Documentation Verification Report

Membership

📁 Source: Mathlib/Algebra/Group/Submonoid/Membership.lean

Statistics

MetricCount
DefinitionsaddGroupMultiples, decidableMemMultiples, fintypeMultiples, multiples, decidableMemPowers, fintypePowers, groupPowers, pow, powLogEquiv, powers
10
Theoremsadd_mem_sup, card_bot, card_le_one_iff_eq_bot, closure_eq_image_sum, closure_eq_mrange, closure_induction_left, closure_induction_right, closure_singleton_eq, closure_singleton_zero, coe_iSup_of_directed, coe_multiples, coe_sSup_of_directedOn, eq_bot_iff_card, eq_bot_of_card_eq, eq_bot_of_card_le, exists_list_of_mem_closure, exists_mem_sup, exists_multiset_of_mem_closure, forall_mem_sup, iSup_induction, iSup_induction', induction_of_closure_eq_top_left, induction_of_closure_eq_top_right, mem_biSup_of_directedOn, mem_closure_pair, mem_closure_singleton, mem_iSup_of_directed, mem_iSup_of_mem, mem_iSup_prop, mem_multiples, mem_multiples_iff, mem_sSup_of_directedOn, mem_sSup_of_mem, mem_sup, mem_sup_left, mem_sup_right, multiples_eq_closure, multiples_le, multiples_zero, sup_eq_range, closure_range_of, mrange_lift, closure_range_of, mrange_lift, coe_powers, of_mclosure_eq_top, addSubmonoidClosure_one, of_mclosure_eq_top, card_bot, card_le_one_iff_eq_bot, closure_eq_image_prod, closure_eq_mrange, closure_induction_left, closure_induction_right, closure_singleton_eq, closure_singleton_one, coe_iSup_of_directed, coe_powers, coe_sSup_of_directedOn, eq_bot_iff_card, eq_bot_of_card_eq, eq_bot_of_card_le, exists_list_of_mem_closure, exists_mem_sup, exists_multiset_of_mem_closure, forall_mem_sup, iSup_induction, iSup_induction', induction_of_closure_eq_top_left, induction_of_closure_eq_top_right, log_mul, log_pow_eq_self, log_pow_int_eq_self, map_powers, mem_biSup_of_directedOn, mem_closure_pair, mem_closure_singleton, mem_closure_singleton_self, mem_iSup_of_directed, mem_iSup_of_mem, mem_iSup_prop, mem_powers, mem_powers_iff, mem_sSup_of_directedOn, mem_sSup_of_mem, mem_sup, mem_sup_left, mem_sup_right, mul_mem_sup, powLogEquiv_apply, powLogEquiv_symm_apply, pow_apply, pow_coe, pow_log_eq_self, pow_right_injective_iff_pow_injective, powers_eq_closure, powers_le, powers_one, sup_eq_range, of_mclosure_eq_top, of_mclosure_eq_top, ofAdd_image_multiples_eq_powers_ofAdd, ofMul_image_powers_eq_multiples_ofMul
103
Total113

AddSubmonoid

Definitions

NameCategoryTheorems
addGroupMultiples 📖CompOp
decidableMemMultiples 📖CompOp
fintypeMultiples 📖CompOp
1 mathmath: addOrderOf_eq_card_multiples
multiples 📖CompOp
35 mathmath: addOrderOf_eq_card_multiples, mem_multiples_iff, multiples_eq_closure, multiples_zero, mem_multiples_of_prime_card, multiples_le, IsAddCyclic.exists_addMonoid_generator, multiplesEquivMultiples_apply, Function.Periodic.map_vadd_multiples, IsOfFinAddOrder.natCard_multiples_le_addOrderOf, LocalizationMap.AwayMap.lift_comp, ofMul_image_powers_eq_multiples_ofMul, finEquivMultiples_symm_apply, mem_multiples, Nat.card_addSubmonoidMultiples, mem_multiples_iff_mem_zmultiples, infinite_multiples, multiples_fg, multiples_eq_zmultiples, AddMonoid.multiples_fg, IsOfFinAddOrder.mem_multiples_iff_mem_range_addOrderOf, multiples_eq_top_of_prime_card, coe_multiples, finite_multiples, multiples_le_zmultiples, finEquivMultiples_apply, IsOfFinAddOrder.multiples_eq_image_range_addOrderOf, AddLocalization.Away.mk_eq_addMonoidOf_mk', IsOfFinAddOrder.finite_multiples, AddSubgroup.toAddSubmonoid_zmultiples, mem_multiples_iff_mem_range_addOrderOf, IsOfFinAddOrder.mem_multiples_iff_mem_zmultiples, IsOfFinAddOrder.multiples_eq_zmultiples, ofAdd_image_multiples_eq_powers_ofAdd, LocalizationMap.AwayMap.lift_eq

Theorems

NameKindAssumesProvesValidatesDepends On
add_mem_sup 📖mathematicalAddSubmonoid
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
AddZero.toAdd
AddZeroClass.toAddZero
add_mem
mem_sup_left
mem_sup_right
card_bot 📖mathematicalFintype.card
AddSubmonoid
AddMonoid.toAddZeroClass
SetLike.instMembership
instSetLike
Bot.bot
instBot
Fintype.card_eq_one_iff
Set.mem_singleton
mem_bot
card_le_one_iff_eq_bot 📖mathematicalFintype.card
AddSubmonoid
AddMonoid.toAddZeroClass
SetLike.instMembership
instSetLike
Bot.bot
instBot
eq_bot_iff_forall
Fintype.card_le_one_iff
Fintype.card_congr'
mem_bot
Fintype.card_unique
le_refl
closure_eq_image_sum 📖mathematicalSetLike.coe
AddSubmonoid
AddMonoid.toAddZeroClass
instSetLike
closure
Set.image
AddZero.toAdd
AddZeroClass.toAddZero
AddZero.toZero
setOf
AddMonoidHom.instAddMonoidHomClass
closure_eq_mrange
AddMonoidHom.coe_mrange
Set.range_list_map_coe
Set.range_comp
FreeAddMonoid.lift_apply
closure_eq_mrange 📖mathematicalclosure
AddMonoid.toAddZeroClass
AddMonoidHom.mrange
FreeAddMonoid
Set
Set.instMembership
AddRightCancelMonoid.toAddMonoid
AddCancelMonoid.toAddRightCancelMonoid
FreeAddMonoid.instAddCancelMonoid
AddMonoidHom
AddZeroClass.toAddZero
AddMonoidHom.instFunLike
AddMonoidHom.instAddMonoidHomClass
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
FreeAddMonoid.lift
AddMonoidHom.instAddMonoidHomClass
FreeAddMonoid.mrange_lift
Subtype.range_coe
closure_induction_left 📖AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ZeroMemClass.zero_mem
AddSubmonoid
instSetLike
AddSubmonoidClass.toZeroMemClass
instAddSubmonoidClass
closure
AddZero.toAdd
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
subset_closure
SetLike.instMembership
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
instAddSubmonoidClass
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
subset_closure
AddMonoidHom.instAddMonoidHomClass
closure_eq_mrange
FreeAddMonoid.inductionOn'
map_add
AddMonoidHomClass.toAddHomClass
Subtype.prop
AddMonoidHom.mem_mrange
closure_induction_right 📖AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ZeroMemClass.zero_mem
AddSubmonoid
instSetLike
AddSubmonoidClass.toZeroMemClass
instAddSubmonoidClass
closure
AddZero.toAdd
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
subset_closure
SetLike.instMembership
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
instAddSubmonoidClass
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
subset_closure
closure_induction_left
op_closure
closure_singleton_eq 📖mathematicalclosure
AddMonoid.toAddZeroClass
Set
Set.instSingletonSet
AddMonoidHom.mrange
Nat.instAddMonoid
AddMonoidHom
AddZeroClass.toAddZero
AddMonoidHom.instFunLike
AddMonoidHom.instAddMonoidHomClass
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
multiplesHom
closure_eq_of_le
AddMonoidHom.instAddMonoidHomClass
Set.singleton_subset_iff
one_nsmul
nsmul_mem
instAddSubmonoidClass
subset_closure
Set.mem_singleton
closure_singleton_zero 📖mathematicalclosure
AddMonoid.toAddZeroClass
Set
Set.instSingletonSet
AddZero.toZero
AddZeroClass.toAddZero
Bot.bot
AddSubmonoid
instBot
nsmul_zero
coe_iSup_of_directed 📖mathematicalDirected
AddSubmonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SetLike.coe
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
Set.iUnion
Set.ext
SetLike.mem_coe
mem_iSup_of_directed
Set.mem_iUnion
coe_multiples 📖mathematicalSetLike.coe
AddSubmonoid
AddMonoid.toAddZeroClass
instSetLike
multiples
Set.range
AddMonoid.toNatSMul
coe_sSup_of_directedOn 📖mathematicalSet.Nonempty
AddSubmonoid
DirectedOn
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SetLike.coe
instSetLike
SupSet.sSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
Set.iUnion
Set
Set.instMembership
Set.ext
SetLike.mem_coe
mem_sSup_of_directedOn
Set.mem_iUnion
eq_bot_iff_card 📖mathematicalBot.bot
AddSubmonoid
AddMonoid.toAddZeroClass
instBot
Fintype.card
SetLike.instMembership
instSetLike
card_bot
eq_bot_of_card_eq
eq_bot_of_card_eq 📖mathematicalFintype.card
AddSubmonoid
AddMonoid.toAddZeroClass
SetLike.instMembership
instSetLike
Bot.bot
instBot
eq_bot_of_card_le
le_of_eq
eq_bot_of_card_le 📖mathematicalFintype.card
AddSubmonoid
AddMonoid.toAddZeroClass
SetLike.instMembership
instSetLike
Bot.bot
instBot
Fintype.card_le_one_iff_subsingleton
eq_bot_of_subsingleton
exists_list_of_mem_closure 📖mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
SetLike.instMembership
instSetLike
closure
Set
Set.instMembership
AddZero.toAdd
AddZeroClass.toAddZero
AddZero.toZero
Set.mem_image
closure_eq_image_sum
SetLike.mem_coe
exists_mem_sup 📖mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
AddZero.toAdd
AddZeroClass.toAddZero
mem_sup
exists_exists_and_exists_and_eq_and
exists_multiset_of_mem_closure 📖mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
SetLike.instMembership
instSetLike
closure
Set
Set.instMembership
Multiset.sum
exists_list_of_mem_closure
Multiset.sum_coe
forall_mem_sup 📖mathematicalAddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
mem_sup
iSup_induction 📖AddSubmonoid
SetLike.instMembership
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
AddZero.toZero
AddZeroClass.toAddZero
AddZero.toAdd
closure_induction
Set.mem_iUnion
iSup_eq_closure
iSup_induction' 📖mem_iSup_of_mem
AddZero.toZero
AddZeroClass.toAddZero
ZeroMemClass.zero_mem
AddSubmonoid
instSetLike
AddSubmonoidClass.toZeroMemClass
instAddSubmonoidClass
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
AddZero.toAdd
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
SetLike.instMembership
mem_iSup_of_mem
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
instAddSubmonoidClass
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
iSup_induction
induction_of_closure_eq_top_left 📖closure
AddMonoid.toAddZeroClass
Top.top
AddSubmonoid
instTop
AddZero.toZero
AddZeroClass.toAddZero
AddZero.toAdd
mem_top
closure_induction_left
induction_of_closure_eq_top_right 📖closure
AddMonoid.toAddZeroClass
Top.top
AddSubmonoid
instTop
AddZero.toZero
AddZeroClass.toAddZero
AddZero.toAdd
mem_top
closure_induction_right
mem_biSup_of_directedOn 📖mathematicalDirectedOn
Function.onFun
AddSubmonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
setOf
SetLike.instMembership
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
closure_induction
Set.mem_iUnion
SetLike.mem_coe
zero_mem
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
instAddSubmonoidClass
closure_iUnion
iSup_congr_Prop
closure_eq
le_iSup
iSup_pos
mem_closure_pair 📖mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
SetLike.instMembership
instSetLike
closure
Set
Set.instInsert
Set.instSingletonSet
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toNatSMul
Set.singleton_union
closure_union
mem_sup
mem_closure_singleton
exists_exists_eq_and
mem_closure_singleton 📖mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
SetLike.instMembership
instSetLike
closure
Set
Set.instSingletonSet
AddMonoid.toNatSMul
AddMonoidHom.instAddMonoidHomClass
closure_singleton_eq
AddMonoidHom.mem_mrange
mem_iSup_of_directed 📖mathematicalDirected
AddSubmonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SetLike.instMembership
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
iSup_pos
iSup_plift_down
mem_biSup_of_directedOn
directedOn_onFun_iff
Set.image_univ
directedOn_range
PLift.exists
mem_iSup_of_mem 📖mathematicalAddSubmonoid
SetLike.instMembership
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
SetLike.le_def
instIsConcreteLE
le_iSup
mem_iSup_prop 📖mathematicalAddSubmonoid
SetLike.instMembership
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
AddZero.toZero
AddZeroClass.toAddZero
iSup_congr_Prop
iSup_pos
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
instAddSubmonoidClass
iSup_neg
mem_bot
IsEmpty.exists_iff
instIsEmptyFalse
mem_multiples 📖mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
SetLike.instMembership
instSetLike
multiples
one_nsmul
mem_multiples_iff 📖mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
SetLike.instMembership
instSetLike
multiples
AddMonoid.toNatSMul
mem_sSup_of_directedOn 📖mathematicalSet.Nonempty
AddSubmonoid
DirectedOn
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SetLike.instMembership
instSetLike
SupSet.sSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
Set
Set.instMembership
sSup_eq_iSup'
mem_iSup_of_directed
Set.Nonempty.to_subtype
DirectedOn.directed_val
mem_sSup_of_mem 📖mathematicalAddSubmonoid
Set
Set.instMembership
SetLike.instMembership
instSetLike
SupSet.sSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
SetLike.le_def
instIsConcreteLE
le_sSup
mem_sup 📖mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoidHom.instAddMonoidHomClass
sup_eq_range
AddMonoidHom.mem_mrange
mem_sup_left 📖mathematicalAddSubmonoid
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
SetLike.le_def
instIsConcreteLE
le_sup_left
mem_sup_right 📖mathematicalAddSubmonoid
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
SetLike.le_def
instIsConcreteLE
le_sup_right
multiples_eq_closure 📖mathematicalmultiples
closure
AddMonoid.toAddZeroClass
Set
Set.instSingletonSet
ext
mem_closure_singleton
multiples_le 📖mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
multiples
SetLike.instMembership
instSetLike
multiples_eq_closure
closure_le
Set.singleton_subset_iff
SetLike.mem_coe
multiples_zero 📖mathematicalmultiples
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
Bot.bot
AddSubmonoid
instBot
bot_unique
multiples_le
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
instAddSubmonoidClass
sup_eq_range 📖mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
AddMonoidHom.mrange
SetLike.instMembership
instSetLike
Prod.instAddZeroClass
toAddZeroClass
AddMonoidHom
AddZeroClass.toAddZero
AddMonoidHom.instFunLike
AddMonoidHom.instAddMonoidHomClass
AddMonoidHom.coprod
subtype
AddMonoidHom.instAddMonoidHomClass
AddMonoidHom.mrange_eq_map
mrange_inl_sup_mrange_inr
map_sup
AddMonoidHom.map_mrange
AddMonoidHom.coprod_comp_inl
AddMonoidHom.coprod_comp_inr
mrange_subtype

FreeAddMonoid

Theorems

NameKindAssumesProvesValidatesDepends On
closure_range_of 📖mathematicalAddSubmonoid.closure
FreeAddMonoid
AddMonoid.toAddZeroClass
AddRightCancelMonoid.toAddMonoid
AddCancelMonoid.toAddRightCancelMonoid
instAddCancelMonoid
Set.range
of
Top.top
AddSubmonoid
AddSubmonoid.instTop
eq_top_iff
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
AddSubmonoid.instAddSubmonoidClass
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
AddSubmonoid.subset_closure
Set.mem_range_self
mrange_lift 📖mathematicalAddMonoidHom.mrange
FreeAddMonoid
AddMonoid.toAddZeroClass
AddRightCancelMonoid.toAddMonoid
AddCancelMonoid.toAddRightCancelMonoid
instAddCancelMonoid
AddMonoidHom
AddZeroClass.toAddZero
AddMonoidHom.instFunLike
AddMonoidHom.instAddMonoidHomClass
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
lift
AddSubmonoid.closure
Set.range
AddMonoidHom.instAddMonoidHomClass
AddMonoidHom.mrange_eq_map
closure_range_of
AddMonoidHom.map_mclosure
Set.range_comp
lift_comp_of

FreeMonoid

Theorems

NameKindAssumesProvesValidatesDepends On
closure_range_of 📖mathematicalSubmonoid.closure
FreeMonoid
Monoid.toMulOneClass
RightCancelMonoid.toMonoid
CancelMonoid.toRightCancelMonoid
instCancelMonoid
Set.range
of
Top.top
Submonoid
Submonoid.instTop
eq_top_iff
OneMemClass.one_mem
SubmonoidClass.toOneMemClass
Submonoid.instSubmonoidClass
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
Submonoid.subset_closure
Set.mem_range_self
mrange_lift 📖mathematicalMonoidHom.mrange
FreeMonoid
Monoid.toMulOneClass
RightCancelMonoid.toMonoid
CancelMonoid.toRightCancelMonoid
instCancelMonoid
MonoidHom
MulOneClass.toMulOne
MonoidHom.instFunLike
MonoidHom.instMonoidHomClass
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
lift
Submonoid.closure
Set.range
MonoidHom.instMonoidHomClass
MonoidHom.mrange_eq_map
closure_range_of
MonoidHom.map_mclosure
Set.range_comp
lift_comp_of

IsIdempotentElem

Theorems

NameKindAssumesProvesValidatesDepends On
coe_powers 📖mathematicalIsIdempotentElem
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
SetLike.coe
Submonoid
Submonoid.instSetLike
Submonoid.powers
Set
Set.instInsert
MulOne.toOne
Set.instSingletonSet
one_mul
mul_one
le_antisymm
Submonoid.powers_le
OneMemClass.one_mem
SubmonoidClass.toOneMemClass
Submonoid.instSubmonoidClass
Submonoid.mem_powers

IsScalarTower

Theorems

NameKindAssumesProvesValidatesDepends On
of_mclosure_eq_top 📖mathematicalSubmonoid.closure
Monoid.toMulOneClass
Top.top
Submonoid
Submonoid.instTop
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
IsScalarTowerSubmonoid.induction_of_closure_eq_top_left
one_smul
SemigroupAction.mul_smul

Nat

Theorems

NameKindAssumesProvesValidatesDepends On
addSubmonoidClosure_one 📖mathematicalAddSubmonoid.closure
AddMonoid.toAddZeroClass
instAddMonoid
Set
Set.instSingletonSet
Top.top
AddSubmonoid
AddSubmonoid.instTop
AddSubmonoid.ext
mul_one

SMulCommClass

Theorems

NameKindAssumesProvesValidatesDepends On
of_mclosure_eq_top 📖mathematicalSubmonoid.closure
Monoid.toMulOneClass
Top.top
Submonoid
Submonoid.instTop
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
SMulCommClassSubmonoid.induction_of_closure_eq_top_left
one_smul
SemigroupAction.mul_smul

Submonoid

Definitions

NameCategoryTheorems
decidableMemPowers 📖CompOp
fintypePowers 📖CompOp
1 mathmath: orderOf_eq_card_powers
groupPowers 📖CompOp
pow 📖CompOp
13 mathmath: selfZPow_of_nonpos, log_pow_eq_self, Surreal.dyadicMap_apply_pow', pow_log_eq_self, selfZPow_of_neg, pow_right_injective_iff_pow_injective, pow_apply, powLogEquiv_apply, selfZPow_sub_natCast, log_pow_int_eq_self, selfZPow_neg_natCast, Surreal.dyadicMap_apply_pow, pow_coe
powLogEquiv 📖CompOp
2 mathmath: powLogEquiv_apply, powLogEquiv_symm_apply
powers 📖CompOp
204 mathmath: AlgebraicGeometry.Proj.awayMap_awayToSection_assoc, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_injective, Algebra.exists_unramified_of_isUnramifiedAt, Localization.Away.mk_eq_monoidOf_mk', AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_apply, LaurentPolynomial.mk'_one_X_pow, HasStandardEtaleSurjectionOn.isStandardEtale, selfZPow_of_nonpos, IsOfFinOrder.natCard_powers_le_orderOf, LocalizedModule.subsingleton_iff_disjoint, Localization.algebraMap_injective_of_span_eq_top, LaurentPolynomial.mk'_mul_T, AlgebraicGeometry.Proj.basicOpenIsoSpec_inv_ι_assoc, IsOfFinOrder.mem_powers_iff_mem_zpowers, powers_one, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_specStalkEquiv, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_fromSpec, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_preimage_basicOpen, finEquivPowers_apply, IsLocalizedModule.exists_subsingleton_away, selfZPow_pow_sub, AlgebraicGeometry.Proj.basicOpenIsoAway_hom, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_hom_apply_asIdeal, LaurentPolynomial.mk'_one_X, Module.FinitePresentation.exists_notMem_bijective, Algebra.basicOpen_subset_etaleLocus_iff_etale, Algebra.IsSmoothAt.exists_notMem_isStandardSmooth, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_surjective, AlgebraicGeometry.Scheme.basic_open_isOpenImmersion, Polynomial.UniversalCoprimeFactorizationRing.exists_liesOver_residueFieldMap_bijective, HomogeneousLocalization.range_awayMapAux_subset, Module.basicOpen_subset_freeLocus_iff, Algebra.Generators.compLocalizationAwayAlgHom_toAlgHom_toComp, Algebra.basicOpen_subset_unramifiedLocus_iff, Submodule.isTorsion'_powers_iff, LocalizationMap.AwayMap.lift_comp, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app_assoc, finite_powers, Polynomial.UniversalCoprimeFactorizationRing.homEquiv_comp_snd, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleObjOppositeOpensCarrierTopValStructureSheafInTypeOpBasicOpenPowersToOpenₗ, HomogeneousLocalization.Away.adjoin_mk_prod_pow_eq_top, instFinitePresentationAway, AlgebraicGeometry.Scheme.isOpenImmersion_SpecMap_localizationAway, AlgebraicGeometry.Proj.basicOpenToSpec_app_top, AlgebraicGeometry.Scheme.instIsOpenImmersionMapOfHomAwayAlgebraMap, HomogeneousLocalization.awayMap_mk, log_mul, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk, Algebra.QuasiFiniteAt.exists_fg_and_exists_notMem_and_awayMap_bijective, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_right_assoc, Algebra.Generators.compLocalizationAwayAlgHom_X_inl, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_snd_assoc, AlgebraicGeometry.Proj.SpecMap_awayMap_awayι_assoc, IsLocalization.Away.instMapRingHomPowersOfCoe, Algebra.IsSmoothAt.exists_isStandardEtale_mvPolynomial, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_fst, Ideal.disjoint_powers_iff_notMem, HomogeneousLocalization.val_awayMap, Localization.awayMap_surjective_iff, AlgebraicGeometry.Proj.basicOpenToSpec_SpecMap_awayMap, IsOfFinOrder.finite_powers, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff', IsOfFinOrder.mem_powers_iff_mem_range_orderOf, Algebra.basicOpen_subset_smoothLocus_iff_smooth, Polynomial.UniversalCoprimeFactorizationRing.homEquiv_comp_fst, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff, AlgebraicGeometry.ProjectiveSpectrum.Proj.isLocalization_atPrime, AlgebraicGeometry.Proj.basicOpenIsoSpec_inv_ι, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec_assoc, LocalizationMap.AwayMap.lift_eq, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_snd, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_fst_assoc, LaurentPolynomial.mk'_eq, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_awayι, Algebra.Generators.sq_ker_comp_le_ker_compLocalizationAwayAlgHom, HomogeneousLocalization.awayMapₐ_apply, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_right, powers_eq_closure, AlgebraicGeometry.ProjIsoSpecTopComponent.fromSpec_toSpec, powers_eq_zpowers, ofMul_image_powers_eq_multiples_ofMul, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem_mul, Module.FinitePresentation.exists_free_localizedModule_powers, IsLocalization.Away.instAlgebraMapSubmonoidPowersOfCoeRingHomAlgebraMap, selfZPow_of_neg, HomogeneousLocalization.awayMap_fromZeroRingHom, Nat.card_submonoidPowers, mem_powers_iff_mem_range_orderOf, Subgroup.toSubmonoid_zpowers, IsLocalizedModule.exists_isLocalizedModule_powers_of_finitePresentation, LaurentSeries.of_powerSeries_localization, powers_le_zpowers, HomogeneousLocalization.Away.span_mk_prod_pow_eq_top, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_left, Algebra.zariskisMainProperty_iff_exists_saturation_eq_top, LocalizedModule.exists_subsingleton_away, Algebra.basicOpen_subset_etaleLocus_iff, AlgebraicGeometry.Proj.awayι_toSpecZero, AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.isPrime_carrier, powers_fg, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq, Algebra.Smooth.exists_span_eq_top_isStandardSmooth, AlgebraicGeometry.Proj.awayι_toSpecZero_assoc, IsCyclic.exists_monoid_generator, Algebra.IsSmoothAt.exists_notMem_smooth, Localization.exists_awayMap_bijective_of_residueField_surjective, pow_right_injective_iff_pow_injective, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_left_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_isIso, pow_apply, IsLocalization.Away.instHMulAwayCoeRingHomAlgebraMap, IsLocalization.Away.instHMulAwayCoeRingHomAlgebraMap_1, Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq, Algebra.IsEtaleAt.exists_isStandardEtale, AlgebraicGeometry.ProjectiveSpectrum.Proj.mk_mem_toSpec_base_apply, AlgebraicGeometry.Proj.opensRange_awayι, Localization.exists_awayMap_injective_of_localRingHom_injective, instFiniteTypeAway, orderOf_eq_card_powers, powers_le_nonZeroDivisors_of_noZeroDivisors, Module.FinitePresentation.exists_lift_equiv_of_isLocalizedModule, AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.mk_mem_carrier, AlgebraicGeometry.ProjectiveSpectrum.Proj.isIso_toSpec, Algebra.exists_formallyUnramified_of_isUnramifiedAt, localization_unit_isIso, HomogeneousLocalization.Away.isLocalization_mul, HomogeneousLocalization.Away.val_mk, HomogeneousLocalization.val_awayMap_eq_aux, LocalizedModule.subsingleton_iff_support_subset, HomogeneousLocalization.val_awayMap_mk, AlgebraicGeometry.Proj.basicOpenToSpec_SpecMap_awayMap_assoc, HomogeneousLocalization.awayMapAux_mk, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_awayι_assoc, powLogEquiv_apply, AlgebraicGeometry.Proj.awayι_preimage_basicOpen, AlgebraicGeometry.Proj.awayMap_awayToSection, Algebra.exists_etale_of_isEtaleAt, AlgebraicGeometry.Proj.SpecMap_awayMap_awayι, AlgebraicGeometry.basicOpenIsoSpecAway_inv_homOfLE_assoc, Surreal.dyadicMap_apply, HomogeneousLocalization.Away.eventually_smul_mem, AlgebraicGeometry.basicOpenIsoSpecAway_inv_homOfLE, mem_powers_iff, selfZPow_sub_natCast, IsIdempotentElem.coe_powers, PrimeSpectrum.range_comap_algebraMap_localization_compl_eq_range_comap_quotientMk, AlgebraicGeometry.ProjectiveSpectrum.Proj.stalkMap_toSpec, AlgebraicGeometry.Scheme.affineOpenCoverOfSpanRangeEqTop_f, AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun_asIdeal, Monoid.powers_fg, mem_powers_of_prime_card, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq_comap, AlgebraicGeometry.Proj.basicOpenIsoSpec_hom, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_germ, Algebra.Generators.compLocalizationAwayAlgHom_relation_eq_zero, IsOfFinOrder.powers_eq_image_range_orderOf, Polynomial.UniversalCoprimeFactorizationRing.factor₁_mul_factor₂, RingHom.RespectsIso.isLocalization_away_iff, mem_powers, Localization.awayMap_injective_iff, AlgebraicGeometry.StructureSheaf.comap_basicOpen, exists_bijective_map_powers, powers_eq_top_of_prime_card, map_powers, selfZPow_neg_natCast, Algebra.algebraMapSubmonoid_powers, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_preimage_basicOpen, RingHom.locally_iff_finite, AlgebraicGeometry.Proj.instIsOpenImmersionAwayι, localization_unit_isIso', Ideal.exists_disjoint_powers_of_span_eq_top, IsLocalization.ideal_eq_iInf_comap_map_away, Localization.exists_awayMap_bijective_of_localRingHom_bijective, mem_powers_iff_mem_zpowers, AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_basicOpen, powersEquivPowers_apply, Algebra.basicOpen_subset_smoothLocus_iff, Localization.awayLift_mk, powLogEquiv_symm_apply, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.num_mem_carrier_iff, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec.image_basicOpen_eq_basicOpen, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app, HomogeneousLocalization.Away.finiteType, isNilpotent_iff_zero_mem_powers, Polynomial.UniversalCoprimeFactorizationRing.isCoprime_factor₁_factor₂, Surreal.dyadicMap_apply_pow, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_bijective, finEquivPowers_symm_apply, ofAdd_image_multiples_eq_powers_ofAdd, powers_le, Algebra.Etale.instAway, AlgebraicGeometry.Proj.lift_awayMapₐ_awayMapₐ_surjective, pow_coe, IsOfFinOrder.powers_eq_zpowers, Module.FinitePresentation.exists_basis_localizedModule_powers, infinite_powers, coe_powers, Polynomial.instEtaleUniversalCoprimeFactorizationRing, AlgebraicGeometry.tilde.instIsLocalizedModuleCarrierCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecModuleCatPresheafModulesSheafModulesSpecToSheafOpBasicOpenPowersHomToOpen, Algebra.instIsStandardOpenImmersionAway, Polynomial.isIntegral_isLocalization_polynomial_quotient, Algebra.ZariskisMainProperty.exists_fg_and_exists_notMem_and_awayMap_bijective

Theorems

NameKindAssumesProvesValidatesDepends On
card_bot 📖mathematicalFintype.card
Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
Bot.bot
instBot
Fintype.card_eq_one_iff
Set.mem_singleton
mem_bot
card_le_one_iff_eq_bot 📖mathematicalFintype.card
Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
Bot.bot
instBot
eq_bot_iff_forall
Fintype.card_le_one_iff
Fintype.card_congr'
Fintype.card_unique
closure_eq_image_prod 📖mathematicalSetLike.coe
Submonoid
Monoid.toMulOneClass
instSetLike
closure
Set.image
MulOne.toMul
MulOneClass.toMulOne
MulOne.toOne
setOf
MonoidHom.instMonoidHomClass
closure_eq_mrange
MonoidHom.coe_mrange
Set.range_list_map_coe
Set.range_comp
FreeMonoid.lift_apply
closure_eq_mrange 📖mathematicalclosure
Monoid.toMulOneClass
MonoidHom.mrange
FreeMonoid
Set
Set.instMembership
RightCancelMonoid.toMonoid
CancelMonoid.toRightCancelMonoid
FreeMonoid.instCancelMonoid
MonoidHom
MulOneClass.toMulOne
MonoidHom.instFunLike
MonoidHom.instMonoidHomClass
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
FreeMonoid.lift
MonoidHom.instMonoidHomClass
FreeMonoid.mrange_lift
Subtype.range_coe
closure_induction_left 📖MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
OneMemClass.one_mem
Submonoid
instSetLike
SubmonoidClass.toOneMemClass
instSubmonoidClass
closure
MulOne.toMul
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
subset_closure
SetLike.instMembership
OneMemClass.one_mem
SubmonoidClass.toOneMemClass
instSubmonoidClass
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
subset_closure
MonoidHom.instMonoidHomClass
closure_eq_mrange
FreeMonoid.inductionOn'
map_mul
MonoidHomClass.toMulHomClass
Subtype.prop
closure_induction_right 📖MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
OneMemClass.one_mem
Submonoid
instSetLike
SubmonoidClass.toOneMemClass
instSubmonoidClass
closure
MulOne.toMul
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
subset_closure
SetLike.instMembership
OneMemClass.one_mem
SubmonoidClass.toOneMemClass
instSubmonoidClass
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
subset_closure
closure_induction_left
op_closure
closure_singleton_eq 📖mathematicalclosure
Monoid.toMulOneClass
Set
Set.instSingletonSet
MonoidHom.mrange
Multiplicative
Multiplicative.mulOneClass
AddMonoid.toAddZeroClass
Nat.instAddMonoid
MonoidHom
MulOneClass.toMulOne
MonoidHom.instFunLike
MonoidHom.instMonoidHomClass
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
powersHom
closure_eq_of_le
MonoidHom.instMonoidHomClass
Set.singleton_subset_iff
pow_one
pow_mem
instSubmonoidClass
subset_closure
Set.mem_singleton
closure_singleton_one 📖mathematicalclosure
Monoid.toMulOneClass
Set
Set.instSingletonSet
MulOne.toOne
MulOneClass.toMulOne
Bot.bot
Submonoid
instBot
one_pow
coe_iSup_of_directed 📖mathematicalDirected
Submonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SetLike.coe
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
Set.iUnion
Set.ext
mem_iSup_of_directed
coe_powers 📖mathematicalSetLike.coe
Submonoid
Monoid.toMulOneClass
instSetLike
powers
Set.range
Monoid.toNatPow
coe_sSup_of_directedOn 📖mathematicalSet.Nonempty
Submonoid
DirectedOn
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SetLike.coe
instSetLike
SupSet.sSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
Set.iUnion
Set
Set.instMembership
Set.ext
mem_sSup_of_directedOn
eq_bot_iff_card 📖mathematicalBot.bot
Submonoid
Monoid.toMulOneClass
instBot
Fintype.card
SetLike.instMembership
instSetLike
card_bot
eq_bot_of_card_eq
eq_bot_of_card_eq 📖mathematicalFintype.card
Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
Bot.bot
instBot
eq_bot_of_card_le
le_of_eq
eq_bot_of_card_le 📖mathematicalFintype.card
Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
Bot.bot
instBot
Fintype.card_le_one_iff_subsingleton
eq_bot_of_subsingleton
exists_list_of_mem_closure 📖mathematicalSubmonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
closure
Set
Set.instMembership
MulOne.toMul
MulOneClass.toMulOne
MulOne.toOne
Set.mem_image
closure_eq_image_prod
SetLike.mem_coe
exists_mem_sup 📖mathematicalSubmonoid
Monoid.toMulOneClass
CommMonoid.toMonoid
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
MulOne.toMul
MulOneClass.toMulOne
exists_multiset_of_mem_closure 📖mathematicalSubmonoid
Monoid.toMulOneClass
CommMonoid.toMonoid
SetLike.instMembership
instSetLike
closure
Set
Set.instMembership
Multiset.prod
exists_list_of_mem_closure
Multiset.prod_coe
forall_mem_sup 📖mathematicalMulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
iSup_induction 📖Submonoid
SetLike.instMembership
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
MulOne.toOne
MulOneClass.toMulOne
MulOne.toMul
closure_induction
Set.mem_iUnion
iSup_eq_closure
iSup_induction' 📖mem_iSup_of_mem
MulOne.toOne
MulOneClass.toMulOne
OneMemClass.one_mem
Submonoid
instSetLike
SubmonoidClass.toOneMemClass
instSubmonoidClass
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
MulOne.toMul
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
SetLike.instMembership
mem_iSup_of_mem
OneMemClass.one_mem
SubmonoidClass.toOneMemClass
instSubmonoidClass
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
iSup_induction
induction_of_closure_eq_top_left 📖closure
Monoid.toMulOneClass
Top.top
Submonoid
instTop
MulOne.toOne
MulOneClass.toMulOne
MulOne.toMul
closure_induction_left
induction_of_closure_eq_top_right 📖closure
Monoid.toMulOneClass
Top.top
Submonoid
instTop
MulOne.toOne
MulOneClass.toMulOne
MulOne.toMul
closure_induction_right
log_mul 📖mathematicalMonoid.toNatPowlog
Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
powers
mul
map_mul
MonoidHomClass.toMulHomClass
MulEquivClass.instMonoidHomClass
MulEquiv.instMulEquivClass
log_pow_eq_self 📖mathematicalMonoid.toNatPowlog
pow
pow_right_injective_iff_pow_injective
pow_log_eq_self
log_pow_int_eq_self 📖mathematicallog
Int.instMonoid
pow
MulEquiv.symm_apply_apply
Int.pow_right_injective
map_powers 📖mathematicalmap
Monoid.toMulOneClass
powers
DFunLike.coe
map.congr_simp
powers_eq_closure
MonoidHom.map_mclosure
Set.image_singleton
mem_biSup_of_directedOn 📖mathematicalDirectedOn
Function.onFun
Submonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
setOf
SetLike.instMembership
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
closure_induction
one_mem
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
instSubmonoidClass
closure_iUnion
iSup_congr_Prop
closure_eq
le_iSup
iSup_pos
mem_closure_pair 📖mathematicalSubmonoid
Monoid.toMulOneClass
CommMonoid.toMonoid
SetLike.instMembership
instSetLike
closure
Set
Set.instInsert
Set.instSingletonSet
MulOne.toMul
MulOneClass.toMulOne
Monoid.toNatPow
Set.singleton_union
closure_union
mem_sup
mem_closure_singleton 📖mathematicalSubmonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
closure
Set
Set.instSingletonSet
Monoid.toNatPow
MonoidHom.instMonoidHomClass
closure_singleton_eq
MonoidHom.mem_mrange
mem_closure_singleton_self 📖mathematicalSubmonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
closure
Set
Set.instSingletonSet
mem_closure_singleton
pow_one
mem_iSup_of_directed 📖mathematicalDirected
Submonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SetLike.instMembership
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
iSup_pos
iSup_plift_down
mem_biSup_of_directedOn
directedOn_onFun_iff
Set.image_univ
directedOn_range
mem_iSup_of_mem 📖mathematicalSubmonoid
SetLike.instMembership
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
SetLike.le_def
instIsConcreteLE
le_iSup
mem_iSup_prop 📖mathematicalSubmonoid
SetLike.instMembership
instSetLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
MulOne.toOne
MulOneClass.toMulOne
iSup_congr_Prop
iSup_pos
SubmonoidClass.toOneMemClass
instSubmonoidClass
iSup_neg
instIsEmptyFalse
mem_powers 📖mathematicalSubmonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
powers
pow_one
mem_powers_iff 📖mathematicalSubmonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
powers
Monoid.toNatPow
mem_sSup_of_directedOn 📖mathematicalSet.Nonempty
Submonoid
DirectedOn
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SetLike.instMembership
instSetLike
SupSet.sSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
Set
Set.instMembership
sSup_eq_iSup'
mem_iSup_of_directed
Set.Nonempty.to_subtype
DirectedOn.directed_val
mem_sSup_of_mem 📖mathematicalSubmonoid
Set
Set.instMembership
SetLike.instMembership
instSetLike
SupSet.sSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
SetLike.le_def
instIsConcreteLE
le_sSup
mem_sup 📖mathematicalSubmonoid
Monoid.toMulOneClass
CommMonoid.toMonoid
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
MulOne.toMul
MulOneClass.toMulOne
MonoidHom.instMonoidHomClass
sup_eq_range
mem_sup_left 📖mathematicalSubmonoid
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
SetLike.le_def
instIsConcreteLE
le_sup_left
mem_sup_right 📖mathematicalSubmonoid
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
SetLike.le_def
instIsConcreteLE
le_sup_right
mul_mem_sup 📖mathematicalSubmonoid
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
MulOne.toMul
MulOneClass.toMulOne
mul_mem
mem_sup_left
mem_sup_right
powLogEquiv_apply 📖mathematicalMonoid.toNatPowDFunLike.coe
MulEquiv
Multiplicative
Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
powers
Multiplicative.mul
mul
EquivLike.toFunLike
MulEquiv.instEquivLike
powLogEquiv
pow
Equiv
Equiv.instEquivLike
Multiplicative.toAdd
powLogEquiv_symm_apply 📖mathematicalMonoid.toNatPowDFunLike.coe
MulEquiv
Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
powers
Multiplicative
mul
Multiplicative.mul
EquivLike.toFunLike
MulEquiv.instEquivLike
MulEquiv.symm
powLogEquiv
Equiv
Equiv.instEquivLike
Multiplicative.ofAdd
log
pow_apply 📖mathematicalpow
Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
powers
Monoid.toNatPow
pow_coe 📖mathematicalSubmonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
powers
pow
Monoid.toNatPow
pow_log_eq_self 📖mathematicalpow
log
Nat.find_spec
Subtype.prop
pow_right_injective_iff_pow_injective 📖mathematicalMonoid.toNatPow
Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
powers
pow
Function.Injective.of_comp_iff
Subtype.coe_injective
powers_eq_closure 📖mathematicalpowers
closure
Monoid.toMulOneClass
Set
Set.instSingletonSet
ext
mem_closure_singleton
powers_le 📖mathematicalSubmonoid
Monoid.toMulOneClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
powers
SetLike.instMembership
instSetLike
powers_eq_closure
powers_one 📖mathematicalpowers
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
Bot.bot
Submonoid
instBot
bot_unique
powers_le
OneMemClass.one_mem
SubmonoidClass.toOneMemClass
instSubmonoidClass
sup_eq_range 📖mathematicalSubmonoid
Monoid.toMulOneClass
CommMonoid.toMonoid
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
MonoidHom.mrange
SetLike.instMembership
instSetLike
Prod.instMulOneClass
toMulOneClass
MonoidHom
MulOneClass.toMulOne
MonoidHom.instFunLike
MonoidHom.instMonoidHomClass
MonoidHom.coprod
subtype
MonoidHom.instMonoidHomClass
MonoidHom.mrange_eq_map
mrange_inl_sup_mrange_inr
map_sup
MonoidHom.map_mrange
MonoidHom.coprod_comp_inl
MonoidHom.coprod_comp_inr
mrange_subtype

VAddAssocClass

Theorems

NameKindAssumesProvesValidatesDepends On
of_mclosure_eq_top 📖mathematicalAddSubmonoid.closure
AddMonoid.toAddZeroClass
Top.top
AddSubmonoid
AddSubmonoid.instTop
HVAdd.hVAdd
instHVAdd
AddSemigroupAction.toVAdd
AddMonoid.toAddSemigroup
AddAction.toAddSemigroupAction
VAddAssocClassAddSubmonoid.induction_of_closure_eq_top_left
zero_vadd
AddSemigroupAction.add_vadd

VAddCommClass

Theorems

NameKindAssumesProvesValidatesDepends On
of_mclosure_eq_top 📖mathematicalAddSubmonoid.closure
AddMonoid.toAddZeroClass
Top.top
AddSubmonoid
AddSubmonoid.instTop
HVAdd.hVAdd
instHVAdd
AddSemigroupAction.toVAdd
AddMonoid.toAddSemigroup
AddAction.toAddSemigroupAction
VAddCommClassAddSubmonoid.induction_of_closure_eq_top_left
zero_vadd
AddSemigroupAction.add_vadd

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
ofAdd_image_multiples_eq_powers_ofAdd 📖mathematicalSet.image
Multiplicative
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Multiplicative.ofAdd
SetLike.coe
AddSubmonoid
AddMonoid.toAddZeroClass
AddSubmonoid.instSetLike
AddSubmonoid.multiples
Submonoid
Monoid.toMulOneClass
Multiplicative.monoid
Submonoid.instSetLike
Submonoid.powers
Equiv.eq_image_iff_symm_image_eq
ofMul_image_powers_eq_multiples_ofMul
ofMul_image_powers_eq_multiples_ofMul 📖mathematicalSet.image
Additive
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Additive.ofMul
SetLike.coe
Submonoid
Monoid.toMulOneClass
Submonoid.instSetLike
Submonoid.powers
AddSubmonoid
AddMonoid.toAddZeroClass
Additive.addMonoid
AddSubmonoid.instSetLike
AddSubmonoid.multiples
Set.ext
Set.mem_image_iff_of_inverse

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