Documentation Verification Report

Basic

πŸ“ Source: Mathlib/Topology/Algebra/Group/Basic.lean

Statistics

MetricCount
DefinitionsaddCommGroupTopologicalClosure, connectedComponentOfZero, topologicalClosure, addLeft, addRight, divLeft, divRight, inv, mulLeft, mulRight, neg, prodAddUnits, prodUnits, shearAddRight, shearMulRight, subLeft, subRight, commGroupTopologicalClosure, connectedComponentOfOne, topologicalClosure, nhdsAddHom, nhdsMulHom, toAddUnits_homeomorph, toUnits_homeomorph
24
TheoremscontinuousConstSMul_int, continuousSMul_int, isOpenQuotientMap_of_isQuotientMap, coe_topologicalClosure_bot, instIsTopologicalAddGroupSubtypeMem, isClosed_topologicalClosure, is_normal_topologicalClosure, le_topologicalClosure, properlyDiscontinuousVAdd_of_tendsto_cofinite, properlyDiscontinuousVAdd_opposite_of_tendsto_cofinite, topologicalClosure_coe, topologicalClosure_minimal, instCompactSpaceOfT1SpaceOfContinuousAdd, instIsTopologicalAddGroupOfContinuousAdd, instLocallyCompactSpaceOfT1SpaceOfContinuousAdd, instWeaklyLocallyCompactSpaceOfT1SpaceOfContinuousAdd, isClosedEmbedding_embedProduct, isEmbedding_val, range_embedProduct, units_continuousConstSMul, of_coeHom_comp, of_inv, of_neg, zpow, zsmul, of_inv, of_neg, zpow, zsmul, induced, of_nhds_one, induced, of_nhds_zero, of_inv, of_neg, zpow, zsmul, zpow, zsmul, topologicalClosure_map_addSubgroup, topologicalClosure_map_subgroup, nhds_of_one, nhds_of_zero, const_div', const_sub, div_const', sub_const, zpow, zsmul, inv_nhdsGT, inv_nhdsLT, inv_nhdsNE, map_add_left_nhdsGT, map_add_left_nhdsLT, map_add_left_nhdsNE, map_add_right_nhdsGT, map_add_right_nhdsLT, map_add_right_nhdsNE, map_mul_left_nhdsGT, map_mul_left_nhdsLT, map_mul_left_nhdsNE, map_mul_right_nhdsGT, map_mul_right_nhdsLT, map_mul_right_nhdsNE, neg_nhdsGT, neg_nhdsLT, neg_nhdsNE, tendsto_add_const_iff, tendsto_const_add_iff, tendsto_const_div_iff, tendsto_const_div_iff', tendsto_const_mul_iff, tendsto_const_sub_iff, tendsto_div_const_iff, tendsto_div_const_iff', tendsto_mul_const_iff, tendsto_sub_const_iff, addLeft_symm, addRight_symm, coe_addLeft, coe_addRight, coe_inv, coe_mulLeft, coe_mulRight, coe_neg, divLeft_apply, divLeft_symm_apply, divRight_apply, divRight_symm_apply, mulLeft_symm, mulRight_symm, shearAddRight_coe, shearAddRight_symm_coe, shearMulRight_coe, shearMulRight_symm_coe, subLeft_apply, subLeft_symm_apply, subRight_apply, subRight_symm_apply, symm_inv, symm_neg, inv, neg, zpow, zsmul, inv, leftCoset, left_addCoset, neg, rightCoset, right_addCoset, inv, neg, inv, leftCoset, left_addCoset, neg, rightCoset, right_addCoset, continuous_addConj_prod, continuous_conj, continuous_conj', exists_antitone_basis_nhds_zero, ext, ext_iff, isInducing_iff_nhds_zero, isOpenMap_iff_nhds_zero, of_comm_of_nhds_zero, of_nhds_zero, of_nhds_zero', t1Space, continuous_conj, continuous_conj', continuous_conj_prod, exists_antitone_basis_nhds_one, ext, ext_iff, isInducing_iff_nhds_one, isOpenMap_iff_nhds_one, of_comm_of_nhds_one, of_nhds_one, of_nhds_one', t1Space, isOpenQuotientMap_of_isQuotientMap, instContinuousInv, instContinuousNeg, instIsTopologicalAddGroup, instIsTopologicalGroup, continuousInv, continuousNeg, has_continuous_inv', has_continuous_neg', topologicalAddGroup, topologicalGroup, continuousInv, continuousNeg, instIsTopologicalAddGroup, instIsTopologicalGroup, sigmaCompactSpace, sigmaCompactSpace, isClosed_centralizer, inv, neg, zpow, zsmul, coe_topologicalClosure_bot, instIsTopologicalGroupSubtypeMem, isClosed_topologicalClosure, is_normal_topologicalClosure, le_topologicalClosure, properlyDiscontinuousSMul_of_tendsto_cofinite, properlyDiscontinuousSMul_opposite_of_tendsto_cofinite, topologicalClosure_coe, topologicalClosure_minimal, units_isCompact, isOpenMap_iff_nhds_one, isOpenMap_iff_nhds_zero, continuousInv, continuousNeg, topologicalAddGroup, topologicalGroup, instCompactSpaceOfT1SpaceOfContinuousMul, instIsTopologicalGroupOfContinuousMul, instLocallyCompactSpaceOfT1SpaceOfContinuousMul, instWeaklyLocallyCompactSpaceOfT1SpaceOfContinuousMul, isClosedEmbedding_embedProduct, isEmbedding_val, range_embedProduct, add_mem_connectedComponent_zero, compact_covered_by_add_left_translates, compact_covered_by_mul_left_translates, compact_open_separated_add_left, compact_open_separated_add_right, compact_open_separated_mul_left, compact_open_separated_mul_right, continuousAt_inv, continuousAt_inv_iff, continuousAt_neg, continuousAt_neg_iff, continuousAt_zpow, continuousAt_zsmul, continuousInv_iInf, continuousInv_inf, continuousInv_of_discreteTopology, continuousInv_sInf, continuousNeg_iInf, continuousNeg_inf, continuousNeg_of_discreteTopology, continuousNeg_sInf, continuousOn_inv, continuousOn_inv_iff, continuousOn_neg, continuousOn_neg_iff, continuousOn_zpow, continuousOn_zsmul, continuousWithinAt_inv, continuousWithinAt_neg, continuous_div_left', continuous_div_right', continuous_inv_iff, continuous_neg_iff, continuous_of_continuousAt_one, continuous_of_continuousAt_oneβ‚‚, continuous_of_continuousAt_zero, continuous_of_continuousAt_zeroβ‚‚, continuous_sub_left, continuous_sub_right, continuous_zpow, continuous_zsmul, discreteTopology_iff_isOpen_singleton_one, discreteTopology_iff_isOpen_singleton_zero, discreteTopology_of_isOpen_singleton_one, discreteTopology_of_isOpen_singleton_zero, exists_disjoint_smul_of_isCompact, exists_disjoint_vadd_of_isCompact, exists_nhds_half_neg, exists_nhds_split_inv, instContinuousInvMulOpposite, instContinuousInvMultiplicativeOfContinuousNeg, instContinuousInvULift, instContinuousNegAddOpposite, instContinuousNegAdditiveOfContinuousInv, instContinuousNegULift, instIsTopologicalAddGroupAddOpposite, instIsTopologicalAddGroupAdditiveOfIsTopologicalGroup, instIsTopologicalGroupMulOpposite, instIsTopologicalGroupMultiplicativeOfIsTopologicalAddGroup, instIsTopologicalGroupULift, inv_closure, inv_mem_connectedComponent_one, inv_mem_nhds_one, isClosedMap_add_left, isClosedMap_add_right, isClosedMap_div_left, isClosedMap_div_right, isClosedMap_inv, isClosedMap_mul_left, isClosedMap_mul_right, isClosedMap_neg, isClosedMap_sub_left, isClosedMap_sub_right, isClosed_setOf_map_inv, isClosed_setOf_map_neg, isOpenMap_add_left, isOpenMap_add_right, isOpenMap_div_left, isOpenMap_div_right, isOpenMap_inv, isOpenMap_mul_left, isOpenMap_mul_right, isOpenMap_neg, isOpenMap_sub_left, isOpenMap_sub_right, map_add_left_nhds, map_add_left_nhds_zero, map_add_right_nhds, map_add_right_nhds_zero, map_mul_left_nhds, map_mul_left_nhds_one, map_mul_right_nhds, map_mul_right_nhds_one, mem_closure_iff_nhds_one, mem_closure_iff_nhds_zero, mul_mem_connectedComponent_one, neg_closure, neg_mem_connectedComponent_zero, neg_mem_nhds_zero, nhdsAddHom_apply, nhdsMulHom_apply, nhds_add, nhds_inv, nhds_mul, nhds_neg, nhds_one_symm, nhds_one_symm', nhds_translation_add_neg, nhds_translation_div, nhds_translation_inv_mul, nhds_translation_mul_inv, nhds_translation_neg_add, nhds_translation_sub, nhds_zero_symm, nhds_zero_symm', tendsto_div_nhds_one_iff, tendsto_inv, tendsto_inv_iff, tendsto_inv_nhdsGE, tendsto_inv_nhdsGE_inv, tendsto_inv_nhdsGT, tendsto_inv_nhdsGT_inv, tendsto_inv_nhdsLE, tendsto_inv_nhdsLE_inv, tendsto_inv_nhdsLT, tendsto_inv_nhdsLT_inv, tendsto_inv_nhdsWithin_Iic_inv, tendsto_neg, tendsto_neg_iff, tendsto_neg_nhdsGE, tendsto_neg_nhdsGE_neg, tendsto_neg_nhdsGT, tendsto_neg_nhdsGT_neg, tendsto_neg_nhdsLE, tendsto_neg_nhdsLE_neg, tendsto_neg_nhdsLT, tendsto_neg_nhdsLT_neg, tendsto_sub_nhds_zero_iff, topologicalAddGroup_iInf, topologicalAddGroup_induced, topologicalAddGroup_inf, topologicalAddGroup_of_discreteTopology, topologicalAddGroup_sInf, topologicalGroup_iInf, topologicalGroup_induced, topologicalGroup_inf, topologicalGroup_of_discreteTopology, topologicalGroup_sInf
336
Total360

AddGroup

Theorems

NameKindAssumesProvesValidatesDepends On
continuousConstSMul_int πŸ“–mathematicalβ€”ContinuousConstSMul
SubNegMonoid.toZSMul
toSubNegMonoid
β€”continuous_zsmul
continuousSMul_int πŸ“–mathematicalβ€”ContinuousSMul
SubNegMonoid.toZSMul
toSubNegMonoid
instTopologicalSpaceInt
β€”continuous_prod_of_discrete_left
instDiscreteTopologyInt
continuous_zsmul

AddMonoidHom

Theorems

NameKindAssumesProvesValidatesDepends On
isOpenQuotientMap_of_isQuotientMap πŸ“–mathematicalTopology.IsQuotientMap
DFunLike.coe
IsOpenQuotientMapβ€”Topology.IsQuotientMap.surjective
Topology.IsQuotientMap.continuous
Topology.IsQuotientMap.isOpen_preimage
Set.ext
Set.mem_iUnion_of_mem
Set.iUnion_congr_Prop
mem_ker
map_add
AddMonoidHomClass.toAddHomClass
map_neg
neg_add_cancel
Set.iUnion_true
Set.mem_preimage
add_neg_cancel_left
add_zero
isOpen_biUnion
Continuous.isOpen_preimage
Continuous.fun_add
continuous_id'
continuous_const

AddSubgroup

Definitions

NameCategoryTheorems
addCommGroupTopologicalClosure πŸ“–CompOpβ€”
connectedComponentOfZero πŸ“–CompOpβ€”
topologicalClosure πŸ“–CompOp
10 mathmath: isClosed_topologicalClosure, is_normal_topologicalClosure, controlled_closure_of_complete, topologicalClosure_minimal, MeasureTheory.Lp.boundedContinuousFunction_topologicalClosure, coe_topologicalClosure_bot, controlled_closure_range_of_complete, le_topologicalClosure, mapClusterPt_atTop_nsmul_iff_mem_topologicalClosure_zmultiples, topologicalClosure_coe

Theorems

NameKindAssumesProvesValidatesDepends On
coe_topologicalClosure_bot πŸ“–mathematicalβ€”SetLike.coe
AddSubgroup
instSetLike
topologicalClosure
Bot.bot
instBot
closure
Set
Set.instSingletonSet
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”β€”
instIsTopologicalAddGroupSubtypeMem πŸ“–mathematicalβ€”IsTopologicalAddGroup
AddSubgroup
SetLike.instMembership
instSetLike
instTopologicalSpaceSubtype
toAddGroup
β€”Topology.IsInducing.topologicalAddGroup
AddMonoidHom.instAddMonoidHomClass
Topology.IsInducing.subtypeVal
isClosed_topologicalClosure πŸ“–mathematicalβ€”IsClosed
SetLike.coe
AddSubgroup
instSetLike
topologicalClosure
β€”isClosed_closure
is_normal_topologicalClosure πŸ“–mathematicalβ€”Normal
topologicalClosure
β€”map_mem_closure
IsTopologicalAddGroup.continuous_conj
IsTopologicalAddGroup.toContinuousAdd
Normal.conj_mem
le_topologicalClosure πŸ“–mathematicalβ€”AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosure
β€”subset_closure
properlyDiscontinuousVAdd_of_tendsto_cofinite πŸ“–mathematicalFilter.Tendsto
AddSubgroup
SetLike.instMembership
instSetLike
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
toAddGroup
AddMonoidHom.instFunLike
subtype
Filter.cofinite
Filter.cocompact
ProperlyDiscontinuousVAdd
AddSemigroupAction.toVAdd
AddMonoid.toAddSemigroup
AddAction.toAddSemigroupAction
instAddAction
AddMonoid.toAddAction
β€”IsCompact.compl_mem_cocompact
IsCompact.image
IsCompact.prod
ContinuousSub.continuous_sub
IsTopologicalAddGroup.to_continuousSub
Set.ext
Set.mem_preimage
Set.mem_image
Set.vadd_inter_nonempty_iff'
compl_compl
Set.preimage_compl
properlyDiscontinuousVAdd_opposite_of_tendsto_cofinite πŸ“–mathematicalFilter.Tendsto
AddSubgroup
SetLike.instMembership
instSetLike
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
toAddGroup
AddMonoidHom.instFunLike
subtype
Filter.cofinite
Filter.cocompact
ProperlyDiscontinuousVAdd
AddOpposite
AddOpposite.instAddGroup
op
instVAdd
β€”Continuous.prodMap
ContinuousNeg.continuous_neg
IsTopologicalAddGroup.toContinuousNeg
continuous_id
IsCompact.compl_mem_cocompact
IsCompact.image
IsCompact.prod
Continuous.comp
continuous_add
IsTopologicalAddGroup.toContinuousAdd
Set.Finite.of_preimage
Set.ext
Set.mem_preimage
Set.mem_image
Set.op_vadd_inter_nonempty_iff
Set.image_congr
compl_compl
Equiv.surjective
topologicalClosure_coe πŸ“–mathematicalβ€”SetLike.coe
AddSubgroup
instSetLike
topologicalClosure
closure
β€”β€”
topologicalClosure_minimal πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosureβ€”closure_minimal

AddUnits

Theorems

NameKindAssumesProvesValidatesDepends On
instCompactSpaceOfT1SpaceOfContinuousAdd πŸ“–mathematicalβ€”CompactSpace
AddUnits
instTopologicalSpaceAddUnits
β€”Topology.IsClosedEmbedding.compactSpace
instCompactSpaceProd
AddOpposite.instCompactSpace
isClosedEmbedding_embedProduct
instIsTopologicalAddGroupOfContinuousAdd πŸ“–mathematicalβ€”IsTopologicalAddGroup
AddUnits
instTopologicalSpaceAddUnits
instAddGroup
β€”instContinuousAdd
continuous_iff
continuous_coe_neg
continuous_val
instLocallyCompactSpaceOfT1SpaceOfContinuousAdd πŸ“–mathematicalβ€”LocallyCompactSpace
AddUnits
instTopologicalSpaceAddUnits
β€”Topology.IsClosedEmbedding.locallyCompactSpace
Prod.locallyCompactSpace
AddOpposite.instLocallyCompactSpace
isClosedEmbedding_embedProduct
instWeaklyLocallyCompactSpaceOfT1SpaceOfContinuousAdd πŸ“–mathematicalβ€”WeaklyLocallyCompactSpace
AddUnits
instTopologicalSpaceAddUnits
β€”Topology.IsClosedEmbedding.weaklyLocallyCompactSpace
instWeaklyLocallyCompactSpaceProd
AddOpposite.instWeaklyLocallyCompactSpace
isClosedEmbedding_embedProduct
isClosedEmbedding_embedProduct πŸ“–mathematicalβ€”Topology.IsClosedEmbedding
AddUnits
AddOpposite
instTopologicalSpaceAddUnits
instTopologicalSpaceProd
AddOpposite.instTopologicalSpaceAddOpposite
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
instAddZeroClass
Prod.instAddZeroClass
AddMonoid.toAddZeroClass
AddOpposite.instAddZeroClass
AddMonoidHom.instFunLike
embedProduct
β€”isEmbedding_embedProduct
range_embedProduct
IsClosed.inter
IsClosed.preimage
Continuous.fun_add
Continuous.fst
continuous_id'
Continuous.comp'
AddOpposite.continuous_unop
Continuous.snd
isClosed_singleton
isEmbedding_val πŸ“–mathematicalβ€”Topology.IsEmbedding
AddUnits
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
instTopologicalSpaceAddUnits
val
β€”Homeomorph.isEmbedding
range_embedProduct πŸ“–mathematicalβ€”Set.range
AddOpposite
AddUnits
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
instAddZeroClass
Prod.instAddZeroClass
AddMonoid.toAddZeroClass
AddOpposite.instAddZeroClass
AddMonoidHom.instFunLike
embedProduct
setOf
AddZero.toAdd
AddOpposite.unop
AddZero.toZero
β€”Set.range_eq_iff
add_neg
neg_add

ConjAct

Theorems

NameKindAssumesProvesValidatesDepends On
units_continuousConstSMul πŸ“–mathematicalβ€”ContinuousConstSMul
ConjAct
Units
unitsScalar
β€”Continuous.mul
continuous_const
continuous_id

Continuous

Theorems

NameKindAssumesProvesValidatesDepends On
of_coeHom_comp πŸ“–mathematicalContinuous
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
MonoidHom.comp
Units
Units.instMulOneClass
Units.coeHom
Units.instTopologicalSpaceUnitsβ€”continuous_induced_rng
continuous_prodMk
MonoidHom.instMonoidHomClass
comp
MulOpposite.continuous_op
ContinuousInv.continuous_inv
of_inv πŸ“–β€”Continuous
Pi.instInv
InvolutiveInv.toInv
β€”β€”continuous_inv_iff
of_neg πŸ“–β€”Continuous
Pi.instNeg
InvolutiveNeg.toNeg
β€”β€”continuous_neg_iff
zpow πŸ“–mathematicalContinuousDivInvMonoid.toZPow
Group.toDivInvMonoid
β€”comp
continuous_zpow
zsmul πŸ“–mathematicalContinuousSubNegMonoid.toZSMul
AddGroup.toSubNegMonoid
β€”comp
continuous_zsmul

ContinuousAt

Theorems

NameKindAssumesProvesValidatesDepends On
of_inv πŸ“–β€”ContinuousAt
Pi.instInv
InvolutiveInv.toInv
β€”β€”continuousAt_inv_iff
of_neg πŸ“–β€”ContinuousAt
Pi.instNeg
InvolutiveNeg.toNeg
β€”β€”continuousAt_neg_iff
zpow πŸ“–mathematicalContinuousAtDivInvMonoid.toZPow
Group.toDivInvMonoid
β€”Filter.Tendsto.zpow
zsmul πŸ“–mathematicalContinuousAtSubNegMonoid.toZSMul
AddGroup.toSubNegMonoid
β€”Filter.Tendsto.zsmul

ContinuousInv

Theorems

NameKindAssumesProvesValidatesDepends On
induced πŸ“–mathematicalβ€”ContinuousInv
TopologicalSpace.induced
DFunLike.coe
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”continuous_induced_rng
map_inv
Continuous.inv
continuous_induced_dom
of_nhds_one πŸ“–mathematicalFilter.Tendsto
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
nhds
InvOneClass.toOne
Filter.map
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
ContinuousInvβ€”continuous_iff_continuousAt
Filter.Tendsto.comp
Filter.tendsto_map
mul_inv_rev
mul_assoc
inv_mul_cancel_left

ContinuousNeg

Theorems

NameKindAssumesProvesValidatesDepends On
induced πŸ“–mathematicalβ€”ContinuousNeg
TopologicalSpace.induced
DFunLike.coe
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”continuous_induced_rng
map_neg
Continuous.neg
continuous_induced_dom
of_nhds_zero πŸ“–mathematicalFilter.Tendsto
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
nhds
NegZeroClass.toZero
Filter.map
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
ContinuousNegβ€”continuous_iff_continuousAt
Filter.Tendsto.comp
Filter.tendsto_map
Filter.tendsto_map'_iff
neg_add_rev
add_assoc
neg_add_cancel_left

ContinuousOn

Theorems

NameKindAssumesProvesValidatesDepends On
of_inv πŸ“–β€”ContinuousOn
Pi.instInv
InvolutiveInv.toInv
β€”β€”continuousOn_inv_iff
of_neg πŸ“–β€”ContinuousOn
Pi.instNeg
InvolutiveNeg.toNeg
β€”β€”continuousOn_neg_iff
zpow πŸ“–mathematicalContinuousOnDivInvMonoid.toZPow
Group.toDivInvMonoid
β€”ContinuousWithinAt.zpow
zsmul πŸ“–mathematicalContinuousOnSubNegMonoid.toZSMul
AddGroup.toSubNegMonoid
β€”ContinuousWithinAt.zsmul

ContinuousWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
zpow πŸ“–mathematicalContinuousWithinAtDivInvMonoid.toZPow
Group.toDivInvMonoid
β€”Filter.Tendsto.zpow
zsmul πŸ“–mathematicalContinuousWithinAtSubNegMonoid.toZSMul
AddGroup.toSubNegMonoid
β€”Filter.Tendsto.zsmul

DenseRange

Theorems

NameKindAssumesProvesValidatesDepends On
topologicalClosure_map_addSubgroup πŸ“–mathematicalContinuous
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
DenseRange
AddSubgroup.topologicalClosure
Top.top
AddSubgroup
AddSubgroup.instTop
AddSubgroup.mapβ€”SetLike.ext'_iff
dense_iff_closure_eq
dense_image
topologicalClosure_map_subgroup πŸ“–mathematicalContinuous
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
DenseRange
Subgroup.topologicalClosure
Top.top
Subgroup
Subgroup.instTop
Subgroup.mapβ€”SetLike.ext'_iff
dense_image

Filter

Theorems

NameKindAssumesProvesValidatesDepends On
inv_nhdsGT πŸ“–mathematicalβ€”Filter
instInv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
Set.Iio
β€”Set.image_congr
Set.image_inv_eq_inv
Set.inv_Ioi
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
inv_nhdsLT πŸ“–mathematicalβ€”Filter
instInv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
nhdsWithin
Set.Iio
PartialOrder.toPreorder
Set.Ioi
β€”Set.image_congr
Set.image_inv_eq_inv
Set.inv_Iio
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
inv_nhdsNE πŸ“–mathematicalβ€”Filter
instInv
InvolutiveInv.toInv
nhdsWithin
Compl.compl
Set
Set.instCompl
Set.instSingletonSet
β€”Set.image_congr
Set.image_inv_eq_inv
Set.compl_inv
Set.inv_singleton
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_add_left_nhdsGT πŸ“–mathematicalβ€”map
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
β€”Set.image_congr
Set.image_add_left
add_comm
Set.preimage_add_const_Ioi
sub_neg_eq_add
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_add_left_nhdsLT πŸ“–mathematicalβ€”map
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
nhdsWithin
Set.Iio
PartialOrder.toPreorder
β€”Set.image_congr
Set.image_add_left
add_comm
Set.preimage_add_const_Iio
sub_neg_eq_add
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_add_left_nhdsNE πŸ“–mathematicalβ€”map
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
nhdsWithin
Compl.compl
Set
Set.instCompl
Set.instSingletonSet
β€”Set.image_congr
Set.image_add_left
Set.preimage_add_left_singleton
neg_neg
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_add_right_nhdsGT πŸ“–mathematicalβ€”map
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
β€”Set.image_congr
Set.image_add_right
Set.preimage_add_const_Ioi
sub_neg_eq_add
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_add_right_nhdsLT πŸ“–mathematicalβ€”map
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
nhdsWithin
Set.Iio
PartialOrder.toPreorder
β€”Set.image_congr
Set.image_add_right
Set.preimage_add_const_Iio
sub_neg_eq_add
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_add_right_nhdsNE πŸ“–mathematicalβ€”map
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
nhdsWithin
Compl.compl
Set
Set.instCompl
Set.instSingletonSet
β€”Set.image_congr
Set.image_add_right
Set.preimage_add_right_singleton
neg_neg
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_mul_left_nhdsGT πŸ“–mathematicalβ€”map
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
CommGroup.toGroup
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
β€”Set.image_congr
Set.image_mul_left
mul_comm
Set.preimage_mul_const_Ioi
div_inv_eq_mul
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_mul_left_nhdsLT πŸ“–mathematicalβ€”map
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
CommGroup.toGroup
nhdsWithin
Set.Iio
PartialOrder.toPreorder
β€”Set.image_congr
Set.image_mul_left
mul_comm
Set.preimage_mul_const_Iio
div_inv_eq_mul
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_mul_left_nhdsNE πŸ“–mathematicalβ€”map
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
nhdsWithin
Compl.compl
Set
Set.instCompl
Set.instSingletonSet
β€”Set.image_congr
Set.image_mul_left
Set.preimage_mul_left_singleton
inv_inv
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_mul_right_nhdsGT πŸ“–mathematicalβ€”map
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
CommGroup.toGroup
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
β€”Set.image_congr
Set.image_mul_right
Set.preimage_mul_const_Ioi
div_inv_eq_mul
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_mul_right_nhdsLT πŸ“–mathematicalβ€”map
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
CommGroup.toGroup
nhdsWithin
Set.Iio
PartialOrder.toPreorder
β€”Set.image_congr
Set.image_mul_right
Set.preimage_mul_const_Iio
div_inv_eq_mul
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
map_mul_right_nhdsNE πŸ“–mathematicalβ€”map
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
nhdsWithin
Compl.compl
Set
Set.instCompl
Set.instSingletonSet
β€”Set.image_congr
Set.image_mul_right
Set.preimage_mul_right_singleton
inv_inv
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
neg_nhdsGT πŸ“–mathematicalβ€”Filter
instNeg
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
Set.Iio
β€”Set.image_congr
Set.image_neg_eq_neg
Set.neg_Ioi
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
neg_nhdsLT πŸ“–mathematicalβ€”Filter
instNeg
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
nhdsWithin
Set.Iio
PartialOrder.toPreorder
Set.Ioi
β€”Set.image_congr
Set.image_neg_eq_neg
Set.neg_Iio
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
neg_nhdsNE πŸ“–mathematicalβ€”Filter
instNeg
InvolutiveNeg.toNeg
nhdsWithin
Compl.compl
Set
Set.instCompl
Set.instSingletonSet
β€”Set.image_congr
Set.image_neg_eq_neg
Set.compl_neg
Set.neg_singleton
Topology.IsEmbedding.map_nhdsWithin_eq
Homeomorph.isEmbedding
tendsto_add_const_iff πŸ“–mathematicalβ€”Tendsto
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
nhds
β€”add_neg_cancel_right
Tendsto.add_const
tendsto_const_add_iff πŸ“–mathematicalβ€”Tendsto
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
nhds
β€”neg_add_cancel_left
Tendsto.const_add
tendsto_const_div_iff πŸ“–mathematicalβ€”Tendsto
DivInvMonoid.toDiv
Group.toDivInvMonoid
nhds
β€”tendsto_const_div_iff'
tendsto_const_div_iff' πŸ“–mathematicalβ€”Tendsto
DivInvMonoid.toDiv
Group.toDivInvMonoid
nhds
β€”inv_div
div_mul_cancel
Tendsto.mul_const
IsTopologicalGroup.toContinuousMul
Tendsto.inv
IsTopologicalGroup.toContinuousInv
Tendsto.const_div'
IsTopologicalGroup.to_continuousDiv
tendsto_const_mul_iff πŸ“–mathematicalβ€”Tendsto
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
nhds
β€”inv_mul_cancel_left
Tendsto.const_mul
tendsto_const_sub_iff πŸ“–mathematicalβ€”Tendsto
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
nhds
β€”neg_sub
sub_add_cancel
Tendsto.add_const
IsTopologicalAddGroup.toContinuousAdd
Tendsto.neg
IsTopologicalAddGroup.toContinuousNeg
Tendsto.const_sub
IsTopologicalAddGroup.to_continuousSub
tendsto_div_const_iff πŸ“–mathematicalβ€”Tendsto
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
nhds
β€”div_mul_eq_div_div_swap
inv_mul_cancelβ‚€
div_one
Tendsto.div_const'
tendsto_div_const_iff' πŸ“–mathematicalβ€”Tendsto
DivInvMonoid.toDiv
Group.toDivInvMonoid
nhds
β€”div_mul_eq_div_div_swap
inv_mul_cancel
div_one
Tendsto.div_const'
tendsto_mul_const_iff πŸ“–mathematicalβ€”Tendsto
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
nhds
β€”mul_inv_cancel_right
Tendsto.mul_const
tendsto_sub_const_iff πŸ“–mathematicalβ€”Tendsto
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
nhds
β€”sub_add_eq_sub_sub_swap
neg_add_cancel
sub_zero
Tendsto.sub_const

Filter.HasBasis

Theorems

NameKindAssumesProvesValidatesDepends On
nhds_of_one πŸ“–mathematicalFilter.HasBasis
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
setOf
Set
Set.instMembership
DivInvMonoid.toDiv
Group.toDivInvMonoid
β€”nhds_translation_mul_inv
div_eq_mul_inv
comap
nhds_of_zero πŸ“–mathematicalFilter.HasBasis
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
setOf
Set
Set.instMembership
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
β€”nhds_translation_add_neg
sub_eq_add_neg
comap

Filter.Tendsto

Theorems

NameKindAssumesProvesValidatesDepends On
const_div' πŸ“–β€”Filter.Tendsto
nhds
β€”β€”div'
tendsto_const_nhds
const_sub πŸ“–β€”Filter.Tendsto
nhds
β€”β€”sub
tendsto_const_nhds
div_const' πŸ“–β€”Filter.Tendsto
nhds
β€”β€”div'
tendsto_const_nhds
sub_const πŸ“–β€”Filter.Tendsto
nhds
β€”β€”sub
tendsto_const_nhds
zpow πŸ“–mathematicalFilter.Tendsto
nhds
DivInvMonoid.toZPow
Group.toDivInvMonoid
β€”comp
ContinuousAt.tendsto
continuousAt_zpow
zsmul πŸ“–mathematicalFilter.Tendsto
nhds
SubNegMonoid.toZSMul
AddGroup.toSubNegMonoid
β€”comp
ContinuousAt.tendsto
continuousAt_zsmul

Homeomorph

Definitions

NameCategoryTheorems
addLeft πŸ“–CompOp
3 mathmath: coe_addLeft, addLeft_symm, MeasureTheory.Content.is_add_left_invariant_innerContent
addRight πŸ“–CompOp
2 mathmath: addRight_symm, coe_addRight
divLeft πŸ“–CompOp
2 mathmath: divLeft_apply, divLeft_symm_apply
divRight πŸ“–CompOp
2 mathmath: divRight_apply, divRight_symm_apply
inv πŸ“–CompOp
2 mathmath: symm_inv, coe_inv
mulLeft πŸ“–CompOp
4 mathmath: ContinuousCohomology.I_obj_ρ_apply, MeasureTheory.Content.is_mul_left_invariant_innerContent, coe_mulLeft, mulLeft_symm
mulRight πŸ“–CompOp
2 mathmath: coe_mulRight, mulRight_symm
neg πŸ“–CompOp
2 mathmath: coe_neg, symm_neg
prodAddUnits πŸ“–CompOpβ€”
prodUnits πŸ“–CompOpβ€”
shearAddRight πŸ“–CompOp
2 mathmath: shearAddRight_symm_coe, shearAddRight_coe
shearMulRight πŸ“–CompOp
2 mathmath: shearMulRight_coe, shearMulRight_symm_coe
subLeft πŸ“–CompOp
2 mathmath: subLeft_symm_apply, subLeft_apply
subRight πŸ“–CompOp
2 mathmath: subRight_symm_apply, subRight_apply

Theorems

NameKindAssumesProvesValidatesDepends On
addLeft_symm πŸ“–mathematicalβ€”symm
addLeft
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”ext
addRight_symm πŸ“–mathematicalβ€”symm
addRight
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”ext
coe_addLeft πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
addLeft
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”β€”
coe_addRight πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
addRight
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”β€”
coe_inv πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
inv
InvolutiveInv.toInv
β€”β€”
coe_mulLeft πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
mulLeft
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”β€”
coe_mulRight πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
mulRight
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”β€”
coe_neg πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
neg
InvolutiveNeg.toNeg
β€”β€”
divLeft_apply πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
divLeft
DivInvMonoid.toDiv
Group.toDivInvMonoid
β€”β€”
divLeft_symm_apply πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
symm
divLeft
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”β€”
divRight_apply πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
divRight
DivInvMonoid.toDiv
Group.toDivInvMonoid
β€”β€”
divRight_symm_apply πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
symm
divRight
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”β€”
mulLeft_symm πŸ“–mathematicalβ€”symm
mulLeft
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”ext
mulRight_symm πŸ“–mathematicalβ€”symm
mulRight
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”ext
shearAddRight_coe πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
instTopologicalSpaceProd
EquivLike.toFunLike
instEquivLike
shearAddRight
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”β€”
shearAddRight_symm_coe πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
instTopologicalSpaceProd
EquivLike.toFunLike
instEquivLike
symm
shearAddRight
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”β€”
shearMulRight_coe πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
instTopologicalSpaceProd
EquivLike.toFunLike
instEquivLike
shearMulRight
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”β€”
shearMulRight_symm_coe πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
instTopologicalSpaceProd
EquivLike.toFunLike
instEquivLike
symm
shearMulRight
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”β€”
subLeft_apply πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
subLeft
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
β€”β€”
subLeft_symm_apply πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
symm
subLeft
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”β€”
subRight_apply πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
subRight
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
β€”β€”
subRight_symm_apply πŸ“–mathematicalβ€”DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
symm
subRight
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”β€”
symm_inv πŸ“–mathematicalβ€”symm
inv
β€”β€”
symm_neg πŸ“–mathematicalβ€”symm
neg
β€”β€”

Inseparable

Theorems

NameKindAssumesProvesValidatesDepends On
inv πŸ“–β€”Inseparableβ€”β€”map
ContinuousInv.continuous_inv
neg πŸ“–β€”Inseparableβ€”β€”map
ContinuousNeg.continuous_neg
zpow πŸ“–mathematicalInseparableDivInvMonoid.toZPowβ€”Specializes.antisymm
Specializes.zpow
specializes
specializes'
zsmul πŸ“–mathematicalInseparableSubNegMonoid.toZSMulβ€”Specializes.antisymm
Specializes.zsmul
specializes
specializes'

IsClosed

Theorems

NameKindAssumesProvesValidatesDepends On
inv πŸ“–mathematicalβ€”IsClosed
Set
Set.inv
InvolutiveInv.toInv
β€”preimage
ContinuousInv.continuous_inv
leftCoset πŸ“–mathematicalβ€”IsClosed
Set
Set.smulSet
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
Monoid.toMulAction
β€”isClosedMap_mul_left
left_addCoset πŸ“–mathematicalβ€”IsClosed
HVAdd.hVAdd
Set
instHVAdd
Set.vaddSet
AddSemigroupAction.toVAdd
AddMonoid.toAddSemigroup
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddAction.toAddSemigroupAction
AddMonoid.toAddAction
β€”isClosedMap_add_left
neg πŸ“–mathematicalβ€”IsClosed
Set
Set.neg
InvolutiveNeg.toNeg
β€”preimage
ContinuousNeg.continuous_neg
rightCoset πŸ“–mathematicalβ€”IsClosed
MulOpposite
Set
Set.smulSet
SemigroupAction.toSMul
Monoid.toSemigroup
MulOpposite.instMonoid
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
Monoid.toOppositeMulAction
MulOpposite.op
β€”isClosedMap_mul_right
right_addCoset πŸ“–mathematicalβ€”IsClosed
HVAdd.hVAdd
AddOpposite
Set
instHVAdd
Set.vaddSet
AddSemigroupAction.toVAdd
AddMonoid.toAddSemigroup
AddOpposite.instAddMonoid
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddAction.toAddSemigroupAction
AddMonoid.toOppositeAddAction
AddOpposite.op
β€”isClosedMap_add_right

IsCompact

Theorems

NameKindAssumesProvesValidatesDepends On
inv πŸ“–mathematicalIsCompactSet
Set.inv
InvolutiveInv.toInv
β€”Set.image_inv_eq_inv
image
ContinuousInv.continuous_inv
neg πŸ“–mathematicalIsCompactSet
Set.neg
InvolutiveNeg.toNeg
β€”Set.image_neg_eq_neg
image
ContinuousNeg.continuous_neg

IsOpen

Theorems

NameKindAssumesProvesValidatesDepends On
inv πŸ“–mathematicalIsOpenSet
Set.inv
InvolutiveInv.toInv
β€”preimage
ContinuousInv.continuous_inv
leftCoset πŸ“–mathematicalIsOpenSet
Set.smulSet
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
Monoid.toMulAction
β€”isOpenMap_mul_left
left_addCoset πŸ“–mathematicalIsOpenHVAdd.hVAdd
Set
instHVAdd
Set.vaddSet
AddSemigroupAction.toVAdd
AddMonoid.toAddSemigroup
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddAction.toAddSemigroupAction
AddMonoid.toAddAction
β€”isOpenMap_add_left
neg πŸ“–mathematicalIsOpenSet
Set.neg
InvolutiveNeg.toNeg
β€”preimage
ContinuousNeg.continuous_neg
rightCoset πŸ“–mathematicalIsOpenMulOpposite
Set
Set.smulSet
SemigroupAction.toSMul
Monoid.toSemigroup
MulOpposite.instMonoid
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
Monoid.toOppositeMulAction
MulOpposite.op
β€”isOpenMap_mul_right
right_addCoset πŸ“–mathematicalIsOpenHVAdd.hVAdd
AddOpposite
Set
instHVAdd
Set.vaddSet
AddSemigroupAction.toVAdd
AddMonoid.toAddSemigroup
AddOpposite.instAddMonoid
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddAction.toAddSemigroupAction
AddMonoid.toOppositeAddAction
AddOpposite.op
β€”isOpenMap_add_right

IsTopologicalAddGroup

Theorems

NameKindAssumesProvesValidatesDepends On
continuous_addConj_prod πŸ“–mathematicalβ€”Continuous
instTopologicalSpaceProd
β€”Continuous.add
continuous_add
Continuous.comp
ContinuousNeg.continuous_neg
continuous_fst
continuous_conj πŸ“–mathematicalβ€”Continuousβ€”Continuous.comp
continuous_add_right
continuous_add_left
continuous_conj' πŸ“–mathematicalβ€”Continuousβ€”Continuous.add
continuous_add_right
ContinuousNeg.continuous_neg
exists_antitone_basis_nhds_zero πŸ“–mathematicalβ€”Filter.HasAntitoneBasis
Nat.instPreorder
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
Set
Set.instHasSubset
Set.add
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”Filter.exists_antitone_basis
FirstCountableTopology.nhds_generated_countable
Filter.HasBasis.tendsto_iff
Filter.HasBasis.prod_nhds
add_zero
Continuous.tendsto
continuous_add
toContinuousAdd
Set.mem_prod
forall_true_left
Filter.HasBasis.eventually_iff
Filter.atTop_basis
instIsDirectedOrder
IsStrictOrderedRing.toIsOrderedRing
Nat.instIsStrictOrderedRing
instArchimedeanNat
LE.le.trans
le_max_left
le_max_right
Filter.HasAntitoneBasis.subbasis_with_rel
ext πŸ“–β€”nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”β€”TopologicalSpace.ext_nhds
nhds_translation_add_neg
ext_iff πŸ“–mathematicalβ€”nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”ext
isInducing_iff_nhds_zero πŸ“–mathematicalβ€”Topology.IsInducing
DFunLike.coe
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
Filter.comap
β€”Topology.isInducing_iff_nhds
map_zero
AddMonoidHomClass.toZeroHomClass
nhds_translation_add_neg
Filter.comap_comap
map_add
AddMonoidHomClass.toAddHomClass
map_neg
isOpenMap_iff_nhds_zero πŸ“–mathematicalβ€”IsOpenMap
DFunLike.coe
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
nhds
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
Filter.map
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”IsOpenMap.nhds_le
map_zero
AddMonoidHomClass.toZeroHomClass
IsOpenMap.of_nhds_le
AddUnits.continuousConstVAdd
add_zero
Homeomorph.map_nhds_eq
map_add_left_nhds_zero
Filter.map_map
LE.le.trans
Filter.map_mono
map_add
AddMonoidHomClass.toAddHomClass
le_refl
of_comm_of_nhds_zero πŸ“–mathematicalFilter.Tendsto
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
SProd.sprod
Filter
Filter.instSProd
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
NegZeroClass.toNeg
Filter.map
IsTopologicalAddGroupβ€”of_nhds_zero
add_neg_cancel_comm
Zero.instNonempty
Filter.tendsto_id
of_nhds_zero πŸ“–mathematicalFilter.Tendsto
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
SProd.sprod
Filter
Filter.instSProd
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
NegZeroClass.toNeg
Filter.map
IsTopologicalAddGroupβ€”of_nhds_zero'
Filter.map_eq_of_inverse
add_assoc
neg_neg
add_neg_cancel
add_zero
add_neg_cancel_left
Filter.map_map
neg_add_cancel_right
of_nhds_zero' πŸ“–mathematicalFilter.Tendsto
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
SProd.sprod
Filter
Filter.instSProd
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
NegZeroClass.toNeg
Filter.map
IsTopologicalAddGroupβ€”ContinuousAdd.of_nhds_zero
ContinuousNeg.of_nhds_zero
le_of_eq
Filter.map_map
comp_add_right
add_neg_cancel
add_zero
Filter.map_id'
t1Space πŸ“–mathematicalβ€”T1Spaceβ€”Set.image_singleton
zero_add
isClosedMap_add_right

IsTopologicalGroup

Theorems

NameKindAssumesProvesValidatesDepends On
continuous_conj πŸ“–mathematicalβ€”Continuousβ€”Continuous.comp
continuous_mul_right
continuous_mul_left
continuous_conj' πŸ“–mathematicalβ€”Continuousβ€”Continuous.mul
continuous_mul_right
ContinuousInv.continuous_inv
continuous_conj_prod πŸ“–mathematicalβ€”Continuous
instTopologicalSpaceProd
β€”Continuous.mul
continuous_mul
Continuous.comp
ContinuousInv.continuous_inv
continuous_fst
exists_antitone_basis_nhds_one πŸ“–mathematicalβ€”Filter.HasAntitoneBasis
Nat.instPreorder
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
Set
Set.instHasSubset
Set.mul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Filter.exists_antitone_basis
FirstCountableTopology.nhds_generated_countable
Filter.HasBasis.tendsto_iff
Filter.HasBasis.prod_nhds
mul_one
Continuous.tendsto
continuous_mul
toContinuousMul
Filter.HasBasis.eventually_iff
Filter.atTop_basis
instIsDirectedOrder
IsStrictOrderedRing.toIsOrderedRing
Nat.instIsStrictOrderedRing
instArchimedeanNat
LE.le.trans
le_max_left
le_max_right
Filter.HasAntitoneBasis.subbasis_with_rel
ext πŸ“–β€”nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”β€”TopologicalSpace.ext_nhds
nhds_translation_mul_inv
ext_iff πŸ“–mathematicalβ€”nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”ext
isInducing_iff_nhds_one πŸ“–mathematicalβ€”Topology.IsInducing
DFunLike.coe
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
Filter.comap
β€”Topology.isInducing_iff_nhds
map_one
MonoidHomClass.toOneHomClass
nhds_translation_mul_inv
Filter.comap_comap
map_mul
MonoidHomClass.toMulHomClass
map_inv
isOpenMap_iff_nhds_one πŸ“–mathematicalβ€”IsOpenMap
DFunLike.coe
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
nhds
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
Filter.map
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”IsOpenMap.nhds_le
map_one
MonoidHomClass.toOneHomClass
IsOpenMap.of_nhds_le
Units.continuousConstSMul
mul_one
Homeomorph.map_nhds_eq
map_mul_left_nhds_one
Filter.map_map
LE.le.trans
Filter.map_mono
map_mul
MonoidHomClass.toMulHomClass
of_comm_of_nhds_one πŸ“–mathematicalFilter.Tendsto
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
CommGroup.toGroup
SProd.sprod
Filter
Filter.instSProd
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
InvOneClass.toInv
Filter.map
IsTopologicalGroupβ€”of_nhds_one
mul_inv_cancel_comm
One.instNonempty
Filter.tendsto_id
of_nhds_one πŸ“–mathematicalFilter.Tendsto
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
SProd.sprod
Filter
Filter.instSProd
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
InvOneClass.toInv
Filter.map
IsTopologicalGroupβ€”of_nhds_one'
Filter.map_eq_of_inverse
mul_assoc
inv_inv
mul_inv_cancel
mul_one
mul_inv_cancel_left
Filter.map_map
inv_mul_cancel_right
of_nhds_one' πŸ“–mathematicalFilter.Tendsto
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
SProd.sprod
Filter
Filter.instSProd
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
InvOneClass.toInv
Filter.map
IsTopologicalGroupβ€”ContinuousMul.of_nhds_one
ContinuousInv.of_nhds_one
le_of_eq
Filter.map_map
comp_mul_right
mul_inv_cancel
mul_one
Filter.map_id'
t1Space πŸ“–mathematicalβ€”T1Spaceβ€”Set.image_singleton
one_mul
isClosedMap_mul_right

MonoidHom

Theorems

NameKindAssumesProvesValidatesDepends On
isOpenQuotientMap_of_isQuotientMap πŸ“–mathematicalTopology.IsQuotientMap
DFunLike.coe
IsOpenQuotientMapβ€”Topology.IsQuotientMap.surjective
Topology.IsQuotientMap.continuous
Topology.IsQuotientMap.isOpen_preimage
Set.ext
Set.mem_iUnion_of_mem
Set.iUnion_congr_Prop
map_mul
MonoidHomClass.toMulHomClass
map_inv
inv_mul_cancel
Set.iUnion_true
mul_inv_cancel_left
mul_one
isOpen_biUnion
Continuous.isOpen_preimage
Continuous.fun_mul
continuous_id'
continuous_const

OrderDual

Theorems

NameKindAssumesProvesValidatesDepends On
instContinuousInv πŸ“–mathematicalβ€”ContinuousInv
OrderDual
instTopologicalSpace
instInvOrderDual
β€”β€”
instContinuousNeg πŸ“–mathematicalβ€”ContinuousNeg
OrderDual
instTopologicalSpace
instNegOrderDual
β€”β€”
instIsTopologicalAddGroup πŸ“–mathematicalβ€”IsTopologicalAddGroup
OrderDual
instTopologicalSpace
instAddGroup
β€”instContinuousAddOrderDual
IsTopologicalAddGroup.toContinuousAdd
instContinuousNeg
IsTopologicalAddGroup.toContinuousNeg
instIsTopologicalGroup πŸ“–mathematicalβ€”IsTopologicalGroup
OrderDual
instTopologicalSpace
instGroup
β€”instContinuousMulOrderDual
IsTopologicalGroup.toContinuousMul
instContinuousInv
IsTopologicalGroup.toContinuousInv

Pi

Theorems

NameKindAssumesProvesValidatesDepends On
continuousInv πŸ“–mathematicalContinuousInvtopologicalSpace
instInv
β€”continuous_pi
Continuous.inv
continuous_apply
continuousNeg πŸ“–mathematicalContinuousNegtopologicalSpace
instNeg
β€”continuous_pi
Continuous.neg
continuous_apply
has_continuous_inv' πŸ“–mathematicalβ€”ContinuousInv
topologicalSpace
instInv
β€”continuousInv
has_continuous_neg' πŸ“–mathematicalβ€”ContinuousNeg
topologicalSpace
instNeg
β€”continuousNeg
topologicalAddGroup πŸ“–mathematicalIsTopologicalAddGrouptopologicalSpace
addGroup
β€”continuousAdd
IsTopologicalAddGroup.toContinuousAdd
continuous_pi
Continuous.neg
IsTopologicalAddGroup.toContinuousNeg
continuous_apply
topologicalGroup πŸ“–mathematicalIsTopologicalGrouptopologicalSpace
group
β€”continuousMul
IsTopologicalGroup.toContinuousMul
continuous_pi
Continuous.inv
IsTopologicalGroup.toContinuousInv
continuous_apply

Prod

Theorems

NameKindAssumesProvesValidatesDepends On
continuousInv πŸ“–mathematicalβ€”ContinuousInv
instTopologicalSpaceProd
instInv
β€”Continuous.prodMk
Continuous.fst'
ContinuousInv.continuous_inv
Continuous.snd'
continuousNeg πŸ“–mathematicalβ€”ContinuousNeg
instTopologicalSpaceProd
instNeg
β€”Continuous.prodMk
Continuous.fst'
ContinuousNeg.continuous_neg
Continuous.snd'
instIsTopologicalAddGroup πŸ“–mathematicalβ€”IsTopologicalAddGroup
instTopologicalSpaceProd
instAddGroup
β€”continuousAdd
IsTopologicalAddGroup.toContinuousAdd
Continuous.prodMap
ContinuousNeg.continuous_neg
IsTopologicalAddGroup.toContinuousNeg
instIsTopologicalGroup πŸ“–mathematicalβ€”IsTopologicalGroup
instTopologicalSpaceProd
instGroup
β€”continuousMul
IsTopologicalGroup.toContinuousMul
Continuous.prodMap
ContinuousInv.continuous_inv
IsTopologicalGroup.toContinuousInv

SeparableWeaklyLocallyCompactAddGroup

Theorems

NameKindAssumesProvesValidatesDepends On
sigmaCompactSpace πŸ“–mathematicalβ€”SigmaCompactSpaceβ€”WeaklyLocallyCompactSpace.exists_compact_mem_nhds
Zero.instNonempty
IsTopologicalAddGroup.toContinuousAdd
TopologicalSpace.exists_dense_seq
Homeomorph.isCompact_preimage
Set.iUnion_eq_univ_iff
Dense.inter_nhds_nonempty
TopologicalSpace.denseRange_denseSeq
Continuous.continuousAt
Homeomorph.continuous
Homeomorph.apply_symm_apply

SeparableWeaklyLocallyCompactGroup

Theorems

NameKindAssumesProvesValidatesDepends On
sigmaCompactSpace πŸ“–mathematicalβ€”SigmaCompactSpaceβ€”WeaklyLocallyCompactSpace.exists_compact_mem_nhds
One.instNonempty
IsTopologicalGroup.toContinuousMul
TopologicalSpace.exists_dense_seq
Homeomorph.isCompact_preimage
Set.iUnion_eq_univ_iff
Dense.inter_nhds_nonempty
TopologicalSpace.denseRange_denseSeq
Continuous.continuousAt
Homeomorph.continuous
Homeomorph.apply_symm_apply

Set

Theorems

NameKindAssumesProvesValidatesDepends On
isClosed_centralizer πŸ“–mathematicalβ€”IsClosed
centralizer
β€”centralizer.eq_1
setOf_forall
isClosed_sInter
isClosed_imp
isClosed_eq
Continuous.fun_mul
continuous_const
continuous_id'

Specializes

Theorems

NameKindAssumesProvesValidatesDepends On
inv πŸ“–β€”Specializesβ€”β€”map
ContinuousInv.continuous_inv
neg πŸ“–β€”Specializesβ€”β€”map
ContinuousNeg.continuous_neg
zpow πŸ“–mathematicalSpecializesDivInvMonoid.toZPowβ€”zpow_natCast
pow
zpow_negSucc
inv
zsmul πŸ“–mathematicalSpecializesSubNegMonoid.toZSMulβ€”natCast_zsmul
nsmul
negSucc_zsmul
neg

Subgroup

Definitions

NameCategoryTheorems
commGroupTopologicalClosure πŸ“–CompOpβ€”
connectedComponentOfOne πŸ“–CompOpβ€”
topologicalClosure πŸ“–CompOp
9 mathmath: isClosed_topologicalClosure, instNormalCommutatorClosure, Field.absoluteGaloisGroup.commutator_closure_isNormal, le_topologicalClosure, topologicalClosure_coe, topologicalClosure_minimal, coe_topologicalClosure_bot, mapClusterPt_atTop_pow_iff_mem_topologicalClosure_zpowers, is_normal_topologicalClosure

Theorems

NameKindAssumesProvesValidatesDepends On
coe_topologicalClosure_bot πŸ“–mathematicalβ€”SetLike.coe
Subgroup
instSetLike
topologicalClosure
Bot.bot
instBot
closure
Set
Set.instSingletonSet
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”β€”
instIsTopologicalGroupSubtypeMem πŸ“–mathematicalβ€”IsTopologicalGroup
Subgroup
SetLike.instMembership
instSetLike
instTopologicalSpaceSubtype
toGroup
β€”Topology.IsInducing.topologicalGroup
MonoidHom.instMonoidHomClass
Topology.IsInducing.subtypeVal
isClosed_topologicalClosure πŸ“–mathematicalβ€”IsClosed
SetLike.coe
Subgroup
instSetLike
topologicalClosure
β€”isClosed_closure
is_normal_topologicalClosure πŸ“–mathematicalβ€”Normal
topologicalClosure
β€”map_mem_closure
IsTopologicalGroup.continuous_conj
IsTopologicalGroup.toContinuousMul
Normal.conj_mem
le_topologicalClosure πŸ“–mathematicalβ€”Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosure
β€”subset_closure
properlyDiscontinuousSMul_of_tendsto_cofinite πŸ“–mathematicalFilter.Tendsto
Subgroup
SetLike.instMembership
instSetLike
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
toGroup
MonoidHom.instFunLike
subtype
Filter.cofinite
Filter.cocompact
ProperlyDiscontinuousSMul
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
instMulAction
Monoid.toMulAction
β€”IsCompact.compl_mem_cocompact
IsCompact.image
IsCompact.prod
ContinuousDiv.continuous_div'
IsTopologicalGroup.to_continuousDiv
Set.ext
Set.smul_inter_nonempty_iff'
compl_compl
Set.preimage_compl
properlyDiscontinuousSMul_opposite_of_tendsto_cofinite πŸ“–mathematicalFilter.Tendsto
Subgroup
SetLike.instMembership
instSetLike
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
toGroup
MonoidHom.instFunLike
subtype
Filter.cofinite
Filter.cocompact
ProperlyDiscontinuousSMul
MulOpposite
MulOpposite.instGroup
op
instSMul
β€”Continuous.prodMap
ContinuousInv.continuous_inv
IsTopologicalGroup.toContinuousInv
continuous_id
IsCompact.compl_mem_cocompact
IsCompact.image
IsCompact.prod
Continuous.comp
continuous_mul
IsTopologicalGroup.toContinuousMul
Set.Finite.of_preimage
Set.ext
Set.op_smul_inter_nonempty_iff
Set.image_congr
compl_compl
Equiv.surjective
topologicalClosure_coe πŸ“–mathematicalβ€”SetLike.coe
Subgroup
instSetLike
topologicalClosure
closure
β€”β€”
topologicalClosure_minimal πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosureβ€”closure_minimal

Submonoid

Theorems

NameKindAssumesProvesValidatesDepends On
units_isCompact πŸ“–mathematicalIsCompact
SetLike.coe
Submonoid
Monoid.toMulOneClass
instSetLike
Units
Units.instTopologicalSpaceUnits
Subgroup
Units.instGroup
Subgroup.instSetLike
units
β€”IsCompact.prod
Homeomorph.isCompact_preimage
Topology.IsClosedEmbedding.isCompact_preimage
Units.isClosedEmbedding_embedProduct

TopologicalGroup

Theorems

NameKindAssumesProvesValidatesDepends On
isOpenMap_iff_nhds_one πŸ“–mathematicalβ€”IsOpenMap
DFunLike.coe
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
nhds
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
Filter.map
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”IsTopologicalGroup.isOpenMap_iff_nhds_one
isOpenMap_iff_nhds_zero πŸ“–mathematicalβ€”IsOpenMap
DFunLike.coe
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
nhds
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
Filter.map
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”IsTopologicalAddGroup.isOpenMap_iff_nhds_zero

Topology.IsInducing

Theorems

NameKindAssumesProvesValidatesDepends On
continuousInv πŸ“–mathematicalTopology.IsInducingContinuousInvβ€”continuous_iff
Continuous.inv
continuous
continuousNeg πŸ“–mathematicalTopology.IsInducingContinuousNegβ€”continuous_iff
Continuous.neg
continuous
topologicalAddGroup πŸ“–mathematicalTopology.IsInducing
DFunLike.coe
IsTopologicalAddGroupβ€”continuousAdd
AddMonoidHomClass.toAddHomClass
IsTopologicalAddGroup.toContinuousAdd
continuousNeg
IsTopologicalAddGroup.toContinuousNeg
map_neg
topologicalGroup πŸ“–mathematicalTopology.IsInducing
DFunLike.coe
IsTopologicalGroupβ€”continuousMul
MonoidHomClass.toMulHomClass
IsTopologicalGroup.toContinuousMul
continuousInv
IsTopologicalGroup.toContinuousInv
map_inv

Units

Theorems

NameKindAssumesProvesValidatesDepends On
instCompactSpaceOfT1SpaceOfContinuousMul πŸ“–mathematicalβ€”CompactSpace
Units
instTopologicalSpaceUnits
β€”Topology.IsClosedEmbedding.compactSpace
instCompactSpaceProd
MulOpposite.instCompactSpace
isClosedEmbedding_embedProduct
instIsTopologicalGroupOfContinuousMul πŸ“–mathematicalβ€”IsTopologicalGroup
Units
instTopologicalSpaceUnits
instGroup
β€”instContinuousMul
continuous_iff
continuous_coe_inv
continuous_val
instLocallyCompactSpaceOfT1SpaceOfContinuousMul πŸ“–mathematicalβ€”LocallyCompactSpace
Units
instTopologicalSpaceUnits
β€”Topology.IsClosedEmbedding.locallyCompactSpace
Prod.locallyCompactSpace
MulOpposite.instLocallyCompactSpace
isClosedEmbedding_embedProduct
instWeaklyLocallyCompactSpaceOfT1SpaceOfContinuousMul πŸ“–mathematicalβ€”WeaklyLocallyCompactSpace
Units
instTopologicalSpaceUnits
β€”Topology.IsClosedEmbedding.weaklyLocallyCompactSpace
instWeaklyLocallyCompactSpaceProd
MulOpposite.instWeaklyLocallyCompactSpace
isClosedEmbedding_embedProduct
isClosedEmbedding_embedProduct πŸ“–mathematicalβ€”Topology.IsClosedEmbedding
Units
MulOpposite
instTopologicalSpaceUnits
instTopologicalSpaceProd
MulOpposite.instTopologicalSpaceMulOpposite
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
instMulOneClass
Prod.instMulOneClass
Monoid.toMulOneClass
MulOpposite.instMulOneClass
MonoidHom.instFunLike
embedProduct
β€”isEmbedding_embedProduct
range_embedProduct
IsClosed.inter
IsClosed.preimage
Continuous.fun_mul
Continuous.fst
continuous_id'
Continuous.comp'
MulOpposite.continuous_unop
Continuous.snd
isClosed_singleton
isEmbedding_val πŸ“–mathematicalβ€”Topology.IsEmbedding
Units
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instTopologicalSpaceUnits
val
β€”Homeomorph.isEmbedding
range_embedProduct πŸ“–mathematicalβ€”Set.range
MulOpposite
Units
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
instMulOneClass
Prod.instMulOneClass
Monoid.toMulOneClass
MulOpposite.instMulOneClass
MonoidHom.instFunLike
embedProduct
setOf
MulOne.toMul
MulOpposite.unop
MulOne.toOne
β€”Set.range_eq_iff
mul_inv
inv_mul

(root)

Definitions

NameCategoryTheorems
nhdsAddHom πŸ“–CompOp
1 mathmath: nhdsAddHom_apply
nhdsMulHom πŸ“–CompOp
1 mathmath: nhdsMulHom_apply
toAddUnits_homeomorph πŸ“–CompOpβ€”
toUnits_homeomorph πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
add_mem_connectedComponent_zero πŸ“–mathematicalSet
Set.instMembership
connectedComponent
AddZero.toZero
AddZeroClass.toAddZero
AddZero.toAddβ€”connectedComponent_eq
Continuous.image_connectedComponent_subset
continuous_add_left
mem_connectedComponent
add_zero
compact_covered_by_add_left_translates πŸ“–mathematicalIsCompact
Set.Nonempty
interior
Set
Set.instHasSubset
Set.iUnion
Finset
Finset.instMembership
Set.preimage
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”IsCompact.elim_finite_subcover
isOpen_interior
Set.mem_iUnion
preimage_interior_subset_interior_preimage
Continuous.fun_add
IsTopologicalAddGroup.toContinuousAdd
continuous_const
continuous_id'
Set.mem_preimage
neg_add_cancel_right
Set.Subset.trans
Set.iUnionβ‚‚_mono
interior_subset
compact_covered_by_mul_left_translates πŸ“–mathematicalIsCompact
Set.Nonempty
interior
Set
Set.instHasSubset
Set.iUnion
Finset
Finset.instMembership
Set.preimage
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”IsCompact.elim_finite_subcover
isOpen_interior
Set.mem_iUnion
preimage_interior_subset_interior_preimage
Continuous.fun_mul
IsTopologicalGroup.toContinuousMul
continuous_const
continuous_id'
Set.mem_preimage
inv_mul_cancel_right
Set.Subset.trans
Set.iUnionβ‚‚_mono
interior_subset
compact_open_separated_add_left πŸ“–mathematicalIsCompact
IsOpen
Set
Set.instHasSubset
Filter
Filter.instMembership
nhds
AddZero.toZero
AddZeroClass.toAddZero
Set.add
AddZero.toAdd
β€”compact_open_separated_add_right
AddOpposite.instContinuousAdd
IsCompact.image
AddOpposite.continuous_op
Homeomorph.isOpenMap
Set.image_mono
Continuous.continuousAt
Set.preimage_image_eq
AddOpposite.op_injective
Set.image_subset_iff
Set.image_op_add
Set.image_preimage_eq
AddOpposite.op_surjective
compact_open_separated_add_right πŸ“–mathematicalIsCompact
IsOpen
Set
Set.instHasSubset
Filter
Filter.instMembership
nhds
AddZero.toZero
AddZeroClass.toAddZero
Set.add
AddZero.toAdd
β€”IsCompact.induction_on
Filter.univ_mem
Set.empty_add
Set.empty_subset
HasSubset.Subset.trans
Set.instIsTransSubset
Set.add_subset_add_right
Filter.inter_mem
Set.union_add
Set.union_subset
Set.add_subset_add_left
Set.inter_subset_left
Set.inter_subset_right
tendsto_add
add_zero
IsOpen.mem_nhds
Filter.mem_prod_iff
Filter.mem_map
nhds_prod_eq
mem_nhdsWithin_of_mem_nhds
Set.add_image_prod
Set.image_subset_iff
compact_open_separated_mul_left πŸ“–mathematicalIsCompact
IsOpen
Set
Set.instHasSubset
Filter
Filter.instMembership
nhds
MulOne.toOne
MulOneClass.toMulOne
Set.mul
MulOne.toMul
β€”compact_open_separated_mul_right
MulOpposite.instContinuousMul
IsCompact.image
MulOpposite.continuous_op
Homeomorph.isOpenMap
Set.image_mono
Continuous.continuousAt
Set.preimage_image_eq
MulOpposite.op_injective
Set.image_subset_iff
Set.image_op_mul
Set.image_preimage_eq
MulOpposite.op_surjective
compact_open_separated_mul_right πŸ“–mathematicalIsCompact
IsOpen
Set
Set.instHasSubset
Filter
Filter.instMembership
nhds
MulOne.toOne
MulOneClass.toMulOne
Set.mul
MulOne.toMul
β€”IsCompact.induction_on
Set.empty_mul
HasSubset.Subset.trans
Set.instIsTransSubset
Set.mul_subset_mul_right
Filter.inter_mem
Set.union_mul
Set.union_subset
Set.mul_subset_mul_left
Set.inter_subset_left
Set.inter_subset_right
tendsto_mul
mul_one
IsOpen.mem_nhds
Filter.mem_prod_iff
Filter.mem_map
nhds_prod_eq
mem_nhdsWithin_of_mem_nhds
Set.image_mul_prod
Set.image_subset_iff
continuousAt_inv πŸ“–mathematicalβ€”ContinuousAtβ€”Continuous.continuousAt
ContinuousInv.continuous_inv
continuousAt_inv_iff πŸ“–mathematicalβ€”ContinuousAt
Pi.instInv
InvolutiveInv.toInv
β€”Homeomorph.comp_continuousAt_iff
continuousAt_neg πŸ“–mathematicalβ€”ContinuousAtβ€”Continuous.continuousAt
ContinuousNeg.continuous_neg
continuousAt_neg_iff πŸ“–mathematicalβ€”ContinuousAt
Pi.instNeg
InvolutiveNeg.toNeg
β€”Homeomorph.comp_continuousAt_iff
continuousAt_zpow πŸ“–mathematicalβ€”ContinuousAt
DivInvMonoid.toZPow
Group.toDivInvMonoid
β€”Continuous.continuousAt
continuous_zpow
continuousAt_zsmul πŸ“–mathematicalβ€”ContinuousAt
SubNegMonoid.toZSMul
AddGroup.toSubNegMonoid
β€”Continuous.continuousAt
continuous_zsmul
continuousInv_iInf πŸ“–mathematicalContinuousInviInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”sInf_range
continuousInv_sInf
Set.forall_mem_range
continuousInv_inf πŸ“–mathematicalβ€”ContinuousInv
TopologicalSpace
SemilatticeInf.toMin
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”inf_eq_iInf
continuousInv_iInf
continuousInv_of_discreteTopology πŸ“–mathematicalβ€”ContinuousInvβ€”continuous_of_discreteTopology
continuousInv_sInf πŸ“–mathematicalContinuousInvInfSet.sInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”continuous_sInf_rng
continuous_sInf_dom
ContinuousInv.continuous_inv
continuousNeg_iInf πŸ“–mathematicalContinuousNegiInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”sInf_range
continuousNeg_sInf
Set.forall_mem_range
continuousNeg_inf πŸ“–mathematicalβ€”ContinuousNeg
TopologicalSpace
SemilatticeInf.toMin
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”inf_eq_iInf
continuousNeg_iInf
continuousNeg_of_discreteTopology πŸ“–mathematicalβ€”ContinuousNegβ€”continuous_of_discreteTopology
continuousNeg_sInf πŸ“–mathematicalContinuousNegInfSet.sInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”continuous_sInf_rng
continuous_sInf_dom
ContinuousNeg.continuous_neg
continuousOn_inv πŸ“–mathematicalβ€”ContinuousOnβ€”Continuous.continuousOn
ContinuousInv.continuous_inv
continuousOn_inv_iff πŸ“–mathematicalβ€”ContinuousOn
Pi.instInv
InvolutiveInv.toInv
β€”Homeomorph.comp_continuousOn_iff
continuousOn_neg πŸ“–mathematicalβ€”ContinuousOnβ€”Continuous.continuousOn
ContinuousNeg.continuous_neg
continuousOn_neg_iff πŸ“–mathematicalβ€”ContinuousOn
Pi.instNeg
InvolutiveNeg.toNeg
β€”Homeomorph.comp_continuousOn_iff
continuousOn_zpow πŸ“–mathematicalβ€”ContinuousOn
DivInvMonoid.toZPow
Group.toDivInvMonoid
β€”Continuous.continuousOn
continuous_zpow
continuousOn_zsmul πŸ“–mathematicalβ€”ContinuousOn
SubNegMonoid.toZSMul
AddGroup.toSubNegMonoid
β€”Continuous.continuousOn
continuous_zsmul
continuousWithinAt_inv πŸ“–mathematicalβ€”ContinuousWithinAtβ€”Continuous.continuousWithinAt
ContinuousInv.continuous_inv
continuousWithinAt_neg πŸ“–mathematicalβ€”ContinuousWithinAtβ€”Continuous.continuousWithinAt
ContinuousNeg.continuous_neg
continuous_div_left' πŸ“–mathematicalβ€”Continuousβ€”Continuous.div'
continuous_const
continuous_id'
continuous_div_right' πŸ“–mathematicalβ€”Continuousβ€”Continuous.div'
continuous_id'
continuous_const
continuous_inv_iff πŸ“–mathematicalβ€”Continuous
Pi.instInv
InvolutiveInv.toInv
β€”Homeomorph.comp_continuous_iff
continuous_neg_iff πŸ“–mathematicalβ€”Continuous
Pi.instNeg
InvolutiveNeg.toNeg
β€”Homeomorph.comp_continuous_iff
continuous_of_continuousAt_one πŸ“–mathematicalContinuousAt
DFunLike.coe
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
Continuousβ€”continuous_iff_continuousAt
map_mul_left_nhds_one
map_mul
MonoidHomClass.toMulHomClass
map_one
MonoidHomClass.toOneHomClass
mul_one
Filter.Tendsto.const_mul
ContinuousAt.tendsto
continuous_of_continuousAt_oneβ‚‚ πŸ“–mathematicalContinuousAt
instTopologicalSpaceProd
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
CommMonoid.toMonoid
MonoidHom.instFunLike
MonoidHom.instCommMonoid
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
Continuousβ€”continuous_iff_continuousAt
nhds_prod_eq
map_mul_left_nhds_one
Filter.prod_map_map_eq
map_mul
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
Filter.Tendsto.mono_right
Filter.Tendsto.mul
tendsto_const_nhds
Filter.Tendsto.comp
Filter.tendsto_fst
Filter.tendsto_snd
le_of_eq
map_one
MonoidHomClass.toOneHomClass
mul_one
continuous_of_continuousAt_zero πŸ“–mathematicalContinuousAt
DFunLike.coe
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
Continuousβ€”continuous_iff_continuousAt
map_add_left_nhds_zero
Filter.tendsto_map'_iff
map_add
AddMonoidHomClass.toAddHomClass
map_zero
AddMonoidHomClass.toZeroHomClass
add_zero
Filter.Tendsto.const_add
ContinuousAt.tendsto
continuous_of_continuousAt_zeroβ‚‚ πŸ“–mathematicalContinuousAt
instTopologicalSpaceProd
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommMonoid.toAddMonoid
AddMonoidHom.instFunLike
AddMonoidHom.instAddCommMonoid
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
Continuousβ€”continuous_iff_continuousAt
nhds_prod_eq
map_add_left_nhds_zero
Filter.prod_map_map_eq
Filter.tendsto_map'_iff
map_add
AddMonoidHomClass.toAddHomClass
AddMonoidHom.instAddMonoidHomClass
Filter.Tendsto.mono_right
Filter.Tendsto.add
tendsto_const_nhds
Filter.Tendsto.comp
Filter.tendsto_fst
Filter.tendsto_snd
le_of_eq
map_zero
AddMonoidHomClass.toZeroHomClass
add_zero
continuous_sub_left πŸ“–mathematicalβ€”Continuousβ€”Continuous.sub
continuous_const
continuous_id'
continuous_sub_right πŸ“–mathematicalβ€”Continuousβ€”Continuous.sub
continuous_id'
continuous_const
continuous_zpow πŸ“–mathematicalβ€”Continuous
DivInvMonoid.toZPow
Group.toDivInvMonoid
β€”zpow_natCast
continuous_pow
IsTopologicalGroup.toContinuousMul
zpow_negSucc
Continuous.inv
IsTopologicalGroup.toContinuousInv
continuous_zsmul πŸ“–mathematicalβ€”Continuous
SubNegMonoid.toZSMul
AddGroup.toSubNegMonoid
β€”natCast_zsmul
continuous_nsmul
IsTopologicalAddGroup.toContinuousAdd
negSucc_zsmul
Continuous.neg
IsTopologicalAddGroup.toContinuousNeg
discreteTopology_iff_isOpen_singleton_one πŸ“–mathematicalβ€”DiscreteTopology
IsOpen
Set
Set.instSingletonSet
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”MulAction.IsPretransitive.discreteTopology_iff
IsScalarTower.continuousConstSMul
IsScalarTower.left
MulAction.Regular.isPretransitive
discreteTopology_iff_isOpen_singleton_zero πŸ“–mathematicalβ€”DiscreteTopology
IsOpen
Set
Set.instSingletonSet
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”AddAction.IsPretransitive.discreteTopology_iff
VAddAssocClass.continuousConstVAdd
VAddAssocClass.left
AddAction.Regular.isPretransitive
discreteTopology_of_isOpen_singleton_one πŸ“–mathematicalIsOpen
Set
Set.instSingletonSet
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
DiscreteTopologyβ€”discreteTopology_iff_isOpen_singleton_one
discreteTopology_of_isOpen_singleton_zero πŸ“–mathematicalIsOpen
Set
Set.instSingletonSet
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
DiscreteTopologyβ€”discreteTopology_iff_isOpen_singleton_zero
exists_disjoint_smul_of_isCompact πŸ“–mathematicalIsCompactDisjoint
Set
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Set.smulSet
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
Monoid.toMulAction
β€”IsCompact.ne_univ
IsCompact.mul
IsTopologicalGroup.toContinuousMul
IsCompact.inv
IsTopologicalGroup.toContinuousInv
Mathlib.Tactic.Contrapose.contraposeβ‚‚
Set.eq_univ_iff_forall
Set.disjoint_left
inv_inv
mul_inv_cancel_right
exists_disjoint_vadd_of_isCompact πŸ“–mathematicalIsCompactDisjoint
Set
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
HVAdd.hVAdd
instHVAdd
Set.vaddSet
AddSemigroupAction.toVAdd
AddMonoid.toAddSemigroup
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddAction.toAddSemigroupAction
AddMonoid.toAddAction
β€”IsCompact.ne_univ
IsCompact.add
IsTopologicalAddGroup.toContinuousAdd
IsCompact.neg
IsTopologicalAddGroup.toContinuousNeg
Mathlib.Tactic.Contrapose.contraposeβ‚‚
Mathlib.Tactic.Push.not_exists
Set.eq_univ_iff_forall
Set.disjoint_left
Set.mem_neg
neg_neg
add_neg_cancel_right
exists_nhds_half_neg πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
Set.instMembership
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
β€”ContinuousAt.add
IsTopologicalAddGroup.toContinuousAdd
continuousAt_fst
ContinuousAt.neg
IsTopologicalAddGroup.toContinuousNeg
continuousAt_snd
neg_zero
add_zero
sub_eq_add_neg
nhds_prod_eq
Filter.mem_prod_self_iff
Set.prod_subset_iff
Set.mem_preimage
exists_nhds_split_inv πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
Set.instMembership
DivInvMonoid.toDiv
Group.toDivInvMonoid
β€”ContinuousAt.mul
IsTopologicalGroup.toContinuousMul
continuousAt_fst
ContinuousAt.inv
IsTopologicalGroup.toContinuousInv
continuousAt_snd
inv_one
mul_one
div_eq_mul_inv
nhds_prod_eq
instContinuousInvMulOpposite πŸ“–mathematicalβ€”ContinuousInv
MulOpposite
MulOpposite.instTopologicalSpaceMulOpposite
MulOpposite.instInv
β€”Topology.IsInducing.continuousInv
Homeomorph.isInducing
MulOpposite.unop_inv
instContinuousInvMultiplicativeOfContinuousNeg πŸ“–mathematicalβ€”ContinuousInv
Multiplicative
instTopologicalSpaceMultiplicative
Multiplicative.inv
β€”ContinuousNeg.continuous_neg
instContinuousInvULift πŸ“–mathematicalβ€”ContinuousInv
ULift.topologicalSpace
ULift.inv
β€”Continuous.comp
continuous_uliftUp
ContinuousInv.continuous_inv
continuous_uliftDown
instContinuousNegAddOpposite πŸ“–mathematicalβ€”ContinuousNeg
AddOpposite
AddOpposite.instTopologicalSpaceAddOpposite
AddOpposite.instNeg
β€”Topology.IsInducing.continuousNeg
Homeomorph.isInducing
AddOpposite.unop_neg
instContinuousNegAdditiveOfContinuousInv πŸ“–mathematicalβ€”ContinuousNeg
Additive
instTopologicalSpaceAdditive
Additive.neg
β€”ContinuousInv.continuous_inv
instContinuousNegULift πŸ“–mathematicalβ€”ContinuousNeg
ULift.topologicalSpace
ULift.neg
β€”Continuous.comp
continuous_uliftUp
ContinuousNeg.continuous_neg
continuous_uliftDown
instIsTopologicalAddGroupAddOpposite πŸ“–mathematicalβ€”IsTopologicalAddGroup
AddOpposite
AddOpposite.instTopologicalSpaceAddOpposite
AddOpposite.instAddGroup
β€”AddOpposite.instContinuousAdd
IsTopologicalAddGroup.toContinuousAdd
instContinuousNegAddOpposite
IsTopologicalAddGroup.toContinuousNeg
instIsTopologicalAddGroupAdditiveOfIsTopologicalGroup πŸ“–mathematicalβ€”IsTopologicalAddGroup
Additive
instTopologicalSpaceAdditive
Additive.addGroup
β€”instContinuousAddAdditiveOfContinuousMul
IsTopologicalGroup.toContinuousMul
ContinuousInv.continuous_inv
IsTopologicalGroup.toContinuousInv
instIsTopologicalGroupMulOpposite πŸ“–mathematicalβ€”IsTopologicalGroup
MulOpposite
MulOpposite.instTopologicalSpaceMulOpposite
MulOpposite.instGroup
β€”MulOpposite.instContinuousMul
IsTopologicalGroup.toContinuousMul
instContinuousInvMulOpposite
IsTopologicalGroup.toContinuousInv
instIsTopologicalGroupMultiplicativeOfIsTopologicalAddGroup πŸ“–mathematicalβ€”IsTopologicalGroup
Multiplicative
instTopologicalSpaceMultiplicative
Multiplicative.group
β€”instContinuousMulMultiplicativeOfContinuousAdd
IsTopologicalAddGroup.toContinuousAdd
ContinuousNeg.continuous_neg
IsTopologicalAddGroup.toContinuousNeg
instIsTopologicalGroupULift πŸ“–mathematicalβ€”IsTopologicalGroup
ULift.topologicalSpace
ULift.group
β€”instContinuousMulULift
IsTopologicalGroup.toContinuousMul
instContinuousInvULift
IsTopologicalGroup.toContinuousInv
inv_closure πŸ“–mathematicalβ€”Set
Set.inv
InvolutiveInv.toInv
closure
β€”Homeomorph.preimage_closure
inv_mem_connectedComponent_one πŸ“–mathematicalSet
Set.instMembership
connectedComponent
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
InvOneClass.toInvβ€”inv_one
Continuous.image_connectedComponent_subset
ContinuousInv.continuous_inv
Set.mem_image
inv_mem_nhds_one πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
Set.inv
InvOneClass.toInv
β€”nhds_one_symm'
isClosedMap_add_left πŸ“–mathematicalβ€”IsClosedMap
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”Homeomorph.isClosedMap
isClosedMap_add_right πŸ“–mathematicalβ€”IsClosedMap
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”Homeomorph.isClosedMap
isClosedMap_div_left πŸ“–mathematicalβ€”IsClosedMap
DivInvMonoid.toDiv
Group.toDivInvMonoid
β€”Homeomorph.isClosedMap
isClosedMap_div_right πŸ“–mathematicalβ€”IsClosedMap
DivInvMonoid.toDiv
Group.toDivInvMonoid
β€”Homeomorph.isClosedMap
isClosedMap_inv πŸ“–mathematicalβ€”IsClosedMap
InvolutiveInv.toInv
β€”Homeomorph.isClosedMap
isClosedMap_mul_left πŸ“–mathematicalβ€”IsClosedMap
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Homeomorph.isClosedMap
isClosedMap_mul_right πŸ“–mathematicalβ€”IsClosedMap
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Homeomorph.isClosedMap
isClosedMap_neg πŸ“–mathematicalβ€”IsClosedMap
InvolutiveNeg.toNeg
β€”Homeomorph.isClosedMap
isClosedMap_sub_left πŸ“–mathematicalβ€”IsClosedMap
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
β€”Homeomorph.isClosedMap
isClosedMap_sub_right πŸ“–mathematicalβ€”IsClosedMap
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
β€”Homeomorph.isClosedMap
isClosed_setOf_map_inv πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
setOf
β€”Set.setOf_forall
isClosed_iInter
isClosed_eq
continuous_apply
Continuous.inv
isClosed_setOf_map_neg πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
setOf
β€”Set.setOf_forall
isClosed_iInter
isClosed_eq
continuous_apply
Continuous.neg
isOpenMap_add_left πŸ“–mathematicalβ€”IsOpenMap
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”Homeomorph.isOpenMap
isOpenMap_add_right πŸ“–mathematicalβ€”IsOpenMap
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”Homeomorph.isOpenMap
isOpenMap_div_left πŸ“–mathematicalβ€”IsOpenMap
DivInvMonoid.toDiv
Group.toDivInvMonoid
β€”Homeomorph.isOpenMap
isOpenMap_div_right πŸ“–mathematicalβ€”IsOpenMap
DivInvMonoid.toDiv
Group.toDivInvMonoid
β€”Homeomorph.isOpenMap
isOpenMap_inv πŸ“–mathematicalβ€”IsOpenMap
InvolutiveInv.toInv
β€”Homeomorph.isOpenMap
isOpenMap_mul_left πŸ“–mathematicalβ€”IsOpenMap
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Homeomorph.isOpenMap
isOpenMap_mul_right πŸ“–mathematicalβ€”IsOpenMap
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Homeomorph.isOpenMap
isOpenMap_neg πŸ“–mathematicalβ€”IsOpenMap
InvolutiveNeg.toNeg
β€”Homeomorph.isOpenMap
isOpenMap_sub_left πŸ“–mathematicalβ€”IsOpenMap
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
β€”Homeomorph.isOpenMap
isOpenMap_sub_right πŸ“–mathematicalβ€”IsOpenMap
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
β€”Homeomorph.isOpenMap
map_add_left_nhds πŸ“–mathematicalβ€”Filter.map
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
nhds
β€”Homeomorph.map_nhds_eq
IsTopologicalAddGroup.toContinuousAdd
map_add_left_nhds_zero πŸ“–mathematicalβ€”Filter.map
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”map_add_left_nhds
add_zero
map_add_right_nhds πŸ“–mathematicalβ€”Filter.map
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
nhds
β€”Homeomorph.map_nhds_eq
IsTopologicalAddGroup.toContinuousAdd
map_add_right_nhds_zero πŸ“–mathematicalβ€”Filter.map
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”map_add_right_nhds
zero_add
map_mul_left_nhds πŸ“–mathematicalβ€”Filter.map
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
nhds
β€”Homeomorph.map_nhds_eq
IsTopologicalGroup.toContinuousMul
map_mul_left_nhds_one πŸ“–mathematicalβ€”Filter.map
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”map_mul_left_nhds
mul_one
map_mul_right_nhds πŸ“–mathematicalβ€”Filter.map
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
nhds
β€”Homeomorph.map_nhds_eq
IsTopologicalGroup.toContinuousMul
map_mul_right_nhds_one πŸ“–mathematicalβ€”Filter.map
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”map_mul_right_nhds
one_mul
mem_closure_iff_nhds_one πŸ“–mathematicalβ€”Set
Set.instMembership
closure
DivInvMonoid.toDiv
Group.toDivInvMonoid
β€”mem_closure_iff_nhds_basis
Filter.HasBasis.nhds_of_one
Filter.basis_sets
mem_closure_iff_nhds_zero πŸ“–mathematicalβ€”Set
Set.instMembership
closure
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
β€”mem_closure_iff_nhds_basis
Filter.HasBasis.nhds_of_zero
Filter.basis_sets
Set.mem_setOf
mul_mem_connectedComponent_one πŸ“–mathematicalSet
Set.instMembership
connectedComponent
MulOne.toOne
MulOneClass.toMulOne
MulOne.toMulβ€”connectedComponent_eq
Continuous.image_connectedComponent_subset
continuous_mul_left
mem_connectedComponent
mul_one
neg_closure πŸ“–mathematicalβ€”Set
Set.neg
InvolutiveNeg.toNeg
closure
β€”Homeomorph.preimage_closure
neg_mem_connectedComponent_zero πŸ“–mathematicalSet
Set.instMembership
connectedComponent
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
NegZeroClass.toNegβ€”neg_zero
Continuous.image_connectedComponent_subset
ContinuousNeg.continuous_neg
Set.mem_image
neg_mem_nhds_zero πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
Set.neg
NegZeroClass.toNeg
β€”nhds_zero_symm'
nhdsAddHom_apply πŸ“–mathematicalβ€”DFunLike.coe
AddHom
Filter
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Filter.instAdd
AddHom.funLike
nhdsAddHom
nhds
β€”β€”
nhdsMulHom_apply πŸ“–mathematicalβ€”DFunLike.coe
MulHom
Filter
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Filter.instMul
MulHom.funLike
nhdsMulHom
nhds
β€”β€”
nhds_add πŸ“–mathematicalβ€”nhds
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Filter
Filter.instAdd
β€”nhds_add_nhds_zero
IsTopologicalAddGroup.toContinuousAdd
map_add_right_nhds
zero_add
map_add_left_nhds
Filter.mapβ‚‚_add
Filter.map_mapβ‚‚
add_assoc
map_add_left_nhds_zero
map_add_right_nhds_zero
Filter.mapβ‚‚_map_left
Filter.mapβ‚‚_map_right
nhds_inv πŸ“–mathematicalβ€”nhds
InvolutiveInv.toInv
Filter
Filter.instInv
β€”Homeomorph.map_nhds_eq
nhds_mul πŸ“–mathematicalβ€”nhds
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Filter
Filter.instMul
β€”nhds_mul_nhds_one
IsTopologicalGroup.toContinuousMul
map_mul_right_nhds
one_mul
map_mul_left_nhds
Filter.mapβ‚‚_mul
Filter.map_mapβ‚‚
mul_assoc
map_mul_left_nhds_one
map_mul_right_nhds_one
Filter.mapβ‚‚_map_left
Filter.mapβ‚‚_map_right
nhds_neg πŸ“–mathematicalβ€”nhds
InvolutiveNeg.toNeg
Filter
Filter.instNeg
β€”Homeomorph.map_nhds_eq
nhds_one_symm πŸ“–mathematicalβ€”Filter.comap
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
nhds
InvOneClass.toOne
β€”IsTopologicalGroup.toContinuousInv
Homeomorph.comap_nhds_eq
inv_one
nhds_one_symm' πŸ“–mathematicalβ€”Filter.map
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
nhds
InvOneClass.toOne
β€”IsTopologicalGroup.toContinuousInv
Homeomorph.map_nhds_eq
inv_one
nhds_translation_add_neg πŸ“–mathematicalβ€”Filter.comap
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
nhds
NegZeroClass.toZero
β€”IsTopologicalAddGroup.toContinuousAdd
Homeomorph.comap_nhds_eq
neg_neg
zero_add
nhds_translation_div πŸ“–mathematicalβ€”Filter.comap
DivInvMonoid.toDiv
Group.toDivInvMonoid
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”div_eq_mul_inv
nhds_translation_mul_inv
nhds_translation_inv_mul πŸ“–mathematicalβ€”Filter.comap
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
nhds
InvOneClass.toOne
β€”IsTopologicalGroup.toContinuousMul
Homeomorph.comap_nhds_eq
inv_inv
mul_one
nhds_translation_mul_inv πŸ“–mathematicalβ€”Filter.comap
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
nhds
InvOneClass.toOne
β€”IsTopologicalGroup.toContinuousMul
Homeomorph.comap_nhds_eq
inv_inv
one_mul
nhds_translation_neg_add πŸ“–mathematicalβ€”Filter.comap
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
nhds
NegZeroClass.toZero
β€”IsTopologicalAddGroup.toContinuousAdd
Homeomorph.comap_nhds_eq
neg_neg
add_zero
nhds_translation_sub πŸ“–mathematicalβ€”Filter.comap
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”sub_eq_add_neg
nhds_translation_add_neg
nhds_zero_symm πŸ“–mathematicalβ€”Filter.comap
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
nhds
NegZeroClass.toZero
β€”IsTopologicalAddGroup.toContinuousNeg
Homeomorph.comap_nhds_eq
neg_zero
nhds_zero_symm' πŸ“–mathematicalβ€”Filter.map
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
nhds
NegZeroClass.toZero
β€”IsTopologicalAddGroup.toContinuousNeg
Homeomorph.map_nhds_eq
neg_zero
tendsto_div_nhds_one_iff πŸ“–mathematicalβ€”Filter.Tendsto
DivInvMonoid.toDiv
Group.toDivInvMonoid
nhds
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”div_mul_cancel
one_mul
Filter.Tendsto.mul
IsTopologicalGroup.toContinuousMul
tendsto_const_nhds
div_self'
Filter.Tendsto.div'
IsTopologicalGroup.to_continuousDiv
tendsto_inv πŸ“–mathematicalβ€”Filter.Tendsto
nhds
β€”continuousAt_inv
tendsto_inv_iff πŸ“–mathematicalβ€”Filter.Tendsto
InvolutiveInv.toInv
nhds
β€”inv_inv
Filter.Tendsto.inv
tendsto_inv_nhdsGE πŸ“–mathematicalβ€”Filter.Tendsto
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
nhdsWithin
Set.Ici
PartialOrder.toPreorder
Set.Iic
β€”Filter.Tendsto.inf
Continuous.tendsto
ContinuousInv.continuous_inv
IsOrderedMonoid.toMulLeftMono
covariant_swap_mul_of_covariant_mul
tendsto_inv_nhdsGE_inv πŸ“–mathematicalβ€”Filter.Tendsto
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
nhdsWithin
Set.Ici
PartialOrder.toPreorder
Set.Iic
β€”inv_inv
tendsto_inv_nhdsGE
tendsto_inv_nhdsGT πŸ“–mathematicalβ€”Filter.Tendsto
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
Set.Iio
β€”Filter.Tendsto.inf
Continuous.tendsto
ContinuousInv.continuous_inv
IsLeftCancelMul.mulLeftStrictMono_of_mulLeftMono
LeftCancelSemigroup.toIsLeftCancelMul
IsOrderedMonoid.toMulLeftMono
IsRightCancelMul.mulRightStrictMono_of_mulRightMono
RightCancelSemigroup.toIsRightCancelMul
covariant_swap_mul_of_covariant_mul
tendsto_inv_nhdsGT_inv πŸ“–mathematicalβ€”Filter.Tendsto
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
Set.Iio
β€”inv_inv
tendsto_inv_nhdsGT
tendsto_inv_nhdsLE πŸ“–mathematicalβ€”Filter.Tendsto
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
nhdsWithin
Set.Iic
PartialOrder.toPreorder
Set.Ici
β€”Filter.Tendsto.inf
Continuous.tendsto
ContinuousInv.continuous_inv
IsOrderedMonoid.toMulLeftMono
covariant_swap_mul_of_covariant_mul
tendsto_inv_nhdsLE_inv πŸ“–mathematicalβ€”Filter.Tendsto
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
nhdsWithin
Set.Iic
PartialOrder.toPreorder
Set.Ici
β€”inv_inv
tendsto_inv_nhdsLE
tendsto_inv_nhdsLT πŸ“–mathematicalβ€”Filter.Tendsto
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
nhdsWithin
Set.Iio
PartialOrder.toPreorder
Set.Ioi
β€”Filter.Tendsto.inf
Continuous.tendsto
ContinuousInv.continuous_inv
IsLeftCancelMul.mulLeftStrictMono_of_mulLeftMono
LeftCancelSemigroup.toIsLeftCancelMul
IsOrderedMonoid.toMulLeftMono
IsRightCancelMul.mulRightStrictMono_of_mulRightMono
RightCancelSemigroup.toIsRightCancelMul
covariant_swap_mul_of_covariant_mul
tendsto_inv_nhdsLT_inv πŸ“–mathematicalβ€”Filter.Tendsto
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
nhdsWithin
Set.Iio
PartialOrder.toPreorder
Set.Ioi
β€”inv_inv
tendsto_inv_nhdsLT
tendsto_inv_nhdsWithin_Iic_inv πŸ“–mathematicalβ€”Filter.Tendsto
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroup.toDivisionCommMonoid
nhdsWithin
Set.Iic
PartialOrder.toPreorder
Set.Ici
β€”tendsto_inv_nhdsLE_inv
tendsto_neg πŸ“–mathematicalβ€”Filter.Tendsto
nhds
β€”continuousAt_neg
tendsto_neg_iff πŸ“–mathematicalβ€”Filter.Tendsto
InvolutiveNeg.toNeg
nhds
β€”neg_neg
Filter.Tendsto.neg
tendsto_neg_nhdsGE πŸ“–mathematicalβ€”Filter.Tendsto
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
nhdsWithin
Set.Ici
PartialOrder.toPreorder
Set.Iic
β€”Filter.Tendsto.inf
Continuous.tendsto
ContinuousNeg.continuous_neg
Filter.tendsto_principal
Set.mem_Iic
neg_le_neg_iff
IsOrderedAddMonoid.toAddLeftMono
covariant_swap_add_of_covariant_add
Filter.eventually_principal
Set.mem_Ici
tendsto_neg_nhdsGE_neg πŸ“–mathematicalβ€”Filter.Tendsto
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
nhdsWithin
Set.Ici
PartialOrder.toPreorder
Set.Iic
β€”neg_neg
tendsto_neg_nhdsGE
tendsto_neg_nhdsGT πŸ“–mathematicalβ€”Filter.Tendsto
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
Set.Iio
β€”Filter.Tendsto.inf
Continuous.tendsto
ContinuousNeg.continuous_neg
Filter.tendsto_principal
Set.mem_Iio
neg_lt_neg_iff
IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono
AddLeftCancelSemigroup.toIsLeftCancelAdd
IsOrderedAddMonoid.toAddLeftMono
IsRightCancelAdd.addRightStrictMono_of_addRightMono
AddRightCancelSemigroup.toIsRightCancelAdd
covariant_swap_add_of_covariant_add
Filter.eventually_principal
Set.mem_Ioi
tendsto_neg_nhdsGT_neg πŸ“–mathematicalβ€”Filter.Tendsto
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
Set.Iio
β€”neg_neg
tendsto_neg_nhdsGT
tendsto_neg_nhdsLE πŸ“–mathematicalβ€”Filter.Tendsto
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
nhdsWithin
Set.Iic
PartialOrder.toPreorder
Set.Ici
β€”Filter.Tendsto.inf
Continuous.tendsto
ContinuousNeg.continuous_neg
Filter.tendsto_principal
Set.mem_Ici
neg_le_neg_iff
IsOrderedAddMonoid.toAddLeftMono
covariant_swap_add_of_covariant_add
Filter.eventually_principal
Set.mem_Iic
tendsto_neg_nhdsLE_neg πŸ“–mathematicalβ€”Filter.Tendsto
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
nhdsWithin
Set.Iic
PartialOrder.toPreorder
Set.Ici
β€”neg_neg
tendsto_neg_nhdsLE
tendsto_neg_nhdsLT πŸ“–mathematicalβ€”Filter.Tendsto
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
nhdsWithin
Set.Iio
PartialOrder.toPreorder
Set.Ioi
β€”Filter.Tendsto.inf
Continuous.tendsto
ContinuousNeg.continuous_neg
Filter.tendsto_principal
Set.mem_Ioi
neg_lt_neg_iff
IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono
AddLeftCancelSemigroup.toIsLeftCancelAdd
IsOrderedAddMonoid.toAddLeftMono
IsRightCancelAdd.addRightStrictMono_of_addRightMono
AddRightCancelSemigroup.toIsRightCancelAdd
covariant_swap_add_of_covariant_add
Filter.eventually_principal
Set.mem_Iio
tendsto_neg_nhdsLT_neg πŸ“–mathematicalβ€”Filter.Tendsto
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
nhdsWithin
Set.Iio
PartialOrder.toPreorder
Set.Ioi
β€”neg_neg
tendsto_neg_nhdsLT
tendsto_sub_nhds_zero_iff πŸ“–mathematicalβ€”Filter.Tendsto
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”sub_add_cancel
zero_add
Filter.Tendsto.add
IsTopologicalAddGroup.toContinuousAdd
tendsto_const_nhds
sub_self
Filter.Tendsto.sub
IsTopologicalAddGroup.to_continuousSub
topologicalAddGroup_iInf πŸ“–mathematicalIsTopologicalAddGroupiInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”sInf_range
topologicalAddGroup_sInf
Set.forall_mem_range
topologicalAddGroup_induced πŸ“–mathematicalβ€”IsTopologicalAddGroup
TopologicalSpace.induced
DFunLike.coe
β€”Topology.IsInducing.topologicalAddGroup
topologicalAddGroup_inf πŸ“–mathematicalβ€”IsTopologicalAddGroup
TopologicalSpace
SemilatticeInf.toMin
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”inf_eq_iInf
topologicalAddGroup_iInf
topologicalAddGroup_of_discreteTopology πŸ“–mathematicalβ€”IsTopologicalAddGroupβ€”continuousAdd_of_discreteTopology
continuousNeg_of_discreteTopology
topologicalAddGroup_sInf πŸ“–mathematicalIsTopologicalAddGroupInfSet.sInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”continuousAdd_sInf
IsTopologicalAddGroup.toContinuousAdd
continuousNeg_sInf
IsTopologicalAddGroup.toContinuousNeg
topologicalGroup_iInf πŸ“–mathematicalIsTopologicalGroupiInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”sInf_range
topologicalGroup_sInf
Set.forall_mem_range
topologicalGroup_induced πŸ“–mathematicalβ€”IsTopologicalGroup
TopologicalSpace.induced
DFunLike.coe
β€”Topology.IsInducing.topologicalGroup
topologicalGroup_inf πŸ“–mathematicalβ€”IsTopologicalGroup
TopologicalSpace
SemilatticeInf.toMin
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”inf_eq_iInf
topologicalGroup_iInf
topologicalGroup_of_discreteTopology πŸ“–mathematicalβ€”IsTopologicalGroupβ€”continuousMul_of_discreteTopology
continuousInv_of_discreteTopology
topologicalGroup_sInf πŸ“–mathematicalIsTopologicalGroupInfSet.sInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”continuousMul_sInf
IsTopologicalGroup.toContinuousMul
continuousInv_sInf
IsTopologicalGroup.toContinuousInv

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