∀i. i ∈ k ⇒
closed_in (subtopology euclidean s) (t i) ∧
f i continuous_on t i) ∧
(∀i j x. i ∈ k ∧ j ∈ k ∧ x ∈ s ∩ t i ∩ t j ⇒ f i x = f j x) ∧
(∀x. x ∈ s ⇒ ∃j. j ∈ k ∧ x ∈ t j ∧ g x = f j x) ⇒
g continuous_on s
[PASTING_LEMMA_EXISTS] Theorem
⊢ ∀f t s k.
s ⊆ BIGUNION {t i | i ∈ k} ∧
(∀i. i ∈ k ⇒
open_in (subtopology euclidean s) (t i) ∧
f i continuous_on t i) ∧
(∀i j x. i ∈ k ∧ j ∈ k ∧ x ∈ s ∩ t i ∩ t j ⇒ f i x = f j x) ⇒
∃g. g continuous_on s ∧ ∀x i. i ∈ k ∧ x ∈ s ∩ t i ⇒ g x = f i x
[PASTING_LEMMA_EXISTS_CLOSED] Theorem
⊢ ∀f t s k.
FINITE k ∧ s ⊆ BIGUNION {t i | i ∈ k} ∧
(∀i. i ∈ k ⇒
closed_in (subtopology euclidean s) (t i) ∧
f i continuous_on t i) ∧
(∀i j x. i ∈ k ∧ j ∈ k ∧ x ∈ s ∩ t i ∩ t j ⇒ f i x = f j x) ⇒
∃g. g continuous_on s ∧ ∀x i. i ∈ k ∧ x ∈ s ∩ t i ⇒ g x = f i x
[PROPER_MAP] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ⇒
((∀k. k ⊆ t ∧ compact k ⇒ compact {x | x ∈ s ∧ f x ∈ k}) ⇔
(∀k. closed_in (subtopology euclidean s) k ⇒
closed_in (subtopology euclidean t) (IMAGE f k)) ∧
∀a. a ∈ t ⇒ compact {x | x ∈ s ∧ f x = a})
[PROPER_MAP_COMPOSE] Theorem
⊢ ∀f g s t u.
IMAGE f s ⊆ t ∧
(∀k. k ⊆ t ∧ compact k ⇒ compact {x | x ∈ s ∧ f x ∈ k}) ∧
(∀k. k ⊆ u ∧ compact k ⇒ compact {x | x ∈ t ∧ g x ∈ k}) ⇒
∀k. k ⊆ u ∧ compact k ⇒ compact {x | x ∈ s ∧ (g ∘ f) x ∈ k}
[PROPER_MAP_FROM_COMPACT] Theorem
⊢ ∀f s k.
f continuous_on s ∧ IMAGE f s ⊆ t ∧ compact s ∧
closed_in (subtopology euclidean t) k ⇒
compact {x | x ∈ s ∧ f x ∈ k}
[PROPER_MAP_FROM_COMPOSITION_LEFT] Theorem
⊢ ∀f g s t u.
f continuous_on s ∧ IMAGE f s = t ∧ g continuous_on t ∧
IMAGE g t ⊆ u ∧
(∀k. k ⊆ u ∧ compact k ⇒ compact {x | x ∈ s ∧ (g ∘ f) x ∈ k}) ⇒
∀k. k ⊆ u ∧ compact k ⇒ compact {x | x ∈ t ∧ g x ∈ k}
[PROPER_MAP_FROM_COMPOSITION_RIGHT] Theorem
⊢ ∀f g s t u.
f continuous_on s ∧ IMAGE f s ⊆ t ∧ g continuous_on t ∧
IMAGE g t ⊆ u ∧
(∀k. k ⊆ u ∧ compact k ⇒ compact {x | x ∈ s ∧ (g ∘ f) x ∈ k}) ⇒
∀k. k ⊆ t ∧ compact k ⇒ compact {x | x ∈ s ∧ f x ∈ k}
[QUASICOMPACT_OPEN_CLOSED] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ⇒
((∀u. u ⊆ t ⇒
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇒
open_in (subtopology euclidean t) u) ⇔
∀u. u ⊆ t ⇒
closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇒
closed_in (subtopology euclidean t) u)
[QUOTIENT_MAP_CLOSED_MAP_EQ] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ∧
(∀u. u ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
open_in (subtopology euclidean t) u)) ⇒
((∀k. closed_in (subtopology euclidean s) k ⇒
closed_in (subtopology euclidean t) (IMAGE f k)) ⇔
∀k. closed_in (subtopology euclidean s) k ⇒
closed_in (subtopology euclidean s)
{x | x ∈ s ∧ f x ∈ IMAGE f k})
[QUOTIENT_MAP_COMPOSE] Theorem
⊢ ∀f g s t u.
IMAGE f s ⊆ t ∧
(∀v. v ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ v} ⇔
open_in (subtopology euclidean t) v)) ∧
(∀v. v ⊆ u ⇒
(open_in (subtopology euclidean t) {x | x ∈ t ∧ g x ∈ v} ⇔
open_in (subtopology euclidean u) v)) ⇒
∀v. v ⊆ u ⇒
(open_in (subtopology euclidean s)
{x | x ∈ s ∧ (g ∘ f) x ∈ v} ⇔
open_in (subtopology euclidean u) v)
[QUOTIENT_MAP_FROM_COMPOSITION] Theorem
⊢ ∀f g s t u.
f continuous_on s ∧ IMAGE f s ⊆ t ∧ g continuous_on t ∧
IMAGE g t ⊆ u ∧
(∀v. v ⊆ u ⇒
(open_in (subtopology euclidean s)
{x | x ∈ s ∧ (g ∘ f) x ∈ v} ⇔
open_in (subtopology euclidean u) v)) ⇒
∀v. v ⊆ u ⇒
(open_in (subtopology euclidean t) {x | x ∈ t ∧ g x ∈ v} ⇔
open_in (subtopology euclidean u) v)
[QUOTIENT_MAP_FROM_SUBSET] Theorem
⊢ ∀f s t u.
f continuous_on t ∧ IMAGE f t ⊆ u ∧ s ⊆ t ∧ IMAGE f s = u ∧
(∀v. v ⊆ u ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ v} ⇔
open_in (subtopology euclidean u) v)) ⇒
∀v. v ⊆ u ⇒
(open_in (subtopology euclidean t) {x | x ∈ t ∧ f x ∈ v} ⇔
open_in (subtopology euclidean u) v)
[QUOTIENT_MAP_IMP_CONTINUOUS_CLOSED] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ∧
(∀u. u ⊆ t ⇒
(closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
closed_in (subtopology euclidean t) u)) ⇒
f continuous_on s
[QUOTIENT_MAP_IMP_CONTINUOUS_OPEN] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ∧
(∀u. u ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
open_in (subtopology euclidean t) u)) ⇒
f continuous_on s
[QUOTIENT_MAP_OPEN_CLOSED] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ⇒
((∀u. u ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
open_in (subtopology euclidean t) u)) ⇔
∀u. u ⊆ t ⇒
(closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
closed_in (subtopology euclidean t) u))
[QUOTIENT_MAP_OPEN_MAP_EQ] Theorem
⊢ ∀f s t.
IMAGE f s ⊆ t ∧
(∀u. u ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
open_in (subtopology euclidean t) u)) ⇒
((∀k. open_in (subtopology euclidean s) k ⇒
open_in (subtopology euclidean t) (IMAGE f k)) ⇔
∀k. open_in (subtopology euclidean s) k ⇒
open_in (subtopology euclidean s)
{x | x ∈ s ∧ f x ∈ IMAGE f k})
[QUOTIENT_MAP_RESTRICT] Theorem
⊢ ∀f s t c.
IMAGE f s ⊆ t ∧
(∀u. u ⊆ t ⇒
(open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
open_in (subtopology euclidean t) u)) ∧
(open_in (subtopology euclidean t) c ∨
closed_in (subtopology euclidean t) c) ⇒
∀u. u ⊆ c ⇒
(open_in (subtopology euclidean {x | x ∈ s ∧ f x ∈ c})
{x | x ∈ {x | x ∈ s ∧ f x ∈ c} ∧ f x ∈ u} ⇔
open_in (subtopology euclidean c) u)
[REAL_AFFINITY_EQ] Theorem
⊢ ∀m c x y. m ≠ 0 ⇒ (m * x + c = y ⇔ x = m⁻¹ * y + -(c / m))
[REAL_AFFINITY_LE] Theorem
⊢ ∀m c x y. 0 < m ⇒ (m * x + c ≤ y ⇔ x ≤ m⁻¹ * y + -(c / m))
[REAL_AFFINITY_LT] Theorem
⊢ ∀m c x y. 0 < m ⇒ (m * x + c < y ⇔ x < m⁻¹ * y + -(c / m))
[REAL_ARCH_RDIV_EQ_0] Theorem
⊢ ∀x c. 0 ≤ x ∧ 0 ≤ c ∧ (∀m. 0 < m ⇒ &m * x ≤ c) ⇒ x = 0
[REAL_CHOOSE_DIST] Theorem
⊢ ∀x e. 0 ≤ e ⇒ ∃y. dist (x,y) = e
[REAL_CHOOSE_SIZE] Theorem
⊢ ∀c. 0 ≤ c ⇒ ∃x. abs x = c
[REAL_CONVEX_BOUND_LE] Theorem
⊢ ∀x y a u v.
x ≤ a ∧ y ≤ a ∧ 0 ≤ u ∧ 0 ≤ v ∧ u + v = 1 ⇒ u * x + v * y ≤ a
[REAL_EQ_AFFINITY] Theorem
⊢ ∀m c x y. m ≠ 0 ⇒ (y = m * x + c ⇔ m⁻¹ * y + -(c / m) = x)
[REAL_EQ_LINV] Theorem
⊢ ∀x. -x = x ⇔ x = 0
[REAL_EQ_RINV] Theorem
⊢ ∀x. x = -x ⇔ x = 0
[REAL_HAUSDIST_LE] Theorem
⊢ ∀s t b.
s ≠ ∅ ∧ t ≠ ∅ ∧ (∀x. x ∈ s ⇒ setdist ({x},t) ≤ b) ∧
(∀y. y ∈ t ⇒ setdist ({y},s) ≤ b) ⇒
hausdist (s,t) ≤ b
[REAL_HAUSDIST_LE_EQ] Theorem
⊢ ∀s t b.
s ≠ ∅ ∧ t ≠ ∅ ∧ bounded s ∧ bounded t ⇒
(hausdist (s,t) ≤ b ⇔
(∀x. x ∈ s ⇒ setdist ({x},t) ≤ b) ∧
∀y. y ∈ t ⇒ setdist ({y},s) ≤ b)
[REAL_HAUSDIST_LE_SUMS] Theorem
⊢ ∀s t b.
s ≠ ∅ ∧ t ≠ ∅ ∧ s ⊆ {y + z | y ∈ t ∧ z ∈ cball (0,b)} ∧
t ⊆ {y + z | y ∈ s ∧ z ∈ cball (0,b)} ⇒
hausdist (s,t) ≤ b
[REAL_LE_AFFINITY] Theorem
⊢ ∀m c x y. 0 < m ⇒ (y ≤ m * x + c ⇔ m⁻¹ * y + -(c / m) ≤ x)
[REAL_LE_HAUSDIST] Theorem
⊢ ∀s t a b c z.
s ≠ ∅ ∧ t ≠ ∅ ∧ (∀x. x ∈ s ⇒ setdist ({x},t) ≤ b) ∧
(∀y. y ∈ t ⇒ setdist ({y},s) ≤ c) ∧
(z ∈ s ∧ a ≤ setdist ({z},t) ∨ z ∈ t ∧ a ≤ setdist ({z},s)) ⇒
a ≤ hausdist (s,t)
[REAL_LE_SETDIST] Theorem
⊢ ∀s t d.
s ≠ ∅ ∧ t ≠ ∅ ∧ (∀x y. x ∈ s ∧ y ∈ t ⇒ d ≤ dist (x,y)) ⇒
d ≤ setdist (s,t)
[REAL_LE_SETDIST_EQ] Theorem
⊢ ∀d s t.
d ≤ setdist (s,t) ⇔
(∀x y. x ∈ s ∧ y ∈ t ⇒ d ≤ dist (x,y)) ∧ (s = ∅ ∨ t = ∅ ⇒ d ≤ 0)
[REAL_LT_AFFINITY] Theorem
⊢ ∀m c x y. 0 < m ⇒ (y < m * x + c ⇔ m⁻¹ * y + -(c / m) < x)
[REAL_LT_HAUSDIST_POINT_EXISTS] Theorem
⊢ ∀s t x d.
bounded s ∧ bounded t ∧ t ≠ ∅ ∧ hausdist (s,t) < d ∧ x ∈ s ⇒
∃y. y ∈ t ∧ dist (x,y) < d
[REAL_SETDIST_LT_EXISTS] Theorem
⊢ ∀s t b.
s ≠ ∅ ∧ t ≠ ∅ ∧ setdist (s,t) < b ⇒
∃x y. x ∈ s ∧ y ∈ t ∧ dist (x,y) < b
[REFLECT_INTERVAL] Theorem
⊢ (∀a b. IMAGE (λx. -x) (interval [(a,b)]) = interval [(-b,-a)]) ∧
∀a b. IMAGE (λx. -x) (interval (a,b)) = interval (-b,-a)
[REGULAR_CLOSED_BIGUNION] Theorem
⊢ ∀f. FINITE f ∧ (∀t. t ∈ f ⇒ closure (interior t) = t) ⇒
closure (interior (BIGUNION f)) = BIGUNION f
[REGULAR_CLOSED_UNION] Theorem
⊢ ∀s t.
closure (interior s) = s ∧ closure (interior t) = t ⇒
closure (interior (s ∪ t)) = s ∪ t
[REGULAR_OPEN_INTER] Theorem
⊢ ∀s t.
interior (closure s) = s ∧ interior (closure t) = t ⇒
interior (closure (s ∩ t)) = s ∩ t
[SEGMENT] Theorem
⊢ (∀a b.
segment [(a,b)] =
if a ≤ b then interval [(a,b)] else interval [(b,a)]) ∧
∀a b.
segment (a,b) = if a ≤ b then interval (a,b) else interval (b,a)
[SEGMENT_CLOSED_OPEN] Theorem
⊢ ∀a b. segment [(a,b)] = segment (a,b) ∪ {a; b}
[SEGMENT_OPEN_SUBSET_CLOSED] Theorem
⊢ ∀a b. segment (a,b) ⊆ segment [(a,b)]
[SEGMENT_REFL] Theorem
⊢ (∀a. segment [(a,a)] = {a}) ∧ ∀a. segment (a,a) = ∅
[SEGMENT_SCALAR_MULTIPLE] Theorem
⊢ (∀a b v.
segment [(a * v,b * v)] =
{x * v | a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a}) ∧
∀a b v.
v ≠ 0 ⇒
segment (a * v,b * v) = {x * v | a < x ∧ x < b ∨ b < x ∧ x < a}
[SEGMENT_SYM] Theorem
⊢ (∀a b. segment [(a,b)] = segment [(b,a)]) ∧
∀a b. segment (a,b) = segment (b,a)
[SEGMENT_TO_CLOSEST_POINT] Theorem
⊢ ∀s a. closed s ∧ s ≠ ∅ ⇒ segment (a,closest_point s a) ∩ s = ∅
[SEGMENT_TO_POINT_EXISTS] Theorem
⊢ ∀s a. closed s ∧ s ≠ ∅ ⇒ ∃b. b ∈ s ∧ segment (a,b) ∩ s = ∅
[SEGMENT_TRANSLATION] Theorem
⊢ (∀c a b.
segment [(c + a,c + b)] = IMAGE (λx. c + x) (segment [(a,b)])) ∧
∀c a b. segment (c + a,c + b) = IMAGE (λx. c + x) (segment (a,b))
[SEPARATE_CLOSED_COMPACT] Theorem
⊢ ∀s t.
closed s ∧ compact t ∧ s ∩ t = ∅ ⇒
∃d. 0 < d ∧ ∀x y. x ∈ s ∧ y ∈ t ⇒ d ≤ dist (x,y)
[SEPARATE_COMPACT_CLOSED] Theorem
⊢ ∀s t.
compact s ∧ closed t ∧ s ∩ t = ∅ ⇒
∃d. 0 < d ∧ ∀x y. x ∈ s ∧ y ∈ t ⇒ d ≤ dist (x,y)
[SEPARATE_POINT_CLOSED] Theorem
⊢ ∀s a. closed s ∧ a ∉ s ⇒ ∃d. 0 < d ∧ ∀x. x ∈ s ⇒ d ≤ dist (a,x)
[SEPARATION_CLOSURES] Theorem
⊢ ∀s t.
s ∩ closure t = ∅ ∧ t ∩ closure s = ∅ ⇒
∃u v. DISJOINT u v ∧ open u ∧ open v ∧ s ⊆ u ∧ t ⊆ v
[SEPARATION_HAUSDORFF] Theorem
⊢ ∀x y. x ≠ y ⇒ ∃u v. open u ∧ open v ∧ x ∈ u ∧ y ∈ v ∧ u ∩ v = ∅
[SEPARATION_NORMAL] Theorem
⊢ ∀s t.
closed s ∧ closed t ∧ s ∩ t = ∅ ⇒
∃u v. open u ∧ open v ∧ s ⊆ u ∧ t ⊆ v ∧ u ∩ v = ∅
[SEPARATION_NORMAL_COMPACT] Theorem
⊢ ∀s t.
compact s ∧ closed t ∧ s ∩ t = ∅ ⇒
∃u v.
open u ∧ compact (closure u) ∧ open v ∧ s ⊆ u ∧ t ⊆ v ∧
u ∩ v = ∅
[SEPARATION_NORMAL_LOCAL] Theorem
⊢ ∀s t u.
closed_in (subtopology euclidean u) s ∧
closed_in (subtopology euclidean u) t ∧ s ∩ t = ∅ ⇒
∃s' t'.
open_in (subtopology euclidean u) s' ∧
open_in (subtopology euclidean u) t' ∧ s ⊆ s' ∧ t ⊆ t' ∧
s' ∩ t' = ∅
[SEPARATION_T0] Theorem
⊢ ∀x y. x ≠ y ⇔ ∃u. open u ∧ (x ∈ u ⇎ y ∈ u)
[SEPARATION_T1] Theorem
⊢ ∀x y. x ≠ y ⇔ ∃u v. open u ∧ open v ∧ x ∈ u ∧ y ∉ u ∧ x ∉ v ∧ y ∈ v
[SEPARATION_T2] Theorem
⊢ ∀x y. x ≠ y ⇔ ∃u v. open u ∧ open v ∧ x ∈ u ∧ y ∈ v ∧ u ∩ v = ∅
[SEQUENCE_CAUCHY_WLOG] Theorem
⊢ ∀P s.
(∀m n. P m ∧ P n ⇒ dist (s m,s n) < e) ⇔
∀m n. P m ∧ P n ∧ m ≤ n ⇒ dist (s m,s n) < e
[SEQUENCE_INFINITE_LEMMA] Theorem
⊢ ∀f l.
(∀n. f n ≠ l) ∧ (f ⟶ l) sequentially ⇒
INFINITE {y | (∃n. y = f n)}
[SEQUENCE_UNIQUE_LIMPT] Theorem
⊢ ∀f l l'.
(f ⟶ l) sequentially ∧ l' limit_point_of {y | ∃n. y = f n} ⇒
l' = l
[SEQUENTIALLY] Theorem
⊢ ∀m n. netord sequentially m n ⇔ m ≥ n
[SEQ_HARMONIC] Theorem
⊢ ((λn. (&n)⁻¹) ⟶ 0) sequentially
[SEQ_HARMONIC_OFFSET] Theorem
⊢ ∀a. ((λn. (&n + a)⁻¹) ⟶ 0) sequentially
[SEQ_OFFSET] Theorem
⊢ ∀f l k. (f ⟶ l) sequentially ⇒ ((λi. f (i + k)) ⟶ l) sequentially
[SEQ_OFFSET_NEG] Theorem
⊢ ∀f l k. (f ⟶ l) sequentially ⇒ ((λi. f (i − k)) ⟶ l) sequentially
[SEQ_OFFSET_REV] Theorem
⊢ ∀f l k. ((λi. f (i + k)) ⟶ l) sequentially ⇒ (f ⟶ l) sequentially
[SERIES_0] Theorem
⊢ ∀s. ((λn. 0) sums 0) s
[SERIES_ABSCONV_IMP_CONV] Theorem
⊢ ∀x k. summable k (λn. abs (x n)) ⇒ summable k x
[SERIES_ADD] Theorem
⊢ ∀x x0 y y0 s.
(x sums x0) s ∧ (y sums y0) s ⇒ ((λn. x n + y n) sums x0 + y0) s
[SERIES_BOUND] Theorem
⊢ ∀f g s a b.
(f sums a) s ∧ (g sums b) s ∧ (∀i. i ∈ s ⇒ abs (f i) ≤ g i) ⇒
abs a ≤ b
[SERIES_CAUCHY] Theorem
⊢ ∀f s.
(∃l. (f sums l) s) ⇔
∀e. 0 < e ⇒ ∃N. ∀m n. m ≥ N ⇒ abs (sum (s ∩ {m .. n}) f) < e
[SERIES_CAUCHY_UNIFORM] Theorem
⊢ ∀P f k.
(∃l. ∀e.
0 < e ⇒
∃N. ∀n x.
N ≤ n ∧ P x ⇒ dist (sum (k ∩ {0 .. n}) (f x),l x) < e) ⇔
∀e. 0 < e ⇒
∃N. ∀m n x. N ≤ m ∧ P x ⇒ abs (sum (k ∩ {m .. n}) (f x)) < e
[SERIES_CMUL] Theorem
⊢ ∀x x0 c s. (x sums x0) s ⇒ ((λn. c * x n) sums c * x0) s
[SERIES_COMPARISON] Theorem
⊢ ∀f g s.
(∃l. (g sums l) s) ∧ (∃N. ∀n. n ≥ N ∧ n ∈ s ⇒ abs (f n) ≤ g n) ⇒
∃l. (f sums l) s
[SERIES_COMPARISON_BOUND] Theorem
⊢ ∀f g s a.
(g sums a) s ∧ (∀i. i ∈ s ⇒ abs (f i) ≤ g i) ⇒
∃l. (f sums l) s ∧ abs l ≤ a
[SERIES_COMPARISON_UNIFORM] Theorem
⊢ ∀f g P s.
(∃l. (g sums l) s) ∧
(∃N. ∀n x. N ≤ n ∧ n ∈ s ∧ P x ⇒ abs (f x n) ≤ g n) ⇒
∃l. ∀e.
0 < e ⇒
∃N. ∀n x. N ≤ n ∧ P x ⇒ dist (sum (s ∩ {0 .. n}) (f x),l x) < e
[SERIES_COMPONENT] Theorem
⊢ ∀f s l. (f sums l) s ⇒ ((λi. f i) sums l) s
[SERIES_DIFFS] Theorem
⊢ ∀f k.
(f ⟶ 0) sequentially ⇒ ((λn. f n − f (n + 1)) sums f k) (from k)
[SERIES_DIRICHLET] Theorem
⊢ ∀f g N k m.
bounded {sum {m .. n} f | n ∈ 𝕌(:num)} ∧
(∀n. N ≤ n ⇒ g (n + 1) ≤ g n) ∧ (g ⟶ 0) sequentially ⇒
summable (from k) (λn. g n * f n)
[SERIES_DIRICHLET_BILINEAR] Theorem
⊢ ∀f g h k m p l.
bilinear h ∧ bounded {sum {m .. n} f | n ∈ 𝕌(:num)} ∧
summable (from p) (λn. abs (g (n + 1) − g n)) ∧
((λn. h (g (n + 1)) (sum {1 .. n} f)) ⟶ l) sequentially ⇒
summable (from k) (λn. h (g n) (f n))
[SERIES_DROP_LE] Theorem
⊢ ∀f g s a b.
(f sums a) s ∧ (g sums b) s ∧ (∀x. x ∈ s ⇒ f x ≤ g x) ⇒ a ≤ b
[SERIES_DROP_POS] Theorem
⊢ ∀f s a. (f sums a) s ∧ (∀x. x ∈ s ⇒ 0 ≤ f x) ⇒ 0 ≤ a
[SERIES_FINITE] Theorem
⊢ ∀f s. FINITE s ⇒ (f sums sum s f) s
[SERIES_FINITE_SUPPORT] Theorem
⊢ ∀f s k.
FINITE (s ∩ k) ∧ (∀x. x ∈ k ∧ x ∉ s ⇒ f x = 0) ⇒
(f sums sum (s ∩ k) f) k
[SERIES_FROM] Theorem
⊢ ∀f l k.
(f sums l) (from k) ⇔ ((λn. sum {k .. n} f) ⟶ l) sequentially
[SERIES_GOESTOZERO] Theorem
⊢ ∀s x.
summable s x ⇒
∀e. 0 < e ⇒ eventually (λn. n ∈ s ⇒ abs (x n) < e) sequentially
[SERIES_INJECTIVE_IMAGE] Theorem
⊢ ∀x s f l.
summable (IMAGE f s) (λn. abs (x n)) ∧
(∀m n. m ∈ s ∧ n ∈ s ∧ f m = f n ⇒ m = n) ⇒
((x ∘ f sums l) s ⇔ (x sums l) (IMAGE f s))
[SERIES_INJECTIVE_IMAGE_STRONG] Theorem
⊢ ∀x s f.
summable (IMAGE f s) (λn. abs (x n)) ∧
(∀m n. m ∈ s ∧ n ∈ s ∧ f m = f n ⇒ m = n) ⇒
((λn. sum (IMAGE f s ∩ {0 .. n}) x − sum (s ∩ {0 .. n}) (x ∘ f)) ⟶
0) sequentially
[SERIES_LINEAR] Theorem
⊢ ∀f h l s. (f sums l) s ∧ linear h ⇒ ((λn. h (f n)) sums h l) s
[SERIES_NEG] Theorem
⊢ ∀x x0 s. (x sums x0) s ⇒ ((λn. -x n) sums -x0) s
[SERIES_RATIO] Theorem
⊢ ∀c a s N.
c < 1 ∧ (∀n. n ≥ N ⇒ abs (a (SUC n)) ≤ c * abs (a n)) ⇒
∃l. (a sums l) s
[SERIES_REARRANGE] Theorem
⊢ ∀x s p l.
summable s (λn. abs (x n)) ∧ p permutes s ∧ (x sums l) s ⇒
(x ∘ p sums l) s
[SERIES_REARRANGE_EQ] Theorem
⊢ ∀x s p l.
summable s (λn. abs (x n)) ∧ p permutes s ⇒
((x ∘ p sums l) s ⇔ (x sums l) s)
[SERIES_RESTRICT] Theorem
⊢ ∀f k l.
((λn. if n ∈ k then f n else 0) sums l) 𝕌(:num) ⇔ (f sums l) k
[SERIES_SUB] Theorem
⊢ ∀x x0 y y0 s.
(x sums x0) s ∧ (y sums y0) s ⇒ ((λn. x n − y n) sums x0 − y0) s
[SERIES_SUBSET] Theorem
⊢ ∀x s t l.
s ⊆ t ∧ ((λi. if i ∈ s then x i else 0) sums l) t ⇒ (x sums l) s
[SERIES_SUM] Theorem
⊢ ∀f l k s.
FINITE s ∧ s ⊆ k ∧ (∀x. x ∉ s ⇒ f x = 0) ∧ sum s f = l ⇒
(f sums l) k
[SERIES_TERMS_TOZERO] Theorem
⊢ ∀f l n. (f sums l) (from n) ⇒ (f ⟶ 0) sequentially
[SERIES_TRIVIAL] Theorem
⊢ ∀f. (f sums 0) ∅
[SERIES_UNIQUE] Theorem
⊢ ∀f l l' s. (f sums l) s ∧ (f sums l') s ⇒ l = l'
[SETDIST_BALLS] Theorem
⊢ (∀a b r s.
setdist (ball (a,r),ball (b,s)) =
if r ≤ 0 ∨ s ≤ 0 then 0 else max 0 (dist (a,b) − (r + s))) ∧
(∀a b r s.
setdist (ball (a,r),cball (b,s)) =
if r ≤ 0 ∨ s < 0 then 0 else max 0 (dist (a,b) − (r + s))) ∧
(∀a b r s.
setdist (cball (a,r),ball (b,s)) =
if r < 0 ∨ s ≤ 0 then 0 else max 0 (dist (a,b) − (r + s))) ∧
∀a b r s.
setdist (cball (a,r),cball (b,s)) =
if r < 0 ∨ s < 0 then 0 else max 0 (dist (a,b) − (r + s))
[SETDIST_CLOSED_COMPACT] Theorem
⊢ ∀s t.
closed s ∧ compact t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
∃x y. x ∈ s ∧ y ∈ t ∧ dist (x,y) = setdist (s,t)
[SETDIST_CLOSEST_POINT] Theorem
⊢ ∀a s.
closed s ∧ s ≠ ∅ ⇒ setdist ({a},s) = dist (a,closest_point s a)
[SETDIST_CLOSURE] Theorem
⊢ (∀s t. setdist (closure s,t) = setdist (s,t)) ∧
∀s t. setdist (s,closure t) = setdist (s,t)
[SETDIST_COMPACT_CLOSED] Theorem
⊢ ∀s t.
compact s ∧ closed t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
∃x y. x ∈ s ∧ y ∈ t ∧ dist (x,y) = setdist (s,t)
[SETDIST_DIFFERENCES] Theorem
⊢ ∀s t. setdist (s,t) = setdist ({0},{x − y | x ∈ s ∧ y ∈ t})
[SETDIST_EMPTY] Theorem
⊢ (∀t. setdist (∅,t) = 0) ∧ ∀s. setdist (s,∅) = 0
[SETDIST_EQ_0_BOUNDED] Theorem
⊢ ∀s t.
bounded s ∨ bounded t ⇒
(setdist (s,t) = 0 ⇔ s = ∅ ∨ t = ∅ ∨ closure s ∩ closure t ≠ ∅)
[SETDIST_EQ_0_CLOSED] Theorem
⊢ ∀s x. closed s ⇒ (setdist ({x},s) = 0 ⇔ s = ∅ ∨ x ∈ s)
[SETDIST_EQ_0_CLOSED_COMPACT] Theorem
⊢ ∀s t.
closed s ∧ compact t ⇒
(setdist (s,t) = 0 ⇔ s = ∅ ∨ t = ∅ ∨ s ∩ t ≠ ∅)
[SETDIST_EQ_0_CLOSED_IN] Theorem
⊢ ∀u s x.
closed_in (subtopology euclidean u) s ∧ x ∈ u ⇒
(setdist ({x},s) = 0 ⇔ s = ∅ ∨ x ∈ s)
[SETDIST_EQ_0_COMPACT_CLOSED] Theorem
⊢ ∀s t.
compact s ∧ closed t ⇒
(setdist (s,t) = 0 ⇔ s = ∅ ∨ t = ∅ ∨ s ∩ t ≠ ∅)
[SETDIST_EQ_0_SING] Theorem
⊢ (∀s x. setdist ({x},s) = 0 ⇔ s = ∅ ∨ x ∈ closure s) ∧
∀s x. setdist (s,{x}) = 0 ⇔ s = ∅ ∨ x ∈ closure s
[SETDIST_FRONTIER] Theorem
⊢ (∀s t. DISJOINT s t ⇒ setdist (frontier s,t) = setdist (s,t)) ∧
∀s t. DISJOINT s t ⇒ setdist (s,frontier t) = setdist (s,t)
[SETDIST_FRONTIERS] Theorem
⊢ ∀s t.
setdist (s,t) =
if DISJOINT s t then setdist (frontier s,frontier t) else 0
[SETDIST_HAUSDIST_TRIANGLE] Theorem
⊢ ∀s t u.
t ≠ ∅ ∧ bounded t ∧ bounded u ⇒
setdist (s,u) ≤ setdist (s,t) + hausdist (t,u)
[SETDIST_LE_DIST] Theorem
⊢ ∀s t x y. x ∈ s ∧ y ∈ t ⇒ setdist (s,t) ≤ dist (x,y)
[SETDIST_LE_HAUSDIST] Theorem
⊢ ∀s t. bounded s ∧ bounded t ⇒ setdist (s,t) ≤ hausdist (s,t)
[SETDIST_LE_SING] Theorem
⊢ ∀s t x. x ∈ s ⇒ setdist (s,t) ≤ setdist ({x},t)
[SETDIST_LINEAR_IMAGE] Theorem
⊢ ∀f s t.
linear f ∧ (∀x. abs (f x) = abs x) ⇒
setdist (IMAGE f s,IMAGE f t) = setdist (s,t)
[SETDIST_LIPSCHITZ] Theorem
⊢ ∀s t x y. abs (setdist ({x},s) − setdist ({y},s)) ≤ dist (x,y)
[SETDIST_POS_LE] Theorem
⊢ ∀s t. 0 ≤ setdist (s,t)
[SETDIST_REFL] Theorem
⊢ ∀s. setdist (s,s) = 0
[SETDIST_SINGS] Theorem
⊢ ∀x y. setdist ({x},{y}) = dist (x,y)
[SETDIST_SING_FRONTIER] Theorem
⊢ ∀s x. x ∉ s ⇒ setdist ({x},frontier s) = setdist ({x},s)
[SETDIST_SING_FRONTIER_CASES] Theorem
⊢ ∀s x.
setdist ({x},s) = if x ∈ s then 0 else setdist ({x},frontier s)
[SETDIST_SING_IN_SET] Theorem
⊢ ∀x s. x ∈ s ⇒ setdist ({x},s) = 0
[SETDIST_SING_LE_HAUSDIST] Theorem
⊢ ∀s t x.
bounded s ∧ bounded t ∧ x ∈ s ⇒ setdist ({x},t) ≤ hausdist (s,t)
[SETDIST_SING_TRIANGLE] Theorem
⊢ ∀s x y. abs (setdist ({x},s) − setdist ({y},s)) ≤ dist (x,y)
[SETDIST_SUBSETS_EQ] Theorem
⊢ ∀s t s' t'.
s' ⊆ s ∧ t' ⊆ t ∧
(∀x y.
x ∈ s ∧ y ∈ t ⇒
∃x' y'. x' ∈ s' ∧ y' ∈ t' ∧ dist (x',y') ≤ dist (x,y)) ⇒
setdist (s',t') = setdist (s,t)
[SETDIST_SUBSET_LEFT] Theorem
⊢ ∀s t u. s ≠ ∅ ∧ s ⊆ t ⇒ setdist (t,u) ≤ setdist (s,u)
[SETDIST_SUBSET_RIGHT] Theorem
⊢ ∀s t u. t ≠ ∅ ∧ t ⊆ u ⇒ setdist (s,u) ≤ setdist (s,t)
[SETDIST_SYM] Theorem
⊢ ∀s t. setdist (s,t) = setdist (t,s)
[SETDIST_TRANSLATION] Theorem
⊢ ∀a s t.
setdist (IMAGE (λx. a + x) s,IMAGE (λx. a + x) t) = setdist (s,t)
[SETDIST_TRIANGLE] Theorem
⊢ ∀s a t. setdist (s,t) ≤ setdist (s,{a}) + setdist ({a},t)
[SETDIST_UNIQUE] Theorem
⊢ ∀s t a b d.
a ∈ s ∧ b ∈ t ∧ dist (a,b) = d ∧
(∀x y. x ∈ s ∧ y ∈ t ⇒ dist (a,b) ≤ dist (x,y)) ⇒
setdist (s,t) = d
[SETDIST_UNIV] Theorem
⊢ (∀s. setdist (s,𝕌(:real)) = 0) ∧ ∀t. setdist (𝕌(:real),t) = 0
[SETDIST_ZERO] Theorem
⊢ ∀s t. ¬DISJOINT s t ⇒ setdist (s,t) = 0
[SETDIST_ZERO_STRONG] Theorem
⊢ ∀s t. ¬DISJOINT (closure s) (closure t) ⇒ setdist (s,t) = 0
[SET_DIFF_FRONTIER] Theorem
⊢ ∀s. s DIFF frontier s = interior s
[SPANNING_SUBSET_INDEPENDENT] Theorem
⊢ ∀s t. t ⊆ s ∧ independent s ∧ s ⊆ span t ⇒ s = t
[SPAN_0] Theorem
⊢ 0 ∈ span s
[SPAN_ADD] Theorem
⊢ ∀x y s. x ∈ span s ∧ y ∈ span s ⇒ x + y ∈ span s
[SPAN_ADD_EQ] Theorem
⊢ ∀s x y. x ∈ span s ⇒ (x + y ∈ span s ⇔ y ∈ span s)
[SPAN_BREAKDOWN] Theorem
⊢ ∀b s a. b ∈ s ∧ a ∈ span s ⇒ ∃k. a − k * b ∈ span (s DELETE b)
[SPAN_BREAKDOWN_EQ] Theorem
⊢ ∀a s. x ∈ span (a INSERT s) ⇔ ∃k. x − k * a ∈ span s
[SPAN_CARD_GE_DIM] Theorem
⊢ ∀v b. v ⊆ span b ∧ FINITE b ⇒ dim v ≤ CARD b
[SPAN_CLAUSES] Theorem
⊢ (∀a s. a ∈ s ⇒ a ∈ span s) ∧ 0 ∈ span s ∧
(∀x y s. x ∈ span s ∧ y ∈ span s ⇒ x + y ∈ span s) ∧
∀x c s. x ∈ span s ⇒ c * x ∈ span s
[SPAN_EMPTY] Theorem
⊢ span ∅ = {0}
[SPAN_EQ_SELF] Theorem
⊢ ∀s. span s = s ⇔ subspace s
[SPAN_EXPLICIT] Theorem
⊢ ∀p. span p = {y | ∃s u. FINITE s ∧ s ⊆ p ∧ sum s (λv. u v * v) = y}
[SPAN_INC] Theorem
⊢ ∀s. s ⊆ span s
[SPAN_INDUCT] Theorem
⊢ ∀s h. (∀x. x ∈ s ⇒ x ∈ h) ∧ subspace h ⇒ ∀x. x ∈ span s ⇒ h x
[SPAN_INDUCT_ALT] Theorem
⊢ ∀s h.
h 0 ∧ (∀c x y. x ∈ s ∧ h y ⇒ h (c * x + y)) ⇒
∀x. x ∈ span s ⇒ h x
[SPAN_LINEAR_IMAGE] Theorem
⊢ ∀f s. linear f ⇒ span (IMAGE f s) = IMAGE f (span s)
[SPAN_MONO] Theorem
⊢ ∀s t. s ⊆ t ⇒ span s ⊆ span t
[SPAN_MUL] Theorem
⊢ ∀x c s. x ∈ span s ⇒ c * x ∈ span s
[SPAN_MUL_EQ] Theorem
⊢ ∀x c s. c ≠ 0 ⇒ (c * x ∈ span s ⇔ x ∈ span s)
[SPAN_NEG] Theorem
⊢ ∀x s. x ∈ span s ⇒ -x ∈ span s
[SPAN_NEG_EQ] Theorem
⊢ ∀x s. -x ∈ span s ⇔ x ∈ span s
[SPAN_SPAN] Theorem
⊢ ∀s. span (span s) = span s
[SPAN_STDBASIS] Theorem
⊢ span {i | 1 ≤ i ∧ i ≤ 1} = 𝕌(:real)
[SPAN_SUB] Theorem
⊢ ∀x y s. x ∈ span s ∧ y ∈ span s ⇒ x − y ∈ span s
[SPAN_SUBSET_SUBSPACE] Theorem
⊢ ∀s t. s ⊆ t ∧ subspace t ⇒ span s ⊆ t
[SPAN_SUBSPACE] Theorem
⊢ ∀b s. b ⊆ s ∧ s ⊆ span b ∧ subspace s ⇒ span b = s
[SPAN_SUM] Theorem
⊢ ∀s f t. FINITE t ∧ (∀x. x ∈ t ⇒ f x ∈ span s) ⇒ sum t f ∈ span s
[SPAN_SUPERSET] Theorem
⊢ ∀x. x ∈ s ⇒ x ∈ span s
[SPAN_TRANS] Theorem
⊢ ∀x y s. x ∈ span s ∧ y ∈ span (x INSERT s) ⇒ y ∈ span s
[SPAN_UNION] Theorem
⊢ ∀s t. span (s ∪ t) = {x + y | x ∈ span s ∧ y ∈ span t}
[SPAN_UNION_SUBSET] Theorem
⊢ ∀s t. span s ∪ span t ⊆ span (s ∪ t)
[SPAN_UNIV] Theorem
⊢ span 𝕌(:real) = 𝕌(:real)
[SPHERE] Theorem
⊢ ∀a r. sphere (a,r) = if r < 0 then ∅ else {a − r; a + r}
[SPHERE_EMPTY] Theorem
⊢ ∀a r. r < 0 ⇒ sphere (a,r) = ∅
[SPHERE_EQ_EMPTY] Theorem
⊢ ∀a r. sphere (a,r) = ∅ ⇔ r < 0
[SPHERE_EQ_SING] Theorem
⊢ ∀a r x. sphere (a,r) = {x} ⇔ x = a ∧ r = 0
[SPHERE_LINEAR_IMAGE] Theorem
⊢ ∀f x r.
linear f ∧ (∀y. ∃x. f x = y) ∧ (∀x. abs (f x) = abs x) ⇒
sphere (f x,r) = IMAGE f (sphere (x,r))
[SPHERE_SING] Theorem
⊢ ∀x e. e = 0 ⇒ sphere (x,e) = {x}
[SPHERE_SUBSET_CBALL] Theorem
⊢ ∀x e. sphere (x,e) ⊆ cball (x,e)
[SPHERE_TRANSLATION] Theorem
⊢ ∀a x r. sphere (a + x,r) = IMAGE (λy. a + y) (sphere (x,r))
[SPHERE_UNION_BALL] Theorem
⊢ ∀a r. sphere (a,r) ∪ ball (a,r) = cball (a,r)
[SUBORDINATE_PARTITION_OF_UNITY] Theorem
⊢ ∀c s.
s ⊆ BIGUNION c ∧ (∀u. u ∈ c ⇒ open u) ∧
(∀x. x ∈ s ⇒ ∃v. open v ∧ x ∈ v ∧ FINITE {u | u ∈ c ∧ u ∩ v ≠ ∅}) ⇒
∃f. (∀u. u ∈ c ⇒ f u continuous_on s ∧ ∀x. x ∈ s ⇒ 0 ≤ f u x) ∧
(∀x u. u ∈ c ∧ x ∈ s ∧ x ∉ u ⇒ f u x = 0) ∧
(∀x. x ∈ s ⇒ sum c (λu. f u x) = 1) ∧
∀x. x ∈ s ⇒
∃n. open n ∧ x ∈ n ∧
FINITE {u | u ∈ c ∧ ¬∀x. x ∈ n ⇒ f u x = 0}
[SUBSET_BALL] Theorem
⊢ ∀x d e. d ≤ e ⇒ ball (x,d) ⊆ ball (x,e)
[SUBSET_BALLS] Theorem
⊢ (∀a a' r r'.
ball (a,r) ⊆ ball (a',r') ⇔ dist (a,a') + r ≤ r' ∨ r ≤ 0) ∧
(∀a a' r r'.
ball (a,r) ⊆ cball (a',r') ⇔ dist (a,a') + r ≤ r' ∨ r ≤ 0) ∧
(∀a a' r r'.
cball (a,r) ⊆ ball (a',r') ⇔ dist (a,a') + r < r' ∨ r < 0) ∧
∀a a' r r'.
cball (a,r) ⊆ cball (a',r') ⇔ dist (a,a') + r ≤ r' ∨ r < 0
[SUBSET_CBALL] Theorem
⊢ ∀x d e. d ≤ e ⇒ cball (x,d) ⊆ cball (x,e)
[SUBSET_CLOSURE] Theorem
⊢ ∀s t. s ⊆ t ⇒ closure s ⊆ closure t
[SUBSET_INTERIOR] Theorem
⊢ ∀s t. s ⊆ t ⇒ interior s ⊆ interior t
[SUBSET_INTERIOR_EQ] Theorem
⊢ ∀s. s ⊆ interior s ⇔ open s
[SUBSET_INTERVAL] Theorem
⊢ (interval [(c,d)] ⊆ interval [(a,b)] ⇔ c ≤ d ⇒ a ≤ c ∧ d ≤ b) ∧
(interval [(c,d)] ⊆ interval (a,b) ⇔ c ≤ d ⇒ a < c ∧ d < b) ∧
(interval (c,d) ⊆ interval [(a,b)] ⇔ c < d ⇒ a ≤ c ∧ d ≤ b) ∧
(interval (c,d) ⊆ interval (a,b) ⇔ c < d ⇒ a ≤ c ∧ d ≤ b)
[SUBSET_INTERVAL_IMP] Theorem
⊢ (a ≤ c ∧ d ≤ b ⇒ interval [(c,d)] ⊆ interval [(a,b)]) ∧
(a < c ∧ d < b ⇒ interval [(c,d)] ⊆ interval (a,b)) ∧
(a ≤ c ∧ d ≤ b ⇒ interval (c,d) ⊆ interval [(a,b)]) ∧
(a ≤ c ∧ d ≤ b ⇒ interval (c,d) ⊆ interval (a,b))
[SUBSPACE_0] Theorem
⊢ subspace s ⇒ 0 ∈ s
[SUBSPACE_ADD] Theorem
⊢ ∀x y s. subspace s ∧ x ∈ s ∧ y ∈ s ⇒ x + y ∈ s
[SUBSPACE_BIGINTER] Theorem
⊢ ∀f. (∀s. s ∈ f ⇒ subspace s) ⇒ subspace (BIGINTER f)
[SUBSPACE_BOUNDED_EQ_TRIVIAL] Theorem
⊢ ∀s. subspace s ⇒ (bounded s ⇔ s = {0})
[SUBSPACE_IMP_NONEMPTY] Theorem
⊢ ∀s. subspace s ⇒ s ≠ ∅
[SUBSPACE_INTER] Theorem
⊢ ∀s t. subspace s ∧ subspace t ⇒ subspace (s ∩ t)
[SUBSPACE_KERNEL] Theorem
⊢ ∀f. linear f ⇒ subspace {x | f x = 0}
[SUBSPACE_LINEAR_IMAGE] Theorem
⊢ ∀f s. linear f ∧ subspace s ⇒ subspace (IMAGE f s)
[SUBSPACE_LINEAR_PREIMAGE] Theorem
⊢ ∀f s. linear f ∧ subspace s ⇒ subspace {x | f x ∈ s}
[SUBSPACE_MUL] Theorem
⊢ ∀x c s. subspace s ∧ x ∈ s ⇒ c * x ∈ s
[SUBSPACE_NEG] Theorem
⊢ ∀x s. subspace s ∧ x ∈ s ⇒ -x ∈ s
[SUBSPACE_SPAN] Theorem
⊢ ∀s. subspace (span s)
[SUBSPACE_SUB] Theorem
⊢ ∀x y s. subspace s ∧ x ∈ s ∧ y ∈ s ⇒ x − y ∈ s
[SUBSPACE_SUBSTANDARD] Theorem
⊢ subspace {x | x = 0}
[SUBSPACE_SUM] Theorem
⊢ ∀s f t. subspace s ∧ FINITE t ∧ (∀x. x ∈ t ⇒ f x ∈ s) ⇒ sum t f ∈ s
[SUBSPACE_SUMS] Theorem
⊢ ∀s t. subspace s ∧ subspace t ⇒ subspace {x + y | x ∈ s ∧ y ∈ t}
[SUBSPACE_TRANSLATION_SELF] Theorem
⊢ ∀s a. subspace s ∧ a ∈ s ⇒ IMAGE (λx. a + x) s = s
[SUBSPACE_TRANSLATION_SELF_EQ] Theorem
⊢ ∀s a. subspace s ⇒ (IMAGE (λx. a + x) s = s ⇔ a ∈ s)
[SUBSPACE_TRIVIAL] Theorem
⊢ subspace {0}
[SUBSPACE_UNION_CHAIN] Theorem
⊢ ∀s t. subspace s ∧ subspace t ∧ subspace (s ∪ t) ⇒ s ⊆ t ∨ t ⊆ s
[SUBSPACE_UNIV] Theorem
⊢ subspace 𝕌(:real)
[SUMMABLE_0] Theorem
⊢ ∀s. summable s (λn. 0)
[SUMMABLE_ADD] Theorem
⊢ ∀x y s. summable s x ∧ summable s y ⇒ summable s (λn. x n + y n)
[SUMMABLE_BILINEAR_PARTIAL_PRE] Theorem
⊢ ∀f g h l k.
bilinear h ∧ ((λn. h (f (n + 1)) (g n)) ⟶ l) sequentially ∧
summable (from k) (λn. h (f (n + 1) − f n) (g n)) ⇒
summable (from k) (λn. h (f n) (g n − g (n − 1)))
[SUMMABLE_CAUCHY] Theorem
⊢ ∀f s.
summable s f ⇔
∀e. 0 < e ⇒ ∃N. ∀m n. m ≥ N ⇒ abs (sum (s ∩ {m .. n}) f) < e
[SUMMABLE_CMUL] Theorem
⊢ ∀s x c. summable s x ⇒ summable s (λn. c * x n)
[SUMMABLE_COMPARISON] Theorem
⊢ ∀f g s.
summable s g ∧ (∃N. ∀n. n ≥ N ∧ n ∈ s ⇒ abs (f n) ≤ g n) ⇒
summable s f
[SUMMABLE_COMPONENT] Theorem
⊢ ∀f s. summable s f ⇒ summable s (λi. f i)
[SUMMABLE_EQ] Theorem
⊢ ∀f g k. (∀x. x ∈ k ⇒ f x = g x) ∧ summable k f ⇒ summable k g
[SUMMABLE_EQ_COFINITE] Theorem
⊢ ∀f s t.
FINITE (s DIFF t ∪ (t DIFF s)) ∧ summable s f ⇒ summable t f
[SUMMABLE_EQ_EVENTUALLY] Theorem
⊢ ∀f g k.
(∃N. ∀n. N ≤ n ∧ n ∈ k ⇒ f n = g n) ∧ summable k f ⇒ summable k g
[SUMMABLE_FROM_ELSEWHERE] Theorem
⊢ ∀f m n. summable (from m) f ⇒ summable (from n) f
[SUMMABLE_IFF] Theorem
⊢ ∀f g k. (∀x. x ∈ k ⇒ f x = g x) ⇒ (summable k f ⇔ summable k g)
[SUMMABLE_IFF_COFINITE] Theorem
⊢ ∀f s t.
FINITE (s DIFF t ∪ (t DIFF s)) ⇒ (summable s f ⇔ summable t f)
[SUMMABLE_IFF_EVENTUALLY] Theorem
⊢ ∀f g k.
(∃N. ∀n. N ≤ n ∧ n ∈ k ⇒ f n = g n) ⇒
(summable k f ⇔ summable k g)
[SUMMABLE_IMP_BOUNDED] Theorem
⊢ ∀f k. summable k f ⇒ bounded (IMAGE f k)
[SUMMABLE_IMP_SUMS_BOUNDED] Theorem
⊢ ∀f k. summable (from k) f ⇒ bounded {sum {k .. n} f | n ∈ 𝕌(:num)}
[SUMMABLE_IMP_TOZERO] Theorem
⊢ ∀f k.
summable k f ⇒ ((λn. if n ∈ k then f n else 0) ⟶ 0) sequentially
[SUMMABLE_LINEAR] Theorem
⊢ ∀f h s. summable s f ∧ linear h ⇒ summable s (λn. h (f n))
[SUMMABLE_NEG] Theorem
⊢ ∀x s. summable s x ⇒ summable s (λn. -x n)
[SUMMABLE_REARRANGE] Theorem
⊢ ∀x s p.
summable s (λn. abs (x n)) ∧ p permutes s ⇒ summable s (x ∘ p)
[SUMMABLE_REINDEX] Theorem
⊢ ∀k a n.
summable (from n) (λx. a (x + k)) ⇔ summable (from (n + k)) a
[SUMMABLE_RESTRICT] Theorem
⊢ ∀f k.
summable 𝕌(:num) (λn. if n ∈ k then f n else 0) ⇔ summable k f
[SUMMABLE_SUB] Theorem
⊢ ∀x y s. summable s x ∧ summable s y ⇒ summable s (λn. x n − y n)
[SUMMABLE_SUBSET] Theorem
⊢ ∀x s t.
s ⊆ t ∧ summable t (λi. if i ∈ s then x i else 0) ⇒ summable s x
[SUMMABLE_SUBSET_ABSCONV] Theorem
⊢ ∀x s t.
summable s (λn. abs (x n)) ∧ t ⊆ s ⇒ summable t (λn. abs (x n))
[SUMMABLE_TRIVIAL] Theorem
⊢ ∀f. summable ∅ f
[SUMS_0] Theorem
⊢ ∀f s. (∀n. n ∈ s ⇒ f n = 0) ⇒ (f sums 0) s
[SUMS_EQ] Theorem
⊢ ∀f g k. (∀x. x ∈ k ⇒ f x = g x) ∧ (f sums l) k ⇒ (g sums l) k
[SUMS_FINITE_DIFF] Theorem
⊢ ∀f t s l.
t ⊆ s ∧ FINITE t ∧ (f sums l) s ⇒ (f sums l − sum t f) (s DIFF t)
[SUMS_FINITE_UNION] Theorem
⊢ ∀f s t l.
FINITE t ∧ (f sums l) s ⇒ (f sums l + sum (t DIFF s) f) (s ∪ t)
[SUMS_IFF] Theorem
⊢ ∀f g k. (∀x. x ∈ k ⇒ f x = g x) ⇒ ((f sums l) k ⇔ (g sums l) k)
[SUMS_INFSUM] Theorem
⊢ ∀f s. (f sums suminf s f) s ⇔ summable s f
[SUMS_INTERVALS] Theorem
⊢ (∀a b c d.
interval [(a,b)] ≠ ∅ ∧ interval [(c,d)] ≠ ∅ ⇒
{x + y | x ∈ interval [(a,b)] ∧ y ∈ interval [(c,d)]} =
interval [(a + c,b + d)]) ∧
∀a b c d.
interval (a,b) ≠ ∅ ∧ interval (c,d) ≠ ∅ ⇒
{x + y | x ∈ interval (a,b) ∧ y ∈ interval (c,d)} =
interval (a + c,b + d)
[SUMS_LIM] Theorem
⊢ ∀f s.
(f sums lim sequentially (λn. sum (s ∩ {0 .. n}) f)) s ⇔
summable s f
[SUMS_OFFSET] Theorem
⊢ ∀f l m n.
(f sums l) (from m) ∧ 0 < n ∧ m ≤ n ⇒
(f sums l − sum {m .. n − 1} f) (from n)
[SUMS_OFFSET_REV] Theorem
⊢ ∀f l m n.
(f sums l) (from m) ∧ 0 < m ∧ n ≤ m ⇒
(f sums l + sum {n .. m − 1} f) (from n)
[SUMS_REINDEX] Theorem
⊢ ∀k a l n.
((λx. a (x + k)) sums l) (from n) ⇔ (a sums l) (from (n + k))
[SUMS_REINDEX_GEN] Theorem
⊢ ∀k a l s.
((λx. a (x + k)) sums l) s ⇔ (a sums l) (IMAGE (λi. i + k) s)
[SUMS_SUMMABLE] Theorem
⊢ ∀f l s. (f sums l) s ⇒ summable s f
[SUM_DIFF_LEMMA] Theorem
⊢ ∀f k m n.
m ≤ n ⇒
sum (k ∩ {0 .. n}) f − sum (k ∩ {0 .. m}) f =
sum (k ∩ {m + 1 .. n}) f
[SUP_INSERT] Theorem
⊢ ∀x s.
bounded s ⇒ sup (x INSERT s) = if s = ∅ then x else max x (sup s)
[SURJECTIVE_IMAGE_EQ] Theorem
⊢ ∀s t.
(∀y. y ∈ t ⇒ ∃x. f x = y) ∧ (∀x. f x ∈ t ⇔ x ∈ s) ⇒ IMAGE f s = t
[SYMMETRIC_CLOSURE] Theorem
⊢ ∀s. (∀x. x ∈ s ⇒ -x ∈ s) ⇒ ∀x. x ∈ closure s ⇒ -x ∈ closure s
[SYMMETRIC_INTERIOR] Theorem
⊢ ∀s. (∀x. x ∈ s ⇒ -x ∈ s) ⇒ ∀x. x ∈ interior s ⇒ -x ∈ interior s
[SYMMETRIC_LINEAR_IMAGE] Theorem
⊢ ∀f s.
(∀x. x ∈ s ⇒ -x ∈ s) ∧ linear f ⇒
∀x. x ∈ IMAGE f s ⇒ -x ∈ IMAGE f s
[TENDSTO_LIM] Theorem
⊢ ∀net f l. ¬trivial_limit net ∧ (f ⟶ l) net ⇒ lim net f = l
[TOPSPACE_EUCLIDEAN] Theorem
⊢ topspace euclidean = 𝕌(:real)
[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] Theorem
⊢ ∀s. topspace (subtopology euclidean s) = s
[TRANSITIVE_STEPWISE_LT] Theorem
⊢ ∀R. (∀x y z. R x y ∧ R y z ⇒ R x z) ∧ (∀n. R n (SUC n)) ⇒
∀m n. m < n ⇒ R m n
[TRANSITIVE_STEPWISE_LT_EQ] Theorem
⊢ ∀R. (∀x y z. R x y ∧ R y z ⇒ R x z) ⇒
((∀m n. m < n ⇒ R m n) ⇔ ∀n. R n (SUC n))
[TRANSLATION_DIFF] Theorem
⊢ ∀s t.
IMAGE (λx. a + x) (s DIFF t) =
IMAGE (λx. a + x) s DIFF IMAGE (λx. a + x) t
[TRIVIAL_LIMIT_AT] Theorem
⊢ ∀a. ¬trivial_limit (at a)
[TRIVIAL_LIMIT_AT_INFINITY] Theorem
⊢ ¬trivial_limit at_infinity
[TRIVIAL_LIMIT_AT_NEGINFINITY] Theorem
⊢ ¬trivial_limit at_neginfinity
[TRIVIAL_LIMIT_AT_POSINFINITY] Theorem
⊢ ¬trivial_limit at_posinfinity
[TRIVIAL_LIMIT_SEQUENTIALLY] Theorem
⊢ ¬trivial_limit sequentially
[TRIVIAL_LIMIT_WITHIN] Theorem
⊢ ∀a. trivial_limit (at a within s) ⇔ ¬(a limit_point_of s)
[UNBOUNDED_HALFSPACE_COMPONENT_GE] Theorem
⊢ ∀a. ¬bounded {x | x ≥ a}
[UNBOUNDED_HALFSPACE_COMPONENT_GT] Theorem
⊢ ∀a. ¬bounded {x | x > a}
[UNBOUNDED_HALFSPACE_COMPONENT_LE] Theorem
⊢ ∀a. ¬bounded {x | x ≤ a}
[UNBOUNDED_HALFSPACE_COMPONENT_LT] Theorem
⊢ ∀a. ¬bounded {x | x < a}
[UNBOUNDED_INTER_COBOUNDED] Theorem
⊢ ∀s t. ¬bounded s ∧ bounded (𝕌(:real) DIFF t) ⇒ s ∩ t ≠ ∅
[UNCOUNTABLE_EUCLIDEAN] Theorem
⊢ uncountable 𝕌(:real)
[UNCOUNTABLE_INTERVAL] Theorem
⊢ (∀a b. interval (a,b) ≠ ∅ ⇒ uncountable (interval [(a,b)])) ∧
∀a b. interval (a,b) ≠ ∅ ⇒ uncountable (interval (a,b))
[UNCOUNTABLE_OPEN] Theorem
⊢ ∀s. open s ∧ s ≠ ∅ ⇒ uncountable s
[UNCOUNTABLE_REAL] Theorem
⊢ uncountable 𝕌(:real)
[UNIFORMLY_CAUCHY_IMP_UNIFORMLY_CONVERGENT] Theorem
⊢ ∀P s l.
(∀e. 0 < e ⇒
∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ P x ⇒ dist (s m x,s n x) < e) ∧
(∀x. P x ⇒ ∀e. 0 < e ⇒ ∃N. ∀n. N ≤ n ⇒ dist (s n x,l x) < e) ⇒
∀e. 0 < e ⇒ ∃N. ∀n x. N ≤ n ∧ P x ⇒ dist (s n x,l x) < e
[UNIFORMLY_CONTINUOUS_EXTENDS_TO_CLOSURE] Theorem
⊢ ∀f s.
f uniformly_continuous_on s ⇒
∃g. g uniformly_continuous_on closure s ∧
(∀x. x ∈ s ⇒ g x = f x) ∧
∀h. h continuous_on closure s ∧ (∀x. x ∈ s ⇒ h x = f x) ⇒
∀x. x ∈ closure s ⇒ h x = g x
[UNIFORMLY_CONTINUOUS_IMP_CAUCHY_CONTINUOUS] Theorem
⊢ ∀f s.
f uniformly_continuous_on s ⇒
∀x. cauchy x ∧ (∀n. x n ∈ s) ⇒ cauchy (f ∘ x)
[UNIFORMLY_CONTINUOUS_IMP_CONTINUOUS] Theorem
⊢ ∀f s. f uniformly_continuous_on s ⇒ f continuous_on s
[UNIFORMLY_CONTINUOUS_ON_ADD] Theorem
⊢ ∀f g s.
f uniformly_continuous_on s ∧ g uniformly_continuous_on s ⇒
(λx. f x + g x) uniformly_continuous_on s
[UNIFORMLY_CONTINUOUS_ON_CLOSURE] Theorem
⊢ ∀f s.
f uniformly_continuous_on s ∧ f continuous_on closure s ⇒
f uniformly_continuous_on closure s
[UNIFORMLY_CONTINUOUS_ON_CMUL] Theorem
⊢ ∀f c s.
f uniformly_continuous_on s ⇒
(λx. c * f x) uniformly_continuous_on s
[UNIFORMLY_CONTINUOUS_ON_COMPOSE] Theorem
⊢ ∀f g s.
f uniformly_continuous_on s ∧ g uniformly_continuous_on IMAGE f s ⇒
g ∘ f uniformly_continuous_on s
[UNIFORMLY_CONTINUOUS_ON_CONST] Theorem
⊢ ∀s c. (λx. c) uniformly_continuous_on s
[UNIFORMLY_CONTINUOUS_ON_DIST_CLOSEST_POINT] Theorem
⊢ ∀s t.
closed s ∧ s ≠ ∅ ⇒
(λx. dist (x,closest_point s x)) uniformly_continuous_on t
[UNIFORMLY_CONTINUOUS_ON_EQ] Theorem
⊢ ∀f g s.
(∀x. x ∈ s ⇒ f x = g x) ∧ f uniformly_continuous_on s ⇒
g uniformly_continuous_on s
[UNIFORMLY_CONTINUOUS_ON_ID] Theorem
⊢ ∀s. (λx. x) uniformly_continuous_on s
[UNIFORMLY_CONTINUOUS_ON_MUL] Theorem
⊢ ∀f g s.
f uniformly_continuous_on s ∧ g uniformly_continuous_on s ∧
bounded (IMAGE f s) ∧ bounded (IMAGE g s) ⇒
(λx. f x * g x) uniformly_continuous_on s
[UNIFORMLY_CONTINUOUS_ON_NEG] Theorem
⊢ ∀f s.
f uniformly_continuous_on s ⇒
(λx. -f x) uniformly_continuous_on s
[UNIFORMLY_CONTINUOUS_ON_SEQUENTIALLY] Theorem
⊢ ∀f s.
f uniformly_continuous_on s ⇔
∀x y.
(∀n. x n ∈ s) ∧ (∀n. y n ∈ s) ∧
((λn. x n − y n) ⟶ 0) sequentially ⇒
((λn. f (x n) − f (y n)) ⟶ 0) sequentially
[UNIFORMLY_CONTINUOUS_ON_SETDIST] Theorem
⊢ ∀s t. (λy. setdist ({y},s)) uniformly_continuous_on t
[UNIFORMLY_CONTINUOUS_ON_SUB] Theorem
⊢ ∀f g s.
f uniformly_continuous_on s ∧ g uniformly_continuous_on s ⇒
(λx. f x − g x) uniformly_continuous_on s
[UNIFORMLY_CONTINUOUS_ON_SUBSET] Theorem
⊢ ∀f s t.
f uniformly_continuous_on s ∧ t ⊆ s ⇒ f uniformly_continuous_on t
[UNIFORMLY_CONTINUOUS_ON_SUM] Theorem
⊢ ∀t f s.
FINITE s ∧ (∀a. a ∈ s ⇒ f a uniformly_continuous_on t) ⇒
(λx. sum s (λa. f a x)) uniformly_continuous_on t
[UNIFORMLY_CONTINUOUS_ON_VMUL] Theorem
⊢ ∀s c v.
c uniformly_continuous_on s ⇒
(λx. c x * v) uniformly_continuous_on s
[UNIFORMLY_CONVERGENT_EQ_CAUCHY] Theorem
⊢ ∀P s.
(∃l. ∀e. 0 < e ⇒ ∃N. ∀n x. N ≤ n ∧ P x ⇒ dist (s n x,l x) < e) ⇔
∀e. 0 < e ⇒
∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ P x ⇒ dist (s m x,s n x) < e
[UNIFORMLY_CONVERGENT_EQ_CAUCHY_ALT] Theorem
⊢ ∀P s.
(∃l. ∀e. 0 < e ⇒ ∃N. ∀n x. N ≤ n ∧ P x ⇒ dist (s n x,l x) < e) ⇔
∀e. 0 < e ⇒
∃N. ∀m n x.
N ≤ m ∧ N ≤ n ∧ m < n ∧ P x ⇒ dist (s m x,s n x) < e
[UNIFORM_LIM_ADD] Theorem
⊢ ∀net P f g l m.
(∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (f n x − l n) < e) net) ∧
(∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (g n x − m n) < e) net) ⇒
∀e. 0 < e ⇒
eventually
(λx. ∀n. P n ⇒ abs (f n x + g n x − (l n + m n)) < e) net
[UNIFORM_LIM_BILINEAR] Theorem
⊢ ∀net P h f g l m b1 b2.
bilinear h ∧ eventually (λx. ∀n. P n ⇒ abs (l n) ≤ b1) net ∧
eventually (λx. ∀n. P n ⇒ abs (m n) ≤ b2) net ∧
(∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (f n x − l n) < e) net) ∧
(∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (g n x − m n) < e) net) ⇒
∀e. 0 < e ⇒
eventually
(λx. ∀n. P n ⇒ abs (h (f n x) (g n x) − h (l n) (m n)) < e)
net
[UNIFORM_LIM_SUB] Theorem
⊢ ∀net P f g l m.
(∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (f n x − l n) < e) net) ∧
(∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (g n x − m n) < e) net) ⇒
∀e. 0 < e ⇒
eventually
(λx. ∀n. P n ⇒ abs (f n x − g n x − (l n − m n)) < e) net
[UNION_FRONTIER] Theorem
⊢ ∀s t.
frontier s ∪ frontier t =
frontier (s ∪ t) ∪ frontier (s ∩ t) ∪ frontier s ∩ frontier t
[UNION_INTERIOR_SUBSET] Theorem
⊢ ∀s t. interior s ∪ interior t ⊆ interior (s ∪ t)
[UNIT_INTERVAL_NONEMPTY] Theorem
⊢ interval [(0,1)] ≠ ∅ ∧ interval (0,1) ≠ ∅
[UPPER_HEMICONTINUOUS] Theorem
⊢ ∀f t s.
(∀x. x ∈ s ⇒ f x ⊆ t) ⇒
((∀u. open_in (subtopology euclidean t) u ⇒
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ⇔
∀u. closed_in (subtopology euclidean t) u ⇒
closed_in (subtopology euclidean s)
{x | x ∈ s ∧ f x ∩ u ≠ ∅})
[UPPER_LOWER_HEMICONTINUOUS] Theorem
⊢ ∀f t s.
(∀x. x ∈ s ⇒ f x ⊆ t) ∧
(∀u. open_in (subtopology euclidean t) u ⇒
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ∧
(∀u. closed_in (subtopology euclidean t) u ⇒
closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ⇒
∀x e.
x ∈ s ∧ 0 < e ∧ bounded (f x) ⇒
∃d. 0 < d ∧
∀x'. x' ∈ s ∧ dist (x,x') < d ⇒ hausdist (f x,f x') < e
[UPPER_LOWER_HEMICONTINUOUS_EXPLICIT] Theorem
⊢ ∀f t s.
(∀x. x ∈ s ⇒ f x ⊆ t) ∧
(∀u. open_in (subtopology euclidean t) u ⇒
open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ∧
(∀u. closed_in (subtopology euclidean t) u ⇒
closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ⇒
∀x e.
x ∈ s ∧ 0 < e ∧ bounded (f x) ∧ f x ≠ ∅ ⇒
∃d. 0 < d ∧
∀x'.
x' ∈ s ∧ dist (x,x') < d ⇒
(∀y. y ∈ f x ⇒ ∃y'. y' ∈ f x' ∧ dist (y,y') < e) ∧
∀y'. y' ∈ f x' ⇒ ∃y. y ∈ f x ∧ dist (y',y) < e
[URYSOHN] Theorem
⊢ ∀s t a b.
closed s ∧ closed t ∧ s ∩ t = ∅ ⇒
∃f. f continuous_on 𝕌(:real) ∧ (∀x. f x ∈ segment [(a,b)]) ∧
(∀x. x ∈ s ⇒ f x = a) ∧ ∀x. x ∈ t ⇒ f x = b
[URYSOHN_LOCAL] Theorem
⊢ ∀s t u a b.
closed_in (subtopology euclidean u) s ∧
closed_in (subtopology euclidean u) t ∧ s ∩ t = ∅ ⇒
∃f. f continuous_on u ∧ (∀x. x ∈ u ⇒ f x ∈ segment [(a,b)]) ∧
(∀x. x ∈ s ⇒ f x = a) ∧ ∀x. x ∈ t ⇒ f x = b
[URYSOHN_LOCAL_STRONG] Theorem
⊢ ∀s t u a b.
closed_in (subtopology euclidean u) s ∧
closed_in (subtopology euclidean u) t ∧ s ∩ t = ∅ ∧ a ≠ b ⇒
∃f. f continuous_on u ∧ (∀x. x ∈ u ⇒ f x ∈ segment [(a,b)]) ∧
(∀x. x ∈ u ⇒ (f x = a ⇔ x ∈ s)) ∧
∀x. x ∈ u ⇒ (f x = b ⇔ x ∈ t)
[URYSOHN_STRONG] Theorem
⊢ ∀s t a b.
closed s ∧ closed t ∧ s ∩ t = ∅ ∧ a ≠ b ⇒
∃f. f continuous_on 𝕌(:real) ∧ (∀x. f x ∈ segment [(a,b)]) ∧
(∀x. f x = a ⇔ x ∈ s) ∧ ∀x. f x = b ⇔ x ∈ t
[WITHIN] Theorem
⊢ ∀n s x y. netord (n within s) x y ⇔ netord n x y ∧ x ∈ s
[WITHIN_UNIV] Theorem
⊢ ∀x. (at x within 𝕌(:real)) = at x
[WITHIN_WITHIN] Theorem
⊢ ∀net s t. ((net within s) within t) = (net within s ∩ t)
[at] Theorem
⊢ ∀a. at a = mk_net (λx y. 0 < dist (x,a) ∧ dist (x,a) ≤ dist (y,a))
[ball] Theorem
⊢ ∀x e. ball (x,e) = {y | dist (x,y) < e}
[closed_def] Theorem
⊢ ∀s. closed s ⇔ open (𝕌(:real) DIFF s)
[continuous_at] Theorem
⊢ f continuous at x ⇔
∀e. 0 < e ⇒ ∃d. 0 < d ∧ ∀x'. dist (x',x) < d ⇒ dist (f x',f x) < e
[continuous_within] Theorem
⊢ f continuous (at x within s) ⇔
∀e. 0 < e ⇒
∃d. 0 < d ∧ ∀x'. x' ∈ s ∧ dist (x',x) < d ⇒ dist (f x',f x) < e
[diameter] Theorem
⊢ ∀s. diameter s =
if s = ∅ then 0 else sup {abs (x − y) | x ∈ s ∧ y ∈ s}
[dist] Theorem
⊢ ∀x y. dist (x,y) = abs (x − y)
[euclidean] Theorem
⊢ euclidean = topology open
[fsigma] Theorem
⊢ ∀s. fsigma s ⇔
∃g. countable g ∧ (∀c. c ∈ g ⇒ closed c) ∧ BIGUNION g = s
[gdelta] Theorem
⊢ ∀s. gdelta s ⇔
∃g. countable g ∧ (∀u. u ∈ g ⇒ open u) ∧ BIGINTER g = s
[interval] Theorem
⊢ interval (a,b) = {x | a < x ∧ x < b} ∧
interval [(a,b)] = {x | a ≤ x ∧ x ≤ b}
[limit_point_of] Theorem
⊢ ∀x s.
x limit_point_of s ⇔
∀t. x ∈ t ∧ open t ⇒ ∃y. y ≠ x ∧ y ∈ s ∧ y ∈ t
[linear_alt_cmul] Theorem
⊢ ∀f. linear f ⇔ ∀c x. f (c * x) = c * f x
[linear_repr] Theorem
⊢ ∀f. linear f ⇔ ∃l. f = (λx. l * x)
[net_tybij] Theorem
⊢ (∀a. mk_net (netord a) = a) ∧
∀r. (∀x y. (∀z. r z x ⇒ r z y) ∨ ∀z. r z y ⇒ r z x) ⇔
netord (mk_net r) = r
[open_def] Theorem
⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ∀x'. dist (x',x) < e ⇒ x' ∈ s
[open_in] Theorem
⊢ ∀u s.
open_in (subtopology euclidean u) s ⇔
s ⊆ u ∧
∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ∀x'. x' ∈ u ∧ dist (x',x) < e ⇒ x' ∈ s
[segment] Theorem
⊢ segment [(a,b)] = {(1 − u) * a + u * b | 0 ≤ u ∧ u ≤ 1} ∧
segment (a,b) = segment [(a,b)] DIFF {a; b}
[setdist] Theorem
⊢ ∀s t.
setdist (s,t) =
if s = ∅ ∨ t = ∅ then 0 else inf {dist (x,y) | x ∈ s ∧ y ∈ t}
*)
end
HOL 4, Trindemossen-1