Documentation Verification Report

Monoid

πŸ“ Source: Mathlib/Topology/Algebra/Monoid.lean

Statistics

MetricCount
DefinitionsaddCommMonoidTopologicalClosure, topologicalClosure, addCommSemigroupTopologicalClosure, topologicalClosure, addLeft, addRight, mulLeft, mulRight, addUnits, units, Monoid, commMonoidTopologicalClosure, topologicalClosure, commSemigroupTopologicalClosure, topologicalClosure, addHomOfMemClosureRangeCoe, addHomOfTendsto, addMonoidHomOfMemClosureRangeCoe, addMonoidHomOfTendsto, monoidHomOfMemClosureRangeCoe, monoidHomOfTendsto, mulHomOfMemClosureRangeCoe, mulHomOfTendsto
23
TheoremsisClosed_range_coe, continuousConstSMul_nat, continuousSMul_nat, isClosed_range_coe, instContinuousAdd, coe_topologicalClosure, continuousAdd, isClosed_topologicalClosure, le_topologicalClosure, mem_nhds_zero, top_closure_add_self_eq, top_closure_add_self_subset, topologicalClosure_minimal, coe_topologicalClosure, continuousAdd, isClosed_topologicalClosure, le_topologicalClosure, top_closure_add_self_subset, topologicalClosure_minimal, instContinuousAdd, addUnits_map, nsmul, pow, units_map, induced, of_nhds_zero, to_continuousVAdd, to_continuousVAdd_op, nsmul, pow, coe_addLeft, coe_addRight, coe_mulLeft, coe_mulRight, induced, of_nhds_one, to_continuousSMul, to_continuousSMul_op, nsmul, pow, nsmul, pow, add_self, mul_self, add_const, const_add, const_mul, mul_const, nsmul, pow, val_addUnits, val_inv_units, val_neg_addUnits, val_units, const_mul, mul_const, const_mul, mul_const, tendsto_cocompact_mul_left, tendsto_cocompact_mul_right, isOpen_singleton_zero, add, mul, nsmul, pow, add, mul, continuousConstSMul, exists_finset_mulSupport, exists_finset_support, isClosed_range_coe, isClosed_range_coe, instContinuousMul, continuousAdd, continuousAdd', continuousMul, continuousMul', continuousAdd, continuousMul, continuousConstSMul, add, mul, nsmul, pow, coe_topologicalClosure, continuousMul, isClosed_topologicalClosure, le_topologicalClosure, mem_nhds_one, top_closure_mul_self_eq, top_closure_mul_self_subset, topologicalClosure_minimal, coe_topologicalClosure, continuousMul, isClosed_topologicalClosure, le_topologicalClosure, top_closure_mul_self_subset, topologicalClosure_minimal, tendsto_mul_zero_of_disjoint_cocompact_left, tendsto_mul_zero_of_disjoint_cocompact_right, continuousAdd, continuousMul, instContinuousMul, continuousConstVAdd, continuousConstVAdd, addHomOfMemClosureRangeCoe_apply, addHomOfTendsto_apply, addMonoidHomOfMemClosureRangeCoe_apply, addMonoidHomOfTendsto_apply, continuousAdd_iInf, continuousAdd_induced, continuousAdd_inf, continuousAdd_of_comm_of_nhds_zero, continuousAdd_of_discreteTopology, continuousAdd_sInf, continuousAt_nsmul, continuousAt_pow, continuousMul_iInf, continuousMul_induced, continuousMul_inf, continuousMul_of_comm_of_nhds_one, continuousMul_of_discreteTopology, continuousMul_sInf, continuousOn_finset_prod, continuousOn_finset_sum, continuousOn_list_prod, continuousOn_list_sum, continuousOn_multiset_prod, continuousOn_multiset_sum, continuousOn_nsmul, continuousOn_pow, continuous_add_left, continuous_add_right, continuous_finprod, continuous_finprod_cond, continuous_finset_prod, continuous_finset_sum, continuous_finsum, continuous_finsum_cond, continuous_list_prod, continuous_list_sum, continuous_mul_left, continuous_mul_right, continuous_multiset_prod, continuous_multiset_sum, continuous_nsmul, continuous_one, continuous_pow, continuous_zero, eventuallyEq_prod, eventuallyEq_sum, exists_mem_nhds_zero_mul_subset, exists_nhds_one_split, exists_nhds_one_split4, exists_nhds_zero_half, exists_nhds_zero_quarter, exists_open_nhds_one_mul_subset, exists_open_nhds_one_split, exists_open_nhds_zero_add_subset, exists_open_nhds_zero_half, finprod_eventually_eq_prod, finsum_eventually_eq_sum, instContinuousAddAdditiveOfContinuousMul, instContinuousAddOrderDual, instContinuousAddULift, instContinuousMulMultiplicativeOfContinuousAdd, instContinuousMulOrderDual, instContinuousMulULift, isClosed_setOf_map_add, isClosed_setOf_map_mul, isClosed_setOf_map_one, isClosed_setOf_map_zero, le_nhds_add, le_nhds_mul, monoidHomOfMemClosureRangeCoe_apply, monoidHomOfTendsto_apply, mulHomOfMemClosureRangeCoe_apply, mulHomOfTendsto_apply, nhds_add_nhds_zero, nhds_mul_nhds_one, nhds_one_mul_nhds, nhds_zero_add_nhds, nonneg_nhds_iff, one_le_nhds_iff, tendsto_add, tendsto_finset_prod, tendsto_finset_sum, tendsto_list_prod, tendsto_list_sum, tendsto_mul, tendsto_mul_cocompact_nhds_zero, tendsto_mul_cofinite_nhds_zero, tendsto_mul_coprod_nhds_zero_inf_of_disjoint_cocompact, tendsto_mul_nhds_zero_of_disjoint_cocompact, tendsto_mul_nhds_zero_prod_of_disjoint_cocompact, tendsto_mul_prod_nhds_zero_of_disjoint_cocompact, tendsto_multiset_prod, tendsto_multiset_sum
198
Total221

AddHom

Theorems

NameKindAssumesProvesValidatesDepends On
isClosed_range_coe πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
Set.range
AddHom
DFunLike.coe
funLike
β€”isClosed_of_closure_subset
addHomClass

AddMonoid

Theorems

NameKindAssumesProvesValidatesDepends On
continuousConstSMul_nat πŸ“–mathematicalβ€”ContinuousConstSMul
toNatSMul
β€”continuous_nsmul
continuousSMul_nat πŸ“–mathematicalβ€”ContinuousSMul
toNatSMul
instTopologicalSpaceNat
β€”continuous_prod_of_discrete_left
instDiscreteTopologyNat
continuous_nsmul

AddMonoidHom

Theorems

NameKindAssumesProvesValidatesDepends On
isClosed_range_coe πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
Set.range
AddMonoidHom
AddZeroClass.toAddZero
DFunLike.coe
instFunLike
β€”isClosed_of_closure_subset
instAddMonoidHomClass

AddOpposite

Theorems

NameKindAssumesProvesValidatesDepends On
instContinuousAdd πŸ“–mathematicalβ€”ContinuousAdd
AddOpposite
instTopologicalSpaceAddOpposite
instAdd
β€”Continuous.comp
continuous_op
Continuous.add
Continuous.snd'
continuous_unop
Continuous.fst'

AddSubmonoid

Definitions

NameCategoryTheorems
addCommMonoidTopologicalClosure πŸ“–CompOpβ€”
topologicalClosure πŸ“–CompOp
5 mathmath: topologicalClosure_minimal, isClosed_topologicalClosure, topologicalClosure_addSubgroupClosure_toAddSubmonoid, le_topologicalClosure, coe_topologicalClosure

Theorems

NameKindAssumesProvesValidatesDepends On
coe_topologicalClosure πŸ“–mathematicalβ€”SetLike.coe
AddSubmonoid
AddMonoid.toAddZeroClass
instSetLike
topologicalClosure
closure
β€”β€”
continuousAdd πŸ“–mathematicalβ€”ContinuousAdd
AddSubmonoid
AddMonoid.toAddZeroClass
SetLike.instMembership
instSetLike
instTopologicalSpaceSubtype
add
β€”AddSubsemigroup.continuousAdd
isClosed_topologicalClosure πŸ“–mathematicalβ€”IsClosed
SetLike.coe
AddSubmonoid
AddMonoid.toAddZeroClass
instSetLike
topologicalClosure
β€”isClosed_closure
le_topologicalClosure πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosure
β€”subset_closure
mem_nhds_zero πŸ“–mathematicalIsOpen
SetLike.coe
AddSubmonoid
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instSetLike
Set
Filter
Filter.instMembership
nhds
AddZero.toZero
AddZeroClass.toAddZero
β€”IsOpen.mem_nhds
zero_mem
top_closure_add_self_eq πŸ“–mathematicalβ€”Set
Set.add
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
closure
SetLike.coe
AddSubmonoid
instSetLike
β€”Set.Subset.antisymm
top_closure_add_self_subset
subset_closure
zero_mem
add_zero
top_closure_add_self_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
Set.add
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
closure
SetLike.coe
AddSubmonoid
instSetLike
β€”Set.image2_subset_iff
map_mem_closureβ‚‚
continuous_add
add_mem
topologicalClosure_minimal πŸ“–mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosureβ€”closure_minimal

AddSubsemigroup

Definitions

NameCategoryTheorems
addCommSemigroupTopologicalClosure πŸ“–CompOpβ€”
topologicalClosure πŸ“–CompOp
4 mathmath: topologicalClosure_minimal, coe_topologicalClosure, isClosed_topologicalClosure, le_topologicalClosure

Theorems

NameKindAssumesProvesValidatesDepends On
coe_topologicalClosure πŸ“–mathematicalβ€”SetLike.coe
AddSubsemigroup
AddSemigroup.toAdd
instSetLike
topologicalClosure
closure
β€”β€”
continuousAdd πŸ“–mathematicalβ€”ContinuousAdd
AddSubsemigroup
AddSemigroup.toAdd
SetLike.instMembership
instSetLike
instTopologicalSpaceSubtype
AddMemClass.toAddSemigroup
instAddMemClass
β€”Topology.IsInducing.continuousAdd
instAddMemClass
AddHom.addHomClass
isClosed_topologicalClosure πŸ“–mathematicalβ€”IsClosed
SetLike.coe
AddSubsemigroup
AddSemigroup.toAdd
instSetLike
topologicalClosure
β€”isClosed_closure
le_topologicalClosure πŸ“–mathematicalβ€”AddSubsemigroup
AddSemigroup.toAdd
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosure
β€”subset_closure
top_closure_add_self_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
Set.add
AddSemigroup.toAdd
closure
SetLike.coe
AddSubsemigroup
instSetLike
β€”Set.image2_subset_iff
map_mem_closureβ‚‚
continuous_add
add_mem
topologicalClosure_minimal πŸ“–mathematicalAddSubsemigroup
AddSemigroup.toAdd
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosureβ€”closure_minimal

AddUnits

Theorems

NameKindAssumesProvesValidatesDepends On
instContinuousAdd πŸ“–mathematicalβ€”ContinuousAdd
AddUnits
instTopologicalSpaceAddUnits
instAdd
β€”Topology.IsInducing.continuousAdd
AddMonoidHomClass.toAddHomClass
AddMonoidHom.instAddMonoidHomClass
Prod.continuousAdd
AddOpposite.instContinuousAdd
isInducing_embedProduct

Continuous

Theorems

NameKindAssumesProvesValidatesDepends On
addUnits_map πŸ“–mathematicalContinuous
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddMonoidHom.instFunLike
AddUnits
AddUnits.instTopologicalSpaceAddUnits
AddUnits.instAddZeroClass
AddUnits.map
β€”AddUnits.continuous_iff
comp
AddUnits.continuous_val
AddUnits.continuous_coe_neg
nsmul πŸ“–mathematicalContinuousAddMonoid.toNatSMulβ€”comp
continuous_nsmul
pow πŸ“–mathematicalContinuousMonoid.toNatPowβ€”comp
continuous_pow
units_map πŸ“–mathematicalContinuous
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
MonoidHom.instFunLike
Units
Units.instTopologicalSpaceUnits
Units.instMulOneClass
Units.map
β€”Units.continuous_iff
comp
Units.continuous_val
Units.continuous_coe_inv

ContinuousAdd

Theorems

NameKindAssumesProvesValidatesDepends On
induced πŸ“–mathematicalβ€”ContinuousAdd
TopologicalSpace.induced
DFunLike.coe
β€”continuous_induced_rng
map_add
Continuous.fun_add
Continuous.comp'
continuous_induced_dom
Continuous.fst
continuous_id'
Continuous.snd
of_nhds_zero πŸ“–mathematicalFilter.Tendsto
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SProd.sprod
Filter
Filter.instSProd
nhds
AddZero.toZero
Filter.map
ContinuousAddβ€”continuous_iff_continuousAt
add_assoc
nhds_prod_eq
Filter.prod_map_map_eq
Filter.map_map
Filter.map_mono
to_continuousVAdd πŸ“–mathematicalβ€”ContinuousVAdd
Add.toVAdd
β€”continuous_add
to_continuousVAdd_op πŸ“–mathematicalβ€”ContinuousVAdd
AddOpposite
Add.toVAddAddOpposite
AddOpposite.instTopologicalSpaceAddOpposite
β€”Continuous.fun_add
Continuous.snd
continuous_id'
Continuous.comp'
AddOpposite.continuous_unop
Continuous.fst

ContinuousAt

Theorems

NameKindAssumesProvesValidatesDepends On
nsmul πŸ“–mathematicalContinuousAtAddMonoid.toNatSMulβ€”Filter.Tendsto.nsmul
pow πŸ“–mathematicalContinuousAtMonoid.toNatPowβ€”Filter.Tendsto.pow

ContinuousMap

Definitions

NameCategoryTheorems
addLeft πŸ“–CompOp
1 mathmath: coe_addLeft
addRight πŸ“–CompOp
3 mathmath: coe_addRight, periodic_tsum_comp_add_zsmul, isBigO_norm_restrict_cocompact
mulLeft πŸ“–CompOp
1 mathmath: coe_mulLeft
mulRight πŸ“–CompOp
1 mathmath: coe_mulRight

Theorems

NameKindAssumesProvesValidatesDepends On
coe_addLeft πŸ“–mathematicalβ€”DFunLike.coe
ContinuousMap
instFunLike
addLeft
β€”β€”
coe_addRight πŸ“–mathematicalβ€”DFunLike.coe
ContinuousMap
instFunLike
addRight
β€”β€”
coe_mulLeft πŸ“–mathematicalβ€”DFunLike.coe
ContinuousMap
instFunLike
mulLeft
β€”β€”
coe_mulRight πŸ“–mathematicalβ€”DFunLike.coe
ContinuousMap
instFunLike
mulRight
β€”β€”

ContinuousMul

Theorems

NameKindAssumesProvesValidatesDepends On
induced πŸ“–mathematicalβ€”ContinuousMul
TopologicalSpace.induced
DFunLike.coe
β€”continuous_induced_rng
map_mul
Continuous.fun_mul
Continuous.comp'
continuous_induced_dom
Continuous.fst
continuous_id'
Continuous.snd
of_nhds_one πŸ“–mathematicalFilter.Tendsto
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
SProd.sprod
Filter
Filter.instSProd
nhds
MulOne.toOne
Filter.map
ContinuousMulβ€”continuous_iff_continuousAt
mul_assoc
nhds_prod_eq
Filter.prod_map_map_eq
Filter.map_map
Filter.map_mono
to_continuousSMul πŸ“–mathematicalβ€”ContinuousSMulβ€”continuous_mul
to_continuousSMul_op πŸ“–mathematicalβ€”ContinuousSMul
MulOpposite
Mul.toSMulMulOpposite
MulOpposite.instTopologicalSpaceMulOpposite
β€”Continuous.fun_mul
Continuous.snd
continuous_id'
Continuous.comp'
MulOpposite.continuous_unop
Continuous.fst

ContinuousOn

Theorems

NameKindAssumesProvesValidatesDepends On
nsmul πŸ“–mathematicalContinuousOnAddMonoid.toNatSMulβ€”ContinuousWithinAt.nsmul
pow πŸ“–mathematicalContinuousOnMonoid.toNatPowβ€”ContinuousWithinAt.pow

ContinuousWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
nsmul πŸ“–mathematicalContinuousWithinAtAddMonoid.toNatSMulβ€”Filter.Tendsto.nsmul
pow πŸ“–mathematicalContinuousWithinAtMonoid.toNatPowβ€”Filter.Tendsto.pow

Filter

Theorems

NameKindAssumesProvesValidatesDepends On
tendsto_cocompact_mul_left πŸ“–mathematicalMulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
MulOne.toOne
Tendsto
cocompact
β€”Tendsto.of_tendsto_comp
one_mul
tendsto_id
comap_cocompact_le
continuous_mul_left
tendsto_cocompact_mul_right πŸ“–mathematicalMulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
MulOne.toOne
Tendsto
cocompact
β€”Tendsto.of_tendsto_comp
comp_mul_right
mul_one
tendsto_id
comap_cocompact_le
continuous_mul_right

Filter.HasBasis

Theorems

NameKindAssumesProvesValidatesDepends On
add_self πŸ“–mathematicalFilter.HasBasis
nhds
AddZero.toZero
AddZeroClass.toAddZero
Set
Set.add
AddZero.toAdd
β€”nhds_add_nhds_zero
Filter.mapβ‚‚_add
Filter.map_uncurry_prod
Set.add_image_prod
map
prod_self
mul_self πŸ“–mathematicalFilter.HasBasis
nhds
MulOne.toOne
MulOneClass.toMulOne
Set
Set.mul
MulOne.toMul
β€”nhds_mul_nhds_one
Filter.mapβ‚‚_mul
Filter.map_uncurry_prod
map
prod_self

Filter.Tendsto

Definitions

NameCategoryTheorems
addUnits πŸ“–CompOp
2 mathmath: val_neg_addUnits, val_addUnits
units πŸ“–CompOp
2 mathmath: val_inv_units, val_units

Theorems

NameKindAssumesProvesValidatesDepends On
add_const πŸ“–β€”Filter.Tendsto
nhds
β€”β€”add
tendsto_const_nhds
const_add πŸ“–β€”Filter.Tendsto
nhds
β€”β€”add
tendsto_const_nhds
const_mul πŸ“–β€”Filter.Tendsto
nhds
β€”β€”mul
tendsto_const_nhds
mul_const πŸ“–β€”Filter.Tendsto
nhds
β€”β€”mul
tendsto_const_nhds
nsmul πŸ“–mathematicalFilter.Tendsto
nhds
AddMonoid.toNatSMulβ€”comp
ContinuousAt.tendsto
continuousAt_nsmul
pow πŸ“–mathematicalFilter.Tendsto
nhds
Monoid.toNatPowβ€”comp
ContinuousAt.tendsto
continuousAt_pow
val_addUnits πŸ“–mathematicalFilter.Tendsto
AddUnits.val
nhds
AddUnits
AddUnits.instNeg
addUnitsβ€”β€”
val_inv_units πŸ“–mathematicalFilter.Tendsto
Units.val
nhds
Units
Units.instInv
unitsβ€”β€”
val_neg_addUnits πŸ“–mathematicalFilter.Tendsto
AddUnits.val
nhds
AddUnits
AddUnits.instNeg
addUnitsβ€”β€”
val_units πŸ“–mathematicalFilter.Tendsto
Units.val
nhds
Units
Units.instInv
unitsβ€”β€”

Filter.TendstoNhdsWithinIio

Theorems

NameKindAssumesProvesValidatesDepends On
const_mul πŸ“–β€”Preorder.toLT
Filter.Tendsto
nhdsWithin
Set.Iio
β€”β€”tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
Filter.Tendsto.const_mul
tendsto_nhds_of_tendsto_nhdsWithin
Filter.Eventually.mono
tendsto_nhdsWithin_iff
Set.mem_Iio
mul_lt_mul_of_pos_left
mul_const πŸ“–β€”Preorder.toLT
Filter.Tendsto
nhdsWithin
Set.Iio
β€”β€”tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
Filter.Tendsto.mul_const
tendsto_nhds_of_tendsto_nhdsWithin
Filter.Eventually.mono
tendsto_nhdsWithin_iff
Set.mem_Iio
mul_lt_mul_of_pos_right

Filter.TendstoNhdsWithinIoi

Theorems

NameKindAssumesProvesValidatesDepends On
const_mul πŸ“–β€”Preorder.toLT
Filter.Tendsto
nhdsWithin
Set.Ioi
β€”β€”tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
Filter.Tendsto.const_mul
tendsto_nhds_of_tendsto_nhdsWithin
Filter.Eventually.mono
tendsto_nhdsWithin_iff
Set.mem_Ioi
mul_lt_mul_of_pos_left
mul_const πŸ“–β€”Preorder.toLT
Filter.Tendsto
nhdsWithin
Set.Ioi
β€”β€”tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
Filter.Tendsto.mul_const
tendsto_nhds_of_tendsto_nhdsWithin
Filter.Eventually.mono
tendsto_nhdsWithin_iff
Set.mem_Ioi
mul_lt_mul_of_pos_right

GroupWithZero

Theorems

NameKindAssumesProvesValidatesDepends On
isOpen_singleton_zero πŸ“–mathematicalβ€”IsOpen
Set
Set.instSingletonSet
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
toMonoidWithZero
β€”t1Space_iff_exists_open
zero_ne_one
NeZero.one
toNontrivial
exists_mem_nhds_zero_mul_subset
isCompact_univ
IsOpen.mem_nhds
inv_mul_cancelβ‚€
Set.mul_mem_mul
Set.mem_univ
subset_antisymm
Set.instAntisymmSubset
mem_of_mem_nhds

Inseparable

Theorems

NameKindAssumesProvesValidatesDepends On
add πŸ“–β€”Inseparableβ€”β€”vadd
ContinuousAdd.to_continuousVAdd
mul πŸ“–β€”Inseparableβ€”β€”smul
ContinuousMul.to_continuousSMul
nsmul πŸ“–mathematicalInseparableAddMonoid.toNatSMulβ€”Specializes.antisymm
Specializes.nsmul
specializes
specializes'
pow πŸ“–mathematicalInseparableMonoid.toNatPowβ€”Specializes.antisymm
Specializes.pow
specializes
specializes'

IsCompact

Theorems

NameKindAssumesProvesValidatesDepends On
add πŸ“–mathematicalIsCompactSet
Set.add
AddSemigroup.toAdd
β€”Set.add_image_prod
image
prod
continuous_add
mul πŸ“–mathematicalIsCompactSet
Set.mul
Semigroup.toMul
β€”Set.image_mul_prod
image
prod
continuous_mul

IsScalarTower

Theorems

NameKindAssumesProvesValidatesDepends On
continuousConstSMul πŸ“–mathematicalβ€”ContinuousConstSMulβ€”smul_one_mul
Continuous.fun_mul
continuous_const
continuous_id'

LocallyFinite

Theorems

NameKindAssumesProvesValidatesDepends On
exists_finset_mulSupport πŸ“–mathematicalLocallyFinite
Function.mulSupport
Filter.Eventually
Set
Set.instHasSubset
SetLike.coe
Finset
Finset.instSetLike
nhds
β€”Filter.mem_of_superset
Set.Finite.coe_toFinset
exists_finset_support πŸ“–mathematicalLocallyFinite
Function.support
Filter.Eventually
Set
Set.instHasSubset
SetLike.coe
Finset
Finset.instSetLike
nhds
β€”Filter.mem_of_superset
Set.Finite.coe_toFinset

MonoidHom

Theorems

NameKindAssumesProvesValidatesDepends On
isClosed_range_coe πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
Set.range
MonoidHom
MulOneClass.toMulOne
DFunLike.coe
instFunLike
β€”isClosed_of_closure_subset
instMonoidHomClass

MulHom

Theorems

NameKindAssumesProvesValidatesDepends On
isClosed_range_coe πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
Set.range
MulHom
DFunLike.coe
funLike
β€”isClosed_of_closure_subset
mulHomClass

MulOpposite

Theorems

NameKindAssumesProvesValidatesDepends On
instContinuousMul πŸ“–mathematicalβ€”ContinuousMul
MulOpposite
instTopologicalSpaceMulOpposite
instMul
β€”Continuous.comp
continuous_op
Continuous.mul
Continuous.snd'
continuous_unop
Continuous.fst'

Pi

Theorems

NameKindAssumesProvesValidatesDepends On
continuousAdd πŸ“–mathematicalContinuousAddtopologicalSpace
instAdd
β€”continuous_pi
Continuous.add
Continuous.fst'
continuous_apply
Continuous.snd'
continuousAdd' πŸ“–mathematicalβ€”ContinuousAdd
topologicalSpace
instAdd
β€”continuousAdd
continuousMul πŸ“–mathematicalContinuousMultopologicalSpace
instMul
β€”continuous_pi
Continuous.mul
Continuous.fst'
continuous_apply
Continuous.snd'
continuousMul' πŸ“–mathematicalβ€”ContinuousMul
topologicalSpace
instMul
β€”continuousMul

Prod

Theorems

NameKindAssumesProvesValidatesDepends On
continuousAdd πŸ“–mathematicalβ€”ContinuousAdd
instTopologicalSpaceProd
instAdd
β€”Continuous.prodMk
Continuous.fun_add
Continuous.fst
continuous_id'
Continuous.snd
continuousMul πŸ“–mathematicalβ€”ContinuousMul
instTopologicalSpaceProd
instMul
β€”Continuous.prodMk
Continuous.fun_mul
Continuous.fst
continuous_id'
Continuous.snd

SMulCommClass

Theorems

NameKindAssumesProvesValidatesDepends On
continuousConstSMul πŸ“–mathematicalβ€”ContinuousConstSMulβ€”mul_smul_one
Continuous.fun_mul
continuous_id'
continuous_const

Specializes

Theorems

NameKindAssumesProvesValidatesDepends On
add πŸ“–β€”Specializesβ€”β€”vadd
ContinuousAdd.to_continuousVAdd
mul πŸ“–β€”Specializesβ€”β€”smul
ContinuousMul.to_continuousSMul
nsmul πŸ“–mathematicalSpecializesAddMonoid.toNatSMulβ€”zero_nsmul
specializes_rfl
succ_nsmul
add
pow πŸ“–mathematicalSpecializesMonoid.toNatPowβ€”pow_zero
pow_succ
mul

Submonoid

Definitions

NameCategoryTheorems
commMonoidTopologicalClosure πŸ“–CompOpβ€”
topologicalClosure πŸ“–CompOp
5 mathmath: isClosed_topologicalClosure, le_topologicalClosure, topologicalClosure_minimal, topologicalClosure_subgroupClosure_toSubmonoid, coe_topologicalClosure

Theorems

NameKindAssumesProvesValidatesDepends On
coe_topologicalClosure πŸ“–mathematicalβ€”SetLike.coe
Submonoid
Monoid.toMulOneClass
instSetLike
topologicalClosure
closure
β€”β€”
continuousMul πŸ“–mathematicalβ€”ContinuousMul
Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
instTopologicalSpaceSubtype
mul
β€”Subsemigroup.continuousMul
isClosed_topologicalClosure πŸ“–mathematicalβ€”IsClosed
SetLike.coe
Submonoid
Monoid.toMulOneClass
instSetLike
topologicalClosure
β€”isClosed_closure
le_topologicalClosure πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosure
β€”subset_closure
mem_nhds_one πŸ“–mathematicalIsOpen
SetLike.coe
Submonoid
Monoid.toMulOneClass
CommMonoid.toMonoid
instSetLike
Set
Filter
Filter.instMembership
nhds
MulOne.toOne
MulOneClass.toMulOne
β€”IsOpen.mem_nhds
one_mem
top_closure_mul_self_eq πŸ“–mathematicalβ€”Set
Set.mul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
closure
SetLike.coe
Submonoid
instSetLike
β€”Set.Subset.antisymm
top_closure_mul_self_subset
subset_closure
one_mem
mul_one
top_closure_mul_self_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
Set.mul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
closure
SetLike.coe
Submonoid
instSetLike
β€”Set.image2_subset_iff
map_mem_closureβ‚‚
continuous_mul
mul_mem
topologicalClosure_minimal πŸ“–mathematicalSubmonoid
Monoid.toMulOneClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosureβ€”closure_minimal

Subsemigroup

Definitions

NameCategoryTheorems
commSemigroupTopologicalClosure πŸ“–CompOpβ€”
topologicalClosure πŸ“–CompOp
4 mathmath: topologicalClosure_minimal, coe_topologicalClosure, isClosed_topologicalClosure, le_topologicalClosure

Theorems

NameKindAssumesProvesValidatesDepends On
coe_topologicalClosure πŸ“–mathematicalβ€”SetLike.coe
Subsemigroup
Semigroup.toMul
instSetLike
topologicalClosure
closure
β€”β€”
continuousMul πŸ“–mathematicalβ€”ContinuousMul
Subsemigroup
Semigroup.toMul
SetLike.instMembership
instSetLike
instTopologicalSpaceSubtype
MulMemClass.toSemigroup
instMulMemClass
β€”Topology.IsInducing.continuousMul
instMulMemClass
MulHom.mulHomClass
isClosed_topologicalClosure πŸ“–mathematicalβ€”IsClosed
SetLike.coe
Subsemigroup
Semigroup.toMul
instSetLike
topologicalClosure
β€”isClosed_closure
le_topologicalClosure πŸ“–mathematicalβ€”Subsemigroup
Semigroup.toMul
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosure
β€”subset_closure
top_closure_mul_self_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
Set.mul
Semigroup.toMul
closure
SetLike.coe
Subsemigroup
instSetLike
β€”Set.image2_subset_iff
map_mem_closureβ‚‚
continuous_mul
mul_mem
topologicalClosure_minimal πŸ“–mathematicalSubsemigroup
Semigroup.toMul
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
topologicalClosureβ€”closure_minimal

Tendsto

Theorems

NameKindAssumesProvesValidatesDepends On
tendsto_mul_zero_of_disjoint_cocompact_left πŸ“–mathematicalDisjoint
Filter
Filter.instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
Filter.instCompleteLatticeFilter
Filter.map
Filter.cocompact
Filter.Tendsto
nhds
MulZeroClass.toZero
MulZeroClass.toMulβ€”Filter.Tendsto.comp
tendsto_mul_prod_nhds_zero_of_disjoint_cocompact
Filter.Tendsto.prodMk
Filter.tendsto_map
tendsto_mul_zero_of_disjoint_cocompact_right πŸ“–mathematicalFilter.Tendsto
nhds
MulZeroClass.toZero
Disjoint
Filter
Filter.instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
Filter.instCompleteLatticeFilter
Filter.map
Filter.cocompact
MulZeroClass.toMulβ€”Filter.Tendsto.comp
tendsto_mul_nhds_zero_prod_of_disjoint_cocompact
Filter.Tendsto.prodMk
Filter.tendsto_map

Topology.IsInducing

Theorems

NameKindAssumesProvesValidatesDepends On
continuousAdd πŸ“–mathematicalTopology.IsInducing
DFunLike.coe
ContinuousAddβ€”ContinuousVAdd.continuous_vadd
continuousVAdd
ContinuousAdd.to_continuousVAdd
continuous
map_add
continuousMul πŸ“–mathematicalTopology.IsInducing
DFunLike.coe
ContinuousMulβ€”ContinuousSMul.continuous_smul
continuousSMul
ContinuousMul.to_continuousSMul
continuous
map_mul

Units

Theorems

NameKindAssumesProvesValidatesDepends On
instContinuousMul πŸ“–mathematicalβ€”ContinuousMul
Units
instTopologicalSpaceUnits
instMul
β€”Topology.IsInducing.continuousMul
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
Prod.continuousMul
MulOpposite.instContinuousMul
isInducing_embedProduct

VAddAssocClass

Theorems

NameKindAssumesProvesValidatesDepends On
continuousConstVAdd πŸ“–mathematicalβ€”ContinuousConstVAddβ€”vadd_zero_add
Continuous.fun_add
continuous_const
continuous_id'

VAddCommClass

Theorems

NameKindAssumesProvesValidatesDepends On
continuousConstVAdd πŸ“–mathematicalβ€”ContinuousConstVAddβ€”add_vadd_zero
Continuous.fun_add
continuous_id'
continuous_const

(root)

Definitions

NameCategoryTheorems
Monoid πŸ“–CompData
3 mathmath: CommMonoid.toMonoid_injective, RightCancelMonoid.toMonoid_injective, LeftCancelMonoid.toMonoid_injective
addHomOfMemClosureRangeCoe πŸ“–CompOp
1 mathmath: addHomOfMemClosureRangeCoe_apply
addHomOfTendsto πŸ“–CompOp
1 mathmath: addHomOfTendsto_apply
addMonoidHomOfMemClosureRangeCoe πŸ“–CompOp
2 mathmath: linearMapOfMemClosureRangeCoe_apply, addMonoidHomOfMemClosureRangeCoe_apply
addMonoidHomOfTendsto πŸ“–CompOp
1 mathmath: addMonoidHomOfTendsto_apply
monoidHomOfMemClosureRangeCoe πŸ“–CompOp
1 mathmath: monoidHomOfMemClosureRangeCoe_apply
monoidHomOfTendsto πŸ“–CompOp
1 mathmath: monoidHomOfTendsto_apply
mulHomOfMemClosureRangeCoe πŸ“–CompOp
1 mathmath: mulHomOfMemClosureRangeCoe_apply
mulHomOfTendsto πŸ“–CompOp
1 mathmath: mulHomOfTendsto_apply

Theorems

NameKindAssumesProvesValidatesDepends On
addHomOfMemClosureRangeCoe_apply πŸ“–mathematicalSet
Set.instMembership
closure
Pi.topologicalSpace
Set.range
DFunLike.coe
AddHom
AddHom.funLike
addHomOfMemClosureRangeCoe
β€”β€”
addHomOfTendsto_apply πŸ“–mathematicalFilter.Tendsto
DFunLike.coe
nhds
Pi.topologicalSpace
AddHom
AddHom.funLike
addHomOfTendsto
β€”β€”
addMonoidHomOfMemClosureRangeCoe_apply πŸ“–mathematicalSet
Set.instMembership
closure
Pi.topologicalSpace
Set.range
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoidHom.instFunLike
addMonoidHomOfMemClosureRangeCoe
β€”β€”
addMonoidHomOfTendsto_apply πŸ“–mathematicalFilter.Tendsto
DFunLike.coe
nhds
Pi.topologicalSpace
AddMonoidHom
AddZeroClass.toAddZero
AddMonoidHom.instFunLike
addMonoidHomOfTendsto
β€”β€”
continuousAdd_iInf πŸ“–mathematicalContinuousAddiInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”sInf_range
continuousAdd_sInf
Set.forall_mem_range
continuousAdd_induced πŸ“–mathematicalβ€”ContinuousAdd
TopologicalSpace.induced
DFunLike.coe
β€”Topology.IsInducing.continuousAdd
continuousAdd_inf πŸ“–mathematicalβ€”ContinuousAdd
TopologicalSpace
SemilatticeInf.toMin
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”inf_eq_iInf
continuousAdd_iInf
continuousAdd_of_comm_of_nhds_zero πŸ“–mathematicalFilter.Tendsto
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
SProd.sprod
Filter
Filter.instSProd
nhds
AddZero.toZero
Filter.map
ContinuousAddβ€”ContinuousAdd.of_nhds_zero
add_comm
continuousAdd_of_discreteTopology πŸ“–mathematicalβ€”ContinuousAddβ€”continuous_of_discreteTopology
instDiscreteTopologyProd
continuousAdd_sInf πŸ“–mathematicalContinuousAddInfSet.sInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”continuous_sInf_rng
continuous_sInf_domβ‚‚
ContinuousAdd.continuous_add
continuousAt_nsmul πŸ“–mathematicalβ€”ContinuousAt
AddMonoid.toNatSMul
β€”Continuous.continuousAt
continuous_nsmul
continuousAt_pow πŸ“–mathematicalβ€”ContinuousAt
Monoid.toNatPow
β€”Continuous.continuousAt
continuous_pow
continuousMul_iInf πŸ“–mathematicalContinuousMuliInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”sInf_range
continuousMul_sInf
Set.forall_mem_range
continuousMul_induced πŸ“–mathematicalβ€”ContinuousMul
TopologicalSpace.induced
DFunLike.coe
β€”Topology.IsInducing.continuousMul
continuousMul_inf πŸ“–mathematicalβ€”ContinuousMul
TopologicalSpace
SemilatticeInf.toMin
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”inf_eq_iInf
continuousMul_iInf
continuousMul_of_comm_of_nhds_one πŸ“–mathematicalFilter.Tendsto
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
SProd.sprod
Filter
Filter.instSProd
nhds
MulOne.toOne
Filter.map
ContinuousMulβ€”ContinuousMul.of_nhds_one
mul_comm
continuousMul_of_discreteTopology πŸ“–mathematicalβ€”ContinuousMulβ€”continuous_of_discreteTopology
instDiscreteTopologyProd
continuousMul_sInf πŸ“–mathematicalContinuousMulInfSet.sInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”continuous_sInf_rng
continuous_sInf_domβ‚‚
ContinuousMul.continuous_mul
continuousOn_finset_prod πŸ“–mathematicalContinuousOnFinset.prodβ€”continuousOn_multiset_prod
continuousOn_finset_sum πŸ“–mathematicalContinuousOnFinset.sumβ€”continuousOn_multiset_sum
continuousOn_list_prod πŸ“–mathematicalContinuousOnMulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
MulOne.toOne
β€”continuousWithinAt_iff_continuousAt_restrict
tendsto_list_prod
continuousOn_list_sum πŸ“–mathematicalContinuousOnAddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddZero.toZero
β€”continuousWithinAt_iff_continuousAt_restrict
tendsto_list_sum
continuousOn_multiset_prod πŸ“–mathematicalContinuousOnMultiset.prod
Multiset.map
β€”continuousOn_list_prod
continuousOn_multiset_sum πŸ“–mathematicalContinuousOnMultiset.sum
Multiset.map
β€”Multiset.mem_coe
continuousOn_list_sum
continuousOn_nsmul πŸ“–mathematicalβ€”ContinuousOn
AddMonoid.toNatSMul
β€”Continuous.continuousOn
continuous_nsmul
continuousOn_pow πŸ“–mathematicalβ€”ContinuousOn
Monoid.toNatPow
β€”Continuous.continuousOn
continuous_pow
continuous_add_left πŸ“–mathematicalβ€”Continuousβ€”Continuous.fun_add
continuous_const
continuous_id'
continuous_add_right πŸ“–mathematicalβ€”Continuousβ€”Continuous.fun_add
continuous_id'
continuous_const
continuous_finprod πŸ“–mathematicalContinuous
LocallyFinite
Function.mulSupport
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
finprodβ€”continuous_iff_continuousAt
finprod_eventually_eq_prod
ContinuousAt.congr
tendsto_finset_prod
Continuous.continuousAt
Filter.EventuallyEq.symm
continuous_finprod_cond πŸ“–mathematicalContinuous
LocallyFinite
Function.mulSupport
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
finprodβ€”continuous_finprod
LocallyFinite.comp_injective
Subtype.coe_injective
continuous_finset_prod πŸ“–mathematicalContinuousFinset.prodβ€”continuous_multiset_prod
continuous_finset_sum πŸ“–mathematicalContinuousFinset.sumβ€”continuous_multiset_sum
continuous_finsum πŸ“–mathematicalContinuous
LocallyFinite
Function.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
finsumβ€”continuous_iff_continuousAt
finsum_eventually_eq_sum
ContinuousAt.congr
tendsto_finset_sum
Continuous.continuousAt
Filter.EventuallyEq.symm
continuous_finsum_cond πŸ“–mathematicalContinuous
LocallyFinite
Function.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
finsumβ€”finsum_subtype_eq_finsum_cond
continuous_finsum
LocallyFinite.comp_injective
Subtype.coe_injective
continuous_list_prod πŸ“–mathematicalContinuousMulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
MulOne.toOne
β€”continuous_iff_continuousAt
tendsto_list_prod
continuous_list_sum πŸ“–mathematicalContinuousAddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddZero.toZero
β€”continuous_iff_continuousAt
tendsto_list_sum
continuous_mul_left πŸ“–mathematicalβ€”Continuousβ€”Continuous.fun_mul
continuous_const
continuous_id'
continuous_mul_right πŸ“–mathematicalβ€”Continuousβ€”Continuous.fun_mul
continuous_id'
continuous_const
continuous_multiset_prod πŸ“–mathematicalContinuousMultiset.prod
Multiset.map
β€”continuous_list_prod
continuous_multiset_sum πŸ“–mathematicalContinuousMultiset.sum
Multiset.map
β€”Multiset.mem_coe
continuous_list_sum
continuous_nsmul πŸ“–mathematicalβ€”Continuous
AddMonoid.toNatSMul
β€”zero_nsmul
continuous_const
succ_nsmul'
Continuous.add
continuous_id
continuous_one πŸ“–mathematicalβ€”Continuous
Pi.instOne
β€”continuous_const
continuous_pow πŸ“–mathematicalβ€”Continuous
Monoid.toNatPow
β€”pow_zero
continuous_const
pow_succ'
Continuous.mul
continuous_id
continuous_zero πŸ“–mathematicalβ€”Continuous
Pi.instZero
β€”continuous_const
eventuallyEq_prod πŸ“–mathematicalFilter.EventuallyEqFinset.prod
Pi.commMonoid
β€”Filter.eventually_all_finset
Filter.mp_mem
Filter.univ_mem'
Finset.prod_apply
Finset.prod_congr
eventuallyEq_sum πŸ“–mathematicalFilter.EventuallyEqFinset.sum
Pi.addCommMonoid
β€”Filter.eventually_all_finset
Filter.mp_mem
Filter.univ_mem'
Finset.sum_apply
Finset.sum_congr
exists_mem_nhds_zero_mul_subset πŸ“–mathematicalIsCompact
Set
Filter
Filter.instMembership
nhds
MulZeroClass.toZero
Set.instHasSubset
Set.mul
MulZeroClass.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
MulZeroClass.mul_zero
Filter.mem_prod_iff
Filter.mem_map
nhds_prod_eq
mem_nhdsWithin_of_mem_nhds
Set.image_mul_prod
Set.image_subset_iff
exists_nhds_one_split πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
MulOne.toOne
MulOneClass.toMulOne
Set.instMembership
MulOne.toMul
β€”exists_open_nhds_one_split
IsOpen.mem_nhds
exists_nhds_one_split4 πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
Set.instMembership
MulOne.toMul
β€”exists_nhds_one_split
mul_assoc
exists_nhds_zero_half πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
AddZero.toZero
AddZeroClass.toAddZero
Set.instMembership
AddZero.toAdd
β€”exists_open_nhds_zero_half
IsOpen.mem_nhds
exists_nhds_zero_quarter πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
Set.instMembership
AddZero.toAdd
β€”exists_nhds_zero_half
add_assoc
exists_open_nhds_one_mul_subset πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
MulOne.toOne
MulOneClass.toMulOne
IsOpen
Set.instMembership
Set.instHasSubset
Set.mul
MulOne.toMul
β€”exists_open_nhds_one_split
exists_open_nhds_one_split πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
MulOne.toOne
MulOneClass.toMulOne
IsOpen
Set.instMembership
MulOne.toMul
β€”tendsto_mul
one_mul
exists_nhds_square
exists_open_nhds_zero_add_subset πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
AddZero.toZero
AddZeroClass.toAddZero
IsOpen
Set.instMembership
Set.instHasSubset
Set.add
AddZero.toAdd
β€”Set.add_subset_iff
exists_open_nhds_zero_half
exists_open_nhds_zero_half πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
AddZero.toZero
AddZeroClass.toAddZero
IsOpen
Set.instMembership
AddZero.toAdd
β€”tendsto_add
zero_add
Set.prod_subset_iff
exists_nhds_square
finprod_eventually_eq_prod πŸ“–mathematicalLocallyFinite
Function.mulSupport
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
Filter.Eventually
finprod
Finset.prod
nhds
β€”LocallyFinite.exists_finset_mulSupport
Filter.Eventually.mono
finprod_eq_prod_of_mulSupport_subset
finsum_eventually_eq_sum πŸ“–mathematicalLocallyFinite
Function.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
Filter.Eventually
finsum
Finset.sum
nhds
β€”LocallyFinite.exists_finset_support
Filter.Eventually.mono
finsum_eq_sum_of_support_subset
instContinuousAddAdditiveOfContinuousMul πŸ“–mathematicalβ€”ContinuousAdd
Additive
instTopologicalSpaceAdditive
Additive.add
β€”continuous_mul
instContinuousAddOrderDual πŸ“–mathematicalβ€”ContinuousAdd
OrderDual
OrderDual.instTopologicalSpace
instAddOrderDual
β€”β€”
instContinuousAddULift πŸ“–mathematicalβ€”ContinuousAdd
ULift.topologicalSpace
ULift.add
β€”Continuous.comp
continuous_uliftUp
Continuous.fun_add
Continuous.comp'
continuous_uliftDown
Continuous.fst
continuous_id'
Continuous.snd
instContinuousMulMultiplicativeOfContinuousAdd πŸ“–mathematicalβ€”ContinuousMul
Multiplicative
instTopologicalSpaceMultiplicative
Multiplicative.mul
β€”continuous_add
instContinuousMulOrderDual πŸ“–mathematicalβ€”ContinuousMul
OrderDual
OrderDual.instTopologicalSpace
instMulOrderDual
β€”β€”
instContinuousMulULift πŸ“–mathematicalβ€”ContinuousMul
ULift.topologicalSpace
ULift.mul
β€”Continuous.comp
continuous_uliftUp
Continuous.fun_mul
Continuous.comp'
continuous_uliftDown
Continuous.fst
continuous_id'
Continuous.snd
isClosed_setOf_map_add πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
setOf
β€”Set.setOf_forall
isClosed_iInter
isClosed_eq
continuous_apply
Continuous.fun_add
isClosed_setOf_map_mul πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
setOf
β€”Set.setOf_forall
isClosed_iInter
isClosed_eq
continuous_apply
Continuous.fun_mul
isClosed_setOf_map_one πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
setOf
β€”isClosed_eq
continuous_apply
continuous_const
isClosed_setOf_map_zero πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
setOf
β€”isClosed_eq
continuous_apply
continuous_const
le_nhds_add πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.instAdd
nhds
β€”Filter.mapβ‚‚_add
Filter.map_uncurry_prod
nhds_prod_eq
Continuous.tendsto
continuous_add
le_nhds_mul πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.instMul
nhds
β€”Filter.mapβ‚‚_mul
Filter.map_uncurry_prod
nhds_prod_eq
Continuous.tendsto
continuous_mul
monoidHomOfMemClosureRangeCoe_apply πŸ“–mathematicalSet
Set.instMembership
closure
Pi.topologicalSpace
Set.range
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
MonoidHom.instFunLike
monoidHomOfMemClosureRangeCoe
β€”β€”
monoidHomOfTendsto_apply πŸ“–mathematicalFilter.Tendsto
DFunLike.coe
nhds
Pi.topologicalSpace
MonoidHom
MulOneClass.toMulOne
MonoidHom.instFunLike
monoidHomOfTendsto
β€”β€”
mulHomOfMemClosureRangeCoe_apply πŸ“–mathematicalSet
Set.instMembership
closure
Pi.topologicalSpace
Set.range
DFunLike.coe
MulHom
MulHom.funLike
mulHomOfMemClosureRangeCoe
β€”β€”
mulHomOfTendsto_apply πŸ“–mathematicalFilter.Tendsto
DFunLike.coe
nhds
Pi.topologicalSpace
MulHom
MulHom.funLike
mulHomOfTendsto
β€”β€”
nhds_add_nhds_zero πŸ“–mathematicalβ€”Filter
Filter.instAdd
AddZero.toAdd
AddZeroClass.toAddZero
nhds
AddZero.toZero
β€”LE.le.antisymm
LE.le.trans_eq
le_nhds_add
add_zero
le_add_of_nonneg_right
Filter.addLeftMono
pure_le_nhds
nhds_mul_nhds_one πŸ“–mathematicalβ€”Filter
Filter.instMul
MulOne.toMul
MulOneClass.toMulOne
nhds
MulOne.toOne
β€”LE.le.antisymm
LE.le.trans_eq
le_nhds_mul
mul_one
le_mul_of_one_le_right'
Filter.mulLeftMono
pure_le_nhds
nhds_one_mul_nhds πŸ“–mathematicalβ€”Filter
Filter.instMul
MulOne.toMul
MulOneClass.toMulOne
nhds
MulOne.toOne
β€”LE.le.antisymm
LE.le.trans_eq
le_nhds_mul
one_mul
le_mul_of_one_le_left'
Filter.mulRightMono
pure_le_nhds
nhds_zero_add_nhds πŸ“–mathematicalβ€”Filter
Filter.instAdd
AddZero.toAdd
AddZeroClass.toAddZero
nhds
AddZero.toZero
β€”LE.le.antisymm
LE.le.trans_eq
le_nhds_add
zero_add
le_add_of_nonneg_left
Filter.addRightMono
pure_le_nhds
nonneg_nhds_iff πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.instZero
nhds
β€”pure_le_nhds_iff
one_le_nhds_iff πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.instOne
nhds
β€”pure_le_nhds_iff
tendsto_add πŸ“–mathematicalβ€”Filter.Tendsto
nhds
instTopologicalSpaceProd
β€”continuous_iff_continuousAt
ContinuousAdd.continuous_add
tendsto_finset_prod πŸ“–mathematicalFilter.Tendsto
nhds
Finset.prodβ€”tendsto_multiset_prod
tendsto_finset_sum πŸ“–mathematicalFilter.Tendsto
nhds
Finset.sumβ€”tendsto_multiset_sum
tendsto_list_prod πŸ“–mathematicalFilter.Tendsto
nhds
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
MulOne.toOne
β€”Filter.Tendsto.mul
tendsto_list_sum πŸ“–mathematicalFilter.Tendsto
nhds
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddZero.toZero
β€”tendsto_const_nhds
Filter.Tendsto.add
tendsto_mul πŸ“–mathematicalβ€”Filter.Tendsto
nhds
instTopologicalSpaceProd
β€”continuous_iff_continuousAt
ContinuousMul.continuous_mul
tendsto_mul_cocompact_nhds_zero πŸ“–mathematicalContinuous
Filter.Tendsto
Filter.cocompact
nhds
MulZeroClass.toZero
MulZeroClass.toMul
instTopologicalSpaceProd
β€”IsCompact.prod
Filter.Tendsto.isCompact_insert_range_of_cocompact
Filter.eventually_map
Filter.Eventually.of_forall
Set.mem_insert_of_mem
Set.mem_range_self
Filter.disjoint_cocompact_right
Filter.coprod_cocompact
Filter.Tendsto.prodMap_coprod
Filter.Tendsto.comp
tendsto_mul_nhds_zero_of_disjoint_cocompact
Filter.tendsto_map
tendsto_mul_cofinite_nhds_zero πŸ“–mathematicalFilter.Tendsto
Filter.cofinite
nhds
MulZeroClass.toZero
MulZeroClass.toMulβ€”Filter.cocompact_eq_cofinite
instDiscreteTopologyProd
discreteTopology_bot
tendsto_mul_cocompact_nhds_zero
continuous_of_discreteTopology
tendsto_mul_coprod_nhds_zero_inf_of_disjoint_cocompact πŸ“–mathematicalDisjoint
Filter
Filter.instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
Filter.instCompleteLatticeFilter
Filter.cocompact
instTopologicalSpaceProd
Filter.Tendsto
MulZeroClass.toMul
Filter.instInf
Filter.coprod
nhds
MulZeroClass.toZero
β€”inf_le_inf_left
Filter.le_prod_map_fst_snd
Filter.coprod_inf_prod_le
Filter.Tendsto.mono_left
Filter.Tendsto.sup
tendsto_mul_nhds_zero_prod_of_disjoint_cocompact
disjoint_map_cocompact
continuous_snd
tendsto_mul_prod_nhds_zero_of_disjoint_cocompact
continuous_fst
tendsto_mul_nhds_zero_of_disjoint_cocompact πŸ“–mathematicalDisjoint
Filter
Filter.instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
Filter.instCompleteLatticeFilter
Filter.cocompact
instTopologicalSpaceProd
Filter.coprod
nhds
MulZeroClass.toZero
Filter.Tendsto
MulZeroClass.toMul
β€”inf_eq_right
tendsto_mul_coprod_nhds_zero_inf_of_disjoint_cocompact
tendsto_mul_nhds_zero_prod_of_disjoint_cocompact πŸ“–mathematicalDisjoint
Filter
Filter.instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
Filter.instCompleteLatticeFilter
Filter.cocompact
Filter.Tendsto
MulZeroClass.toMul
SProd.sprod
Filter.instSProd
nhds
MulZeroClass.toZero
β€”Filter.map_mono
nhds_prod_le_of_disjoint_cocompact
Continuous.tendsto_nhdsSet_nhds
continuous_mul
mul_eq_zero_of_left
tendsto_mul_prod_nhds_zero_of_disjoint_cocompact πŸ“–mathematicalDisjoint
Filter
Filter.instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
Filter.instCompleteLatticeFilter
Filter.cocompact
Filter.Tendsto
MulZeroClass.toMul
SProd.sprod
Filter.instSProd
nhds
MulZeroClass.toZero
β€”Filter.map_mono
prod_nhds_le_of_disjoint_cocompact
Continuous.tendsto_nhdsSet_nhds
continuous_mul
mul_eq_zero_of_right
tendsto_multiset_prod πŸ“–mathematicalFilter.Tendsto
nhds
Multiset.prod
Multiset.map
β€”tendsto_list_prod
tendsto_multiset_sum πŸ“–mathematicalFilter.Tendsto
nhds
Multiset.sum
Multiset.map
β€”Multiset.mem_coe
tendsto_list_sum

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