Documentation Verification Report

Basic

📁 Source: Mathlib/Algebra/Group/Subgroup/Basic.lean

Statistics

MetricCount
DefinitionsliftOfRightInverse, liftOfRightInverseAux, liftOfSurjective, Characteristic, inertia, pi, prod, prodEquiv, noncenter, conjugatesOfSet, liftOfRightInverse, liftOfRightInverseAux, liftOfSurjective, NormalizerCondition, Characteristic, normalClosure, normalCore, pi, prod, prodEquiv
20
Theoremseq_liftOfRightInverse, ker_fst, ker_prodMap, ker_snd, liftOfRightInverseAux_comp_apply, liftOfRightInverse_comp, liftOfRightInverse_comp_apply, prodMap_comap_prod, fixed, addSubgroupOf, comap, map, mem_comm, addCommute_of_normal_of_disjoint, addSubgroupOf_normalizer_eq, botCharacteristic, bot_prod_bot, characteristic_iff_comap_eq, characteristic_iff_comap_le, characteristic_iff_le_comap, characteristic_iff_le_map, characteristic_iff_map_eq, characteristic_iff_map_le, closure_prod, coe_pi, coe_prod, comap_normalizer_eq_of_le_range, comap_normalizer_eq_of_surjective, inertia_map_subtype, inf_addSubgroupOf_inf_normal_of_left, inf_addSubgroupOf_inf_normal_of_right, inf_normalizer_le_normalizer_inf, instIsAddTorsionFree, le_normalizer_comap, le_normalizer_map, le_normalizer_of_normal, le_normalizer_of_normal_addSubgroupOf, le_pi_iff, le_prod_iff, map_equiv_normalizer_eq, map_normalizer_eq_of_bijective, mem_inertia, mem_pi, mem_prod, normal_addSubgroupOf, normal_addSubgroupOf_iff, normal_addSubgroupOf_iff_le_normalizer, normal_addSubgroupOf_iff_le_normalizer_inf, normal_addSubgroupOf_of_le_normalizer, normal_addSubgroupOf_sup_of_le_normalizer, normal_bot, normal_comap, normal_iInf_normal, normal_in_normalizer, normal_inf_normal, normal_of_characteristic, normal_top, normalizer_eq_top, normalizer_eq_top_iff, pi_bot, pi_empty, pi_eq_bot_iff, pi_top, prod_addSubgroupOf_prod_normal, prod_eq_bot_iff, prod_le_iff, prod_mono, prod_mono_left, prod_mono_right, prod_normal, prod_top, single_mem_pi, sub_mem_comm_iff, subgroupOf_inertia, subset_normalizer_of_normal, topCharacteristic, top_prod, top_prod_top, mem_noncenter, conj_mem_conjugatesOfSet, conjugatesOfSet_mono, conjugatesOfSet_subset, conjugates_subset_normal, mem_conjugatesOfSet_iff, subset_conjugatesOfSet, normalClosure_eq_top_of, eq_liftOfRightInverse, ker_fst, ker_prodMap, ker_snd, liftOfRightInverseAux_comp_apply, liftOfRightInverse_comp, liftOfRightInverse_comp_apply, prodMap_comap_prod, fixed, comap, map, of_map_injective, of_map_subtype, subgroupOf, mem_comm, botCharacteristic, bot_prod_bot, characteristic_iff_comap_eq, characteristic_iff_comap_le, characteristic_iff_le_comap, characteristic_iff_le_map, characteristic_iff_map_eq, characteristic_iff_map_le, closure_le_normalClosure, closure_prod, coe_pi, coe_prod, comap_normalClosure, comap_normalizer_eq_of_le_range, comap_normalizer_eq_of_surjective, commute_of_normal_of_disjoint, conjugatesOfSet_subset_normalClosure, div_mem_comm_iff, inf_normalizer_le_normalizer_inf, inf_subgroupOf_inf_normal_of_left, inf_subgroupOf_inf_normal_of_right, instIsMulTorsionFree, le_normalClosure, le_normalizer_comap, le_normalizer_map, le_normalizer_of_normal, le_normalizer_of_normal_subgroupOf, le_pi_iff, le_prod_iff, map_equiv_normalizer_eq, map_normalClosure, map_normalizer_eq_of_bijective, mem_pi, mem_prod, mulSingle_mem_pi, normalClosure_closure_eq_normalClosure, normalClosure_eq_iInf, normalClosure_eq_self, normalClosure_idempotent, normalClosure_le_normal, normalClosure_mono, normalClosure_normal, normalClosure_subset_iff, normalCore_eq_iSup, normalCore_eq_self, normalCore_idempotent, normalCore_le, normalCore_mono, normalCore_normal, normal_bot, normal_comap, normal_iInf_normal, normal_in_normalizer, normal_inf_normal, normal_le_normalCore, normal_of_characteristic, normal_subgroupOf, normal_subgroupOf_iff, normal_subgroupOf_iff_le_normalizer, normal_subgroupOf_iff_le_normalizer_inf, normal_subgroupOf_of_le_normalizer, normal_subgroupOf_sup_of_le_normalizer, normal_top, normalizer_eq_top, normalizer_eq_top_iff, pi_bot, pi_empty, pi_eq_bot_iff, pi_top, prod_eq_bot_iff, prod_le_iff, prod_mono, prod_mono_left, prod_mono_right, prod_normal, prod_subgroupOf_prod_normal, prod_top, subgroupOf_normalizer_eq, subset_normalClosure, subset_normalizer_of_normal, topCharacteristic, top_prod, top_prod_top, div_mem_comm_iff, normalizerCondition_iff_only_full_group_self_normalizing, sub_mem_comm_iff
187
Total207

AddMonoidHom

Definitions

NameCategoryTheorems
liftOfRightInverse 📖CompOp
3 mathmath: liftOfRightInverse_comp_apply, liftOfRightInverse_comp, eq_liftOfRightInverse
liftOfRightInverseAux 📖CompOp
1 mathmath: liftOfRightInverseAux_comp_apply
liftOfSurjective 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
eq_liftOfRightInverse 📖mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
instFunLike
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
AddSubgroup.instPartialOrder
ker
comp
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
liftOfRightInverse
Equiv.apply_symm_apply
ker_fst 📖mathematicalker
Prod.instAddGroup
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
fst
AddSubgroup.prod
Bot.bot
AddSubgroup
AddSubgroup.instBot
Top.top
AddSubgroup.instTop
SetLike.ext
ker_prodMap 📖mathematicalker
Prod.instAddGroup
Prod.instAddZeroClass
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
prodMap
AddSubgroup.prod
comap_bot
prodMap_comap_prod
AddSubgroup.bot_prod_bot
ker_snd 📖mathematicalker
Prod.instAddGroup
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
snd
AddSubgroup.prod
Top.top
AddSubgroup
AddSubgroup.instTop
Bot.bot
AddSubgroup.instBot
SetLike.ext
liftOfRightInverseAux_comp_apply 📖mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
instFunLike
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
AddSubgroup.instPartialOrder
ker
liftOfRightInverseAuxadd_neg_eq_zero
map_neg
map_add
mem_ker
liftOfRightInverse_comp 📖mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
instFunLike
comp
Equiv
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
AddSubgroup.instPartialOrder
ker
EquivLike.toFunLike
Equiv.instEquivLike
liftOfRightInverse
ext
liftOfRightInverse_comp_apply
liftOfRightInverse_comp_apply 📖mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
instFunLike
Equiv
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
AddSubgroup.instPartialOrder
ker
EquivLike.toFunLike
Equiv.instEquivLike
liftOfRightInverse
liftOfRightInverseAux_comp_apply
prodMap_comap_prod 📖mathematicalAddSubgroup.comap
Prod.instAddGroup
prodMap
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddSubgroup.prod
SetLike.coe_injective
Set.preimage_prod_map_prod

AddSubgroup

Definitions

NameCategoryTheorems
Characteristic 📖CompData
11 mathmath: characteristic_iff_comap_le, characteristic_centralizer, characteristic_iff_le_map, topCharacteristic, Characteristic.comap_quotient_mk, characteristic_iff_map_eq, characteristic_iff_le_comap, botCharacteristic, characteristic_iff_comap_eq, centerCharacteristic, characteristic_iff_map_le
inertia 📖CompOp
3 mathmath: subgroupOf_inertia, mem_inertia, inertia_map_subtype
pi 📖CompOp
18 mathmath: DFinsupp.range_mapRangeAddMonoidHom, mem_pi, pi_top, pi_bot, single_mem_pi, pi_empty, index_pi, QuotientAddGroup.rightRel_pi, pi_le_iff, pi_eq_bot_iff, DFinsupp.ker_mapRangeAddMonoidHom, DirectSum.ker_map, DirectSum.range_map, FG.pi, le_pi_iff, coe_pi, closure_pi, QuotientAddGroup.leftRel_pi
prod 📖CompOp
25 mathmath: coe_prod, le_prod_iff, OpenAddSubgroup.toAddSubgroup_prod, prod_mono_left, bot_prod_bot, mem_prod, index_prod, AddMonoidHom.ker_fst, prod_addSubgroupOf_prod_normal, FG.prod, prod_top, QuotientAddGroup.leftRel_prod, AddMonoidHom.ker_snd, AddMonoidHom.prodMap_comap_prod, top_prod_top, closure_prod, top_prod, prod_le_iff, prod_mono, prod_mono_right, AddMonoidHom.ker_prodMap, goursatFst_prod_goursatSnd_le, prod_eq_bot_iff, prod_normal, QuotientAddGroup.rightRel_prod
prodEquiv 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
addCommute_of_normal_of_disjoint 📖mathematicalDisjoint
AddSubgroup
instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
instCompleteLattice
SetLike.instMembership
instSetLike
AddCommute
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Disjoint.le_bot
add_mem
Normal.conj_mem
neg_mem
add_assoc
SetLike.mem_coe
neg_add_rev
neg_neg
add_eq_zero_iff_eq_neg
addSubgroupOf_normalizer_eq 📖mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
addSubgroupOf
normalizer
SetLike.instMembership
instSetLike
toAddGroup
comap_normalizer_eq_of_le_range
LE.le.trans_eq
range_subtype
botCharacteristic 📖mathematicalCharacteristic
Bot.bot
AddSubgroup
instBot
characteristic_iff_le_map
bot_le
bot_prod_bot 📖mathematicalprod
Bot.bot
AddSubgroup
instBot
Prod.instAddGroup
SetLike.coe_injective
Set.singleton_prod_singleton
Set.singleton_eq_singleton_iff
Prod.mk_eq_zero
characteristic_iff_comap_eq 📖mathematicalCharacteristic
comap
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Characteristic.fixed
characteristic_iff_comap_le 📖mathematicalCharacteristic
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
characteristic_iff_comap_eq
le_of_eq
le_antisymm
AddEquiv.symm_apply_apply
characteristic_iff_le_comap 📖mathematicalCharacteristic
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
characteristic_iff_comap_eq
ge_of_eq
le_antisymm
AddEquiv.symm_apply_apply
characteristic_iff_le_map 📖mathematicalCharacteristic
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
map_equiv_eq_comap_symm'
characteristic_iff_le_comap
characteristic_iff_map_eq 📖mathematicalCharacteristic
map
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
map_equiv_eq_comap_symm'
characteristic_iff_comap_eq
characteristic_iff_map_le 📖mathematicalCharacteristic
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
map_equiv_eq_comap_symm'
characteristic_iff_comap_le
closure_prod 📖mathematicalSet
Set.instMembership
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
closure
Prod.instAddGroup
SProd.sprod
Set.instSProd
prod
le_antisymm
closure_le
Set.prod_subset_prod_iff
subset_closure
prod_le_iff
map_le_iff_le_comap
coe_pi 📖mathematicalSetLike.coe
AddSubgroup
Pi.addGroup
instSetLike
pi
Set.pi
coe_prod 📖mathematicalSetLike.coe
AddSubgroup
Prod.instAddGroup
instSetLike
prod
SProd.sprod
Set
Set.instSProd
comap_normalizer_eq_of_le_range 📖mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddMonoidHom.range
comap
normalizer
le_antisymm
le_normalizer_comap
map_le_iff_le_comap
LE.le.trans
le_normalizer_map
map_comap_eq_self
le_refl
comap_normalizer_eq_of_surjective 📖mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
comap
normalizer
comap_normalizer_eq_of_le_range
inertia_map_subtype 📖mathematicalSubgroup.map
Subgroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.toGroup
Subgroup.subtype
inertia
Submonoid.instMulActionSubtypeMem
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Subgroup.instSubgroupClass
Subgroup.instMin
Subgroup.instSubgroupClass
subgroupOf_inertia
Subgroup.subgroupOf_map_subtype
inf_addSubgroupOf_inf_normal_of_left 📖mathematicalNormal
AddSubgroup
SetLike.instMembership
instSetLike
instMin
toAddGroup
addSubgroupOf
normal_addSubgroupOf_iff_le_normalizer_inf
inf_inf_inf_comm
inf_idem
le_trans
inf_le_inf
le_normalizer
inf_normalizer_le_normalizer_inf
inf_addSubgroupOf_inf_normal_of_right 📖mathematicalNormal
AddSubgroup
SetLike.instMembership
instSetLike
instMin
toAddGroup
addSubgroupOf
normal_addSubgroupOf_iff_le_normalizer_inf
inf_inf_inf_comm
inf_idem
le_trans
inf_le_inf
le_normalizer
inf_normalizer_le_normalizer_inf
inf_normalizer_le_normalizer_inf 📖mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
instMin
normalizer
instIsAddTorsionFree 📖mathematicalIsAddTorsionFree
AddSubgroup
SetLike.instMembership
instSetLike
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
toAddGroup
nsmul_right_injective
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
SetLike.coe_eq_coe
le_normalizer_comap 📖mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
normalizer
mem_comap
mem_normalizer_iff
map_add
AddMonoidHomClass.toAddHomClass
AddMonoidHom.instAddMonoidHomClass
map_neg
le_normalizer_map 📖mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
normalizer
mem_map
mem_normalizer_iff
map_add
AddMonoidHomClass.toAddHomClass
AddMonoidHom.instAddMonoidHomClass
map_neg
add_assoc
add_neg_cancel_left
add_neg_cancel
add_zero
neg_add_cancel
neg_add_cancel_left
le_normalizer_of_normal 📖mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
subset_normalizer_of_normal
le_normalizer_of_normal_addSubgroupOf 📖mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizernormal_addSubgroupOf_iff_le_normalizer
le_pi_iff 📖mathematicalAddSubgroup
Pi.addGroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
pi
comap
Pi.evalAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Set.subset_pi_iff
le_prod_iff 📖mathematicalAddSubgroup
Prod.instAddGroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
prod
map
AddMonoidHom.fst
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.snd
toAddSubmonoid_le
AddSubmonoid.le_prod_iff
map_equiv_normalizer_eq 📖mathematicalmap
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
normalizer
ext
mem_map_equiv
mem_normalizer_iff
Equiv.forall_congr
AddEquiv.symm_apply_apply
map_add
AddMonoidHomClass.toAddHomClass
AddEquivClass.instAddMonoidHomClass
AddEquiv.instAddEquivClass
map_neg
map_normalizer_eq_of_bijective 📖mathematicalFunction.Bijective
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
map
normalizer
map_equiv_normalizer_eq
AddMonoidHomClass.toAddHomClass
AddMonoidHom.instAddMonoidHomClass
mem_inertia 📖mathematicalSubgroup
SetLike.instMembership
Subgroup.instSetLike
inertia
AddSubgroup
instSetLike
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
mem_pi 📖mathematicalAddSubgroup
Pi.addGroup
SetLike.instMembership
instSetLike
pi
mem_prod 📖mathematicalAddSubgroup
Prod.instAddGroup
SetLike.instMembership
instSetLike
prod
normal_addSubgroupOf 📖mathematicalNormal
AddSubgroup
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
normal_comap
normal_addSubgroupOf_iff 📖mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Normal
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
Normal.conj_mem
normal_addSubgroupOf_iff_le_normalizer 📖mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Normal
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
normalizer
addSubgroupOf_eq_top
addSubgroupOf_normalizer_eq
normalizer_eq_top_iff
normal_addSubgroupOf_iff_le_normalizer_inf 📖mathematicalNormal
AddSubgroup
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
instMin
normal_addSubgroupOf_iff_le_normalizer
inf_le_right
inf_addSubgroupOf_right
normal_addSubgroupOf_of_le_normalizer 📖mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
Normal
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
normal_addSubgroupOf_iff_le_normalizer_inf
LE.le.trans
le_inf
le_normalizer
inf_normalizer_le_normalizer_inf
normal_addSubgroupOf_sup_of_le_normalizer 📖mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
Normal
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
toAddGroup
addSubgroupOf
normal_addSubgroupOf_iff_le_normalizer
le_sup_right
sup_le
le_normalizer
normal_bot 📖mathematicalNormal
Bot.bot
AddSubgroup
instBot
mem_bot
conj_eq_zero_iff
Zero.instNonempty
normal_comap 📖mathematicalNormal
comap
Normal.comap
normal_iInf_normal 📖mathematicalNormaliInf
AddSubgroup
instInfSet
mem_iInf
Normal.conj_mem
normal_in_normalizer 📖mathematicalNormal
AddSubgroup
SetLike.instMembership
instSetLike
normalizer
toAddGroup
addSubgroupOf
normal_addSubgroupOf_iff_le_normalizer
le_normalizer
le_rfl
normal_inf_normal 📖mathematicalNormal
AddSubgroup
instMin
Normal.conj_mem
normal_of_characteristic 📖mathematicalNormalSetLike.ext_iff
Characteristic.fixed
normal_top 📖mathematicalNormal
Top.top
AddSubgroup
instTop
normalizer_eq_top 📖mathematicalnormalizer
Top.top
AddSubgroup
instTop
normalizer_eq_top_iff
normalizer_eq_top_iff 📖mathematicalnormalizer
Top.top
AddSubgroup
instTop
Normal
eq_top_iff
mem_top
Normal.conj_mem
neg_add_cancel_left
Normal.mem_comm_iff
pi_bot 📖mathematicalpi
Set.univ
Bot.bot
AddSubgroup
instBot
Pi.addGroup
ext
mem_pi
Set.mem_univ
mem_bot
pi_empty 📖mathematicalpi
Set
Set.instEmptyCollection
Top.top
AddSubgroup
Pi.addGroup
instTop
ext
mem_pi
Set.mem_empty_iff_false
IsEmpty.forall_iff
instIsEmptyFalse
mem_top
pi_eq_bot_iff 📖mathematicalpi
Set.univ
Bot.bot
AddSubgroup
Pi.addGroup
instBot
SetLike.ext'_iff
Set.univ_pi_eq_singleton_iff
pi_top 📖mathematicalpi
Top.top
AddSubgroup
instTop
Pi.addGroup
ext
mem_pi
mem_top
prod_addSubgroupOf_prod_normal 📖mathematicalNormal
AddSubgroup
Prod.instAddGroup
SetLike.instMembership
instSetLike
prod
toAddGroup
addSubgroupOf
Normal.conj_mem
mem_prod
prod_eq_bot_iff 📖mathematicalprod
Bot.bot
AddSubgroup
Prod.instAddGroup
instBot
AddSubmonoid.prod_eq_bot_iff
prod_le_iff 📖mathematicalAddSubgroup
Prod.instAddGroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
prod
map
AddMonoidHom.inl
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.inr
toAddSubmonoid_le
AddSubmonoid.prod_le_iff
prod_mono 📖mathematicalRelator.LiftFun
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Prod.instAddGroup
prod
Set.prod_mono
prod_mono_left 📖mathematicalMonotone
AddSubgroup
Prod.instAddGroup
PartialOrder.toPreorder
instPartialOrder
prod
prod_mono
le_refl
prod_mono_right 📖mathematicalMonotone
AddSubgroup
Prod.instAddGroup
PartialOrder.toPreorder
instPartialOrder
prod
prod_mono
le_refl
prod_normal 📖mathematicalNormal
Prod.instAddGroup
prod
Normal.conj_mem
mem_prod
prod_top 📖mathematicalprod
Top.top
AddSubgroup
instTop
comap
Prod.instAddGroup
AddMonoidHom.fst
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
ext
mem_prod
mem_top
mem_comap
single_mem_pi 📖mathematicalAddSubgroup
Pi.addGroup
SetLike.instMembership
instSetLike
pi
Pi.single
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
Set.update_mem_pi_iff_of_mem
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
sub_mem_comm_iff 📖mathematicalAddSubgroup
SetLike.instMembership
instSetLike
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
sub_mem_comm_iff
instAddSubgroupClass
subgroupOf_inertia 📖mathematicalSubgroup.subgroupOf
inertia
Subgroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.toGroup
Submonoid.instMulActionSubtypeMem
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Subgroup.instSubgroupClass
subset_normalizer_of_normal 📖mathematicalSet
Set.instHasSubset
SetLike.coe
AddSubgroup
instSetLike
normalizer
le_top
normalizer_eq_top
topCharacteristic 📖mathematicalCharacteristic
Top.top
AddSubgroup
instTop
characteristic_iff_map_le
le_top
top_prod 📖mathematicalprod
Top.top
AddSubgroup
instTop
comap
Prod.instAddGroup
AddMonoidHom.snd
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
ext
mem_prod
mem_top
mem_comap
top_prod_top 📖mathematicalprod
Top.top
AddSubgroup
instTop
Prod.instAddGroup
top_prod
comap_top

AddSubgroup.Characteristic

Theorems

NameKindAssumesProvesValidatesDepends On
fixed 📖mathematicalAddSubgroup.comap
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid

AddSubgroup.Normal

Theorems

NameKindAssumesProvesValidatesDepends On
addSubgroupOf 📖mathematicalAddSubgroup.Normal
AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
AddSubgroup.toAddGroup
AddSubgroup.addSubgroupOf
comap
comap 📖mathematicalAddSubgroup.Normal
AddSubgroup.comap
AddSubgroup.mem_comap
map_add
AddMonoidHomClass.toAddHomClass
AddMonoidHom.instAddMonoidHomClass
map_neg
conj_mem
map 📖mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
AddSubgroup.Normal
AddSubgroup.map
AddSubgroup.normalizer_eq_top_iff
top_le_iff
AddMonoidHom.range_eq_top_of_surjective
AddMonoidHom.range_eq_map
AddSubgroup.normalizer_eq_top
AddSubgroup.le_normalizer_map

AddSubgroup.SubgroupNormal

Theorems

NameKindAssumesProvesValidatesDepends On
mem_comm 📖AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
AddSubgroup.instPartialOrder
SetLike.instMembership
AddSubgroup.instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddSubgroup.normal_addSubgroupOf_iff
add_zero
add_neg_cancel
add_assoc

ConjClasses

Definitions

NameCategoryTheorems
noncenter 📖CompOp
4 mathmath: mem_noncenter, mk_bijOn, Group.nat_card_center_add_sum_card_noncenter_eq_card, Group.card_center_add_sum_card_noncenter_eq_card

Theorems

NameKindAssumesProvesValidatesDepends On
mem_noncenter 📖mathematicalConjClasses
Set
Set.instMembership
noncenter
Set.Nontrivial
carrier

Group

Definitions

NameCategoryTheorems
conjugatesOfSet 📖CompOp
5 mathmath: Subgroup.conjugatesOfSet_subset_normalClosure, conjugatesOfSet_mono, mem_conjugatesOfSet_iff, subset_conjugatesOfSet, conjugatesOfSet_subset

Theorems

NameKindAssumesProvesValidatesDepends On
conj_mem_conjugatesOfSet 📖mathematicalSet
Set.instMembership
conjugatesOfSet
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
toDivisionMonoid
mem_conjugatesOfSet_iff
IsConj.trans
isConj_iff
conjugatesOfSet_mono 📖mathematicalSet
Set.instHasSubset
conjugatesOfSetSet.biUnion_subset_biUnion_left
conjugatesOfSet_subset 📖mathematicalSet
Set.instHasSubset
SetLike.coe
Subgroup
Subgroup.instSetLike
conjugatesOfSetSet.iUnion₂_subset
conjugates_subset_normal
conjugates_subset_normal 📖mathematicalSubgroup
SetLike.instMembership
Subgroup.instSetLike
Set
Set.instHasSubset
conjugatesOf
DivInvMonoid.toMonoid
toDivInvMonoid
SetLike.coe
isConj_iff
Subgroup.Normal.conj_mem
mem_conjugatesOfSet_iff 📖mathematicalSet
Set.instMembership
conjugatesOfSet
IsConj
DivInvMonoid.toMonoid
toDivInvMonoid
conjugatesOfSet.eq_1
Set.mem_iUnion₂
subset_conjugatesOfSet 📖mathematicalSet
Set.instHasSubset
conjugatesOfSet
mem_conjugatesOfSet_iff
IsConj.refl

IsConj

Theorems

NameKindAssumesProvesValidatesDepends On
normalClosure_eq_top_of 📖Subgroup
SetLike.instMembership
Subgroup.instSetLike
IsConj
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Subgroup.normalClosure
Subgroup.toGroup
Set
Set.instSingletonSet
Top.top
Subgroup.instTop
isConj_iff
Subgroup.Normal.conj_mem
SubgroupClass.toSubmonoidClass
Subgroup.instSubgroupClass
inv_inv
mul_inv_cancel
one_mul
mul_assoc
mul_one
eq_top_iff
MonoidHom.range_eq_top
MonoidHom.range_eq_map
le_imp_le_of_le_of_le
Subgroup.map_mono
le_refl
Subgroup.map_le_iff_le_comap
Subgroup.normalClosure_le_normal
Subgroup.normal_comap
Subgroup.normalClosure_normal
Set.singleton_subset_iff
SetLike.mem_coe
Subgroup.subset_normalClosure
Set.mem_singleton

MonoidHom

Definitions

NameCategoryTheorems
liftOfRightInverse 📖CompOp
3 mathmath: eq_liftOfRightInverse, liftOfRightInverse_comp, liftOfRightInverse_comp_apply
liftOfRightInverseAux 📖CompOp
1 mathmath: liftOfRightInverseAux_comp_apply
liftOfSurjective 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
eq_liftOfRightInverse 📖mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instFunLike
Subgroup
Preorder.toLE
PartialOrder.toPreorder
Subgroup.instPartialOrder
ker
comp
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
liftOfRightInverse
Equiv.apply_symm_apply
ker_fst 📖mathematicalker
Prod.instGroup
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
fst
Subgroup.prod
Bot.bot
Subgroup
Subgroup.instBot
Top.top
Subgroup.instTop
SetLike.ext
ker_prodMap 📖mathematicalker
Prod.instGroup
Prod.instMulOneClass
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
prodMap
Subgroup.prod
comap_bot
prodMap_comap_prod
Subgroup.bot_prod_bot
ker_snd 📖mathematicalker
Prod.instGroup
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
snd
Subgroup.prod
Top.top
Subgroup
Subgroup.instTop
Bot.bot
Subgroup.instBot
SetLike.ext
liftOfRightInverseAux_comp_apply 📖mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instFunLike
Subgroup
Preorder.toLE
PartialOrder.toPreorder
Subgroup.instPartialOrder
ker
liftOfRightInverseAuxmul_inv_eq_one
map_inv
map_mul
mem_ker
liftOfRightInverse_comp 📖mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instFunLike
comp
Equiv
Subgroup
Preorder.toLE
PartialOrder.toPreorder
Subgroup.instPartialOrder
ker
EquivLike.toFunLike
Equiv.instEquivLike
liftOfRightInverse
ext
liftOfRightInverse_comp_apply
liftOfRightInverse_comp_apply 📖mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instFunLike
Equiv
Subgroup
Preorder.toLE
PartialOrder.toPreorder
Subgroup.instPartialOrder
ker
EquivLike.toFunLike
Equiv.instEquivLike
liftOfRightInverse
liftOfRightInverseAux_comp_apply
prodMap_comap_prod 📖mathematicalSubgroup.comap
Prod.instGroup
prodMap
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Subgroup.prod
SetLike.coe_injective
Set.preimage_prod_map_prod

Subgroup

Definitions

NameCategoryTheorems
Characteristic 📖CompData
21 mathmath: characteristic_centralizer, centerCharacteristic, characteristic_iff_comap_eq, alternatingGroup.characteristic_kleinFour, characteristic_iff_comap_le, characteristic_iff_map_le, Sylow.characteristic_of_subsingleton, frattini_characteristic, Sylow.characteristic_of_normal, characteristic_iff_le_comap, instCharacteristicPermAlternatingGroup, derivedSeries_characteristic, instCharacteristicUpperCentralSeries, instCharacteristicUpperCentralSeriesStep, botCharacteristic, characteristic_iff_map_eq, topCharacteristic, commutator_characteristic, Characteristic.comap_quotient_mk, instCharacteristicLowerCentralSeries, characteristic_iff_le_map
normalClosure 📖CompOp
22 mathmath: le_normalClosure, normalClosure_of_stabilizer_eq_top, conjugatesOfSet_subset_normalClosure, commutator_def', commutator_eq_normalClosure, alternatingGroup.normalClosure_swap_mul_swap_five, alternatingGroup.normalClosure_finRotate_five, normalClosure_subset_iff, PresentedGroup.closure_rels_subset_ker, PresentedGroup.mk_eq_one_iff, normalClosure_eq_iInf, map_normalClosure, normalClosure_mono, Equiv.Perm.IsThreeCycle.alternating_normalClosure, normalClosure_normal, normalClosure_le_normal, normalClosure_closure_eq_normalClosure, closure_le_normalClosure, subset_normalClosure, normalClosure_idempotent, normalClosure_eq_self, comap_normalClosure
normalCore 📖CompOp
11 mathmath: normalCore_eq_iSup, normalCore_isClosed, normalCore_eq_iInf_conjAct, normal_le_normalCore, normalCore_mono, normalCore_eq_ker, normalCore_eq_self, normalCore_normal, normalCore_idempotent, normalCore_le, finiteIndex_normalCore
pi 📖CompOp
18 mathmath: mulSingle_mem_pi, pi_le_iff, pi_bot, coe_pi, pi_top, mem_pi, pi_empty, le_pi_iff, FG.pi, pi_eq_bot_iff, commutator_pi_pi_le, lowerCentralSeries_pi_le, index_pi, closure_pi, lowerCentralSeries_pi_of_finite, QuotientGroup.rightRel_pi, commutator_pi_pi_of_finite, QuotientGroup.leftRel_pi
prod 📖CompOp
27 mathmath: prod_subgroupOf_prod_normal, QuotientGroup.rightRel_prod, closure_prod, commutator_prod_prod, QuotientGroup.leftRel_prod, top_prod, MonoidHom.ker_prodMap, prod_eq_bot_iff, prod_mono, top_prod_top, le_prod_iff, prod_top, MonoidHom.ker_fst, prod_le_iff, prod_mono_left, goursatFst_prod_goursatSnd_le, index_prod, MonoidHom.prodMap_comap_prod, prod_mono_right, FG.prod, mem_prod, bot_prod_bot, MonoidHom.ker_snd, lowerCentralSeries_prod, OpenSubgroup.toSubgroup_prod, coe_prod, prod_normal
prodEquiv 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
botCharacteristic 📖mathematicalCharacteristic
Bot.bot
Subgroup
instBot
characteristic_iff_le_map
bot_le
bot_prod_bot 📖mathematicalprod
Bot.bot
Subgroup
instBot
Prod.instGroup
SetLike.coe_injective
Set.singleton_prod_singleton
characteristic_iff_comap_eq 📖mathematicalCharacteristic
comap
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Characteristic.fixed
characteristic_iff_comap_le 📖mathematicalCharacteristic
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
characteristic_iff_comap_eq
le_of_eq
le_antisymm
MulEquiv.symm_apply_apply
characteristic_iff_le_comap 📖mathematicalCharacteristic
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
characteristic_iff_comap_eq
ge_of_eq
le_antisymm
MulEquiv.symm_apply_apply
characteristic_iff_le_map 📖mathematicalCharacteristic
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
map_equiv_eq_comap_symm'
characteristic_iff_le_comap
characteristic_iff_map_eq 📖mathematicalCharacteristic
map
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
map_equiv_eq_comap_symm'
characteristic_iff_comap_eq
characteristic_iff_map_le 📖mathematicalCharacteristic
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
map_equiv_eq_comap_symm'
characteristic_iff_comap_le
closure_le_normalClosure 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
closure
normalClosure
closure_prod 📖mathematicalSet
Set.instMembership
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
closure
Prod.instGroup
SProd.sprod
Set.instSProd
prod
le_antisymm
closure_le
Set.prod_subset_prod_iff
subset_closure
prod_le_iff
map_le_iff_le_comap
coe_pi 📖mathematicalSetLike.coe
Subgroup
Pi.group
instSetLike
pi
Set.pi
coe_prod 📖mathematicalSetLike.coe
Subgroup
Prod.instGroup
instSetLike
prod
SProd.sprod
Set
Set.instSProd
comap_normalClosure 📖mathematicalnormalClosure
Set.preimage
DFunLike.coe
MulEquiv
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
EquivLike.toFunLike
MulEquiv.instEquivLike
comap
MonoidHomClass.toMonoidHom
MulEquivClass.instMonoidHomClass
MulEquiv.instMulEquivClass
Equiv.image_symm_eq_preimage
MulEquivClass.instMonoidHomClass
MulEquiv.instMulEquivClass
comap_equiv_eq_map_symm
map_normalClosure
MulEquiv.surjective
Set.image_congr
comap_normalizer_eq_of_le_range 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
MonoidHom.range
comap
normalizer
le_antisymm
le_normalizer_comap
map_le_iff_le_comap
LE.le.trans
le_normalizer_map
map_comap_eq_self
le_refl
comap_normalizer_eq_of_surjective 📖mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
comap
normalizer
comap_normalizer_eq_of_le_range
commute_of_normal_of_disjoint 📖mathematicalDisjoint
Subgroup
instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
instCompleteLattice
SetLike.instMembership
instSetLike
Commute
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Disjoint.le_bot
mul_mem
Normal.conj_mem
inv_mem
mul_assoc
mul_inv_rev
inv_inv
mul_eq_one_iff_eq_inv
conjugatesOfSet_subset_normalClosure 📖mathematicalSet
Set.instHasSubset
Group.conjugatesOfSet
SetLike.coe
Subgroup
instSetLike
normalClosure
subset_closure
div_mem_comm_iff 📖mathematicalSubgroup
SetLike.instMembership
instSetLike
DivInvMonoid.toDiv
Group.toDivInvMonoid
div_mem_comm_iff
instSubgroupClass
inf_normalizer_le_normalizer_inf 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
instMin
normalizer
inf_subgroupOf_inf_normal_of_left 📖mathematicalNormal
Subgroup
SetLike.instMembership
instSetLike
instMin
toGroup
subgroupOf
normal_subgroupOf_iff_le_normalizer_inf
inf_inf_inf_comm
inf_idem
le_trans
inf_le_inf
le_normalizer
inf_normalizer_le_normalizer_inf
inf_subgroupOf_inf_normal_of_right 📖mathematicalNormal
Subgroup
SetLike.instMembership
instSetLike
instMin
toGroup
subgroupOf
normal_subgroupOf_iff_le_normalizer_inf
inf_inf_inf_comm
inf_idem
le_trans
inf_le_inf
le_normalizer
inf_normalizer_le_normalizer_inf
instIsMulTorsionFree 📖mathematicalIsMulTorsionFree
Subgroup
SetLike.instMembership
instSetLike
DivInvMonoid.toMonoid
Group.toDivInvMonoid
toGroup
pow_left_injective
SubgroupClass.toSubmonoidClass
instSubgroupClass
le_normalClosure 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
SetLike.coe
instSetLike
subset_normalClosure
le_normalizer_comap 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
normalizer
map_mul
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
map_inv
le_normalizer_map 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
normalizer
map_mul
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
map_inv
mul_assoc
mul_inv_cancel_left
mul_inv_cancel
mul_one
inv_mul_cancel
inv_mul_cancel_left
le_normalizer_of_normal 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
subset_normalizer_of_normal
le_normalizer_of_normal_subgroupOf 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizernormal_subgroupOf_iff_le_normalizer
le_pi_iff 📖mathematicalSubgroup
Pi.group
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
pi
comap
Pi.evalMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Set.subset_pi_iff
le_prod_iff 📖mathematicalSubgroup
Prod.instGroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
prod
map
MonoidHom.fst
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.snd
Submonoid.le_prod_iff
map_equiv_normalizer_eq 📖mathematicalmap
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
normalizer
ext
Equiv.forall_congr
MulEquiv.symm_apply_apply
map_mul
MonoidHomClass.toMulHomClass
MulEquivClass.instMonoidHomClass
MulEquiv.instMulEquivClass
map_inv
map_normalClosure 📖mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
map
normalClosure
Set.image
Normal.map
normalClosure_normal
le_antisymm
normal_comap
normalClosure_le_normal
Set.image_mono
subset_normalClosure
map_normalizer_eq_of_bijective 📖mathematicalFunction.Bijective
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
map
normalizer
map_equiv_normalizer_eq
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
mem_pi 📖mathematicalSubgroup
Pi.group
SetLike.instMembership
instSetLike
pi
mem_prod 📖mathematicalSubgroup
Prod.instGroup
SetLike.instMembership
instSetLike
prod
mulSingle_mem_pi 📖mathematicalSubgroup
Pi.group
SetLike.instMembership
instSetLike
pi
Pi.mulSingle
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
Set.update_mem_pi_iff_of_mem
OneMemClass.one_mem
SubmonoidClass.toOneMemClass
SubgroupClass.toSubmonoidClass
instSubgroupClass
normalClosure_closure_eq_normalClosure 📖mathematicalnormalClosure
SetLike.coe
Subgroup
instSetLike
closure
le_antisymm
normalClosure_le_normal
normalClosure_normal
closure_le_normalClosure
normalClosure_mono
subset_closure
normalClosure_eq_iInf 📖mathematicalnormalClosure
iInf
Subgroup
instInfSet
Normal
Set
Set.instHasSubset
SetLike.coe
instSetLike
le_antisymm
le_iInf
normalClosure_le_normal
iInf_le_of_le
normalClosure_normal
subset_normalClosure
le_rfl
normalClosure_eq_self 📖mathematicalnormalClosure
SetLike.coe
Subgroup
instSetLike
le_antisymm
normalClosure_le_normal
Eq.subset
Set.instReflSubset
le_normalClosure
normalClosure_idempotent 📖mathematicalnormalClosure
SetLike.coe
Subgroup
instSetLike
normalClosure_eq_self
normalClosure_normal
normalClosure_le_normal 📖mathematicalSet
Set.instHasSubset
SetLike.coe
Subgroup
instSetLike
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
closure_induction
Group.conjugatesOfSet_subset
OneMemClass.one_mem
SubmonoidClass.toOneMemClass
SubgroupClass.toSubmonoidClass
instSubgroupClass
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
InvMemClass.inv_mem
SubgroupClass.toInvMemClass
normalClosure_mono 📖mathematicalSet
Set.instHasSubset
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
normalClosure_le_normal
normalClosure_normal
Set.Subset.trans
subset_normalClosure
normalClosure_normal 📖mathematicalNormal
normalClosure
closure_induction
conjugatesOfSet_subset_normalClosure
Group.conj_mem_conjugatesOfSet
mul_one
mul_inv_cancel
SubmonoidClass.toOneMemClass
SubgroupClass.toSubmonoidClass
instSubgroupClass
conj_mul
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
conj_inv
InvMemClass.inv_mem
SubgroupClass.toInvMemClass
normalClosure_subset_iff 📖mathematicalSet
Set.instHasSubset
SetLike.coe
Subgroup
instSetLike
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
normalClosure_le_normal
Set.Subset.trans
subset_normalClosure
normalCore_eq_iSup 📖mathematicalnormalCore
iSup
Subgroup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
Normal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
le_antisymm
le_iSup_of_le
normalCore_normal
normalCore_le
le_rfl
iSup_le
normal_le_normalCore
normalCore_eq_self 📖mathematicalnormalCorele_antisymm
normalCore_le
normal_le_normalCore
le_rfl
normalCore_idempotent 📖mathematicalnormalCorenormalCore_eq_self
normalCore_normal
normalCore_le 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalCore
mul_one
inv_one
one_mul
normalCore_mono 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalCorenormal_le_normalCore
normalCore_normal
LE.le.trans
normalCore_le
normalCore_normal 📖mathematicalNormal
normalCore
mul_assoc
mul_inv_rev
normal_bot 📖mathematicalNormal
Bot.bot
Subgroup
instBot
One.instNonempty
normal_comap 📖mathematicalNormal
comap
Normal.comap
normal_iInf_normal 📖mathematicalNormaliInf
Subgroup
instInfSet
mem_iInf
Normal.conj_mem
normal_in_normalizer 📖mathematicalNormal
Subgroup
SetLike.instMembership
instSetLike
normalizer
toGroup
subgroupOf
normal_subgroupOf_iff_le_normalizer
le_normalizer
le_rfl
normal_inf_normal 📖mathematicalNormal
Subgroup
instMin
Normal.conj_mem
normal_le_normalCore 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalCore
ge_trans
normalCore_le
Normal.conj_mem
normal_of_characteristic 📖mathematicalNormalSetLike.ext_iff
Characteristic.fixed
normal_subgroupOf 📖mathematicalNormal
Subgroup
SetLike.instMembership
instSetLike
toGroup
subgroupOf
normal_comap
normal_subgroupOf_iff 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Normal
SetLike.instMembership
instSetLike
toGroup
subgroupOf
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
Normal.conj_mem
normal_subgroupOf_iff_le_normalizer 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Normal
SetLike.instMembership
instSetLike
toGroup
subgroupOf
normalizer
subgroupOf_eq_top
subgroupOf_normalizer_eq
normalizer_eq_top_iff
normal_subgroupOf_iff_le_normalizer_inf 📖mathematicalNormal
Subgroup
SetLike.instMembership
instSetLike
toGroup
subgroupOf
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
instMin
normal_subgroupOf_iff_le_normalizer
inf_le_right
inf_subgroupOf_right
normal_subgroupOf_of_le_normalizer 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
Normal
SetLike.instMembership
instSetLike
toGroup
subgroupOf
normal_subgroupOf_iff_le_normalizer_inf
LE.le.trans
le_inf
le_normalizer
inf_normalizer_le_normalizer_inf
normal_subgroupOf_sup_of_le_normalizer 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
Normal
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
toGroup
subgroupOf
normal_subgroupOf_iff_le_normalizer
le_sup_right
sup_le
le_normalizer
normal_top 📖mathematicalNormal
Top.top
Subgroup
instTop
normalizer_eq_top 📖mathematicalnormalizer
Top.top
Subgroup
instTop
normalizer_eq_top_iff
normalizer_eq_top_iff 📖mathematicalnormalizer
Top.top
Subgroup
instTop
Normal
eq_top_iff
mem_top
Normal.conj_mem
inv_mul_cancel_left
Normal.mem_comm_iff
pi_bot 📖mathematicalpi
Set.univ
Bot.bot
Subgroup
instBot
Pi.group
ext
pi_empty 📖mathematicalpi
Set
Set.instEmptyCollection
Top.top
Subgroup
Pi.group
instTop
ext
instIsEmptyFalse
pi_eq_bot_iff 📖mathematicalpi
Set.univ
Bot.bot
Subgroup
Pi.group
instBot
Set.univ_pi_eq_singleton_iff
pi_top 📖mathematicalpi
Top.top
Subgroup
instTop
Pi.group
ext
prod_eq_bot_iff 📖mathematicalprod
Bot.bot
Subgroup
Prod.instGroup
instBot
Submonoid.prod_eq_bot_iff
prod_le_iff 📖mathematicalSubgroup
Prod.instGroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
prod
map
MonoidHom.inl
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.inr
Submonoid.prod_le_iff
prod_mono 📖mathematicalRelator.LiftFun
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Prod.instGroup
prod
Set.prod_mono
prod_mono_left 📖mathematicalMonotone
Subgroup
Prod.instGroup
PartialOrder.toPreorder
instPartialOrder
prod
prod_mono
le_refl
prod_mono_right 📖mathematicalMonotone
Subgroup
Prod.instGroup
PartialOrder.toPreorder
instPartialOrder
prod
prod_mono
le_refl
prod_normal 📖mathematicalNormal
Prod.instGroup
prod
Normal.conj_mem
mem_prod
prod_subgroupOf_prod_normal 📖mathematicalNormal
Subgroup
Prod.instGroup
SetLike.instMembership
instSetLike
prod
toGroup
subgroupOf
Normal.conj_mem
mem_prod
prod_top 📖mathematicalprod
Top.top
Subgroup
instTop
comap
Prod.instGroup
MonoidHom.fst
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
ext
subgroupOf_normalizer_eq 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
subgroupOf
normalizer
SetLike.instMembership
instSetLike
toGroup
comap_normalizer_eq_of_le_range
LE.le.trans_eq
range_subtype
subset_normalClosure 📖mathematicalSet
Set.instHasSubset
SetLike.coe
Subgroup
instSetLike
normalClosure
Set.Subset.trans
Group.subset_conjugatesOfSet
conjugatesOfSet_subset_normalClosure
subset_normalizer_of_normal 📖mathematicalSet
Set.instHasSubset
SetLike.coe
Subgroup
instSetLike
normalizer
le_top
normalizer_eq_top
topCharacteristic 📖mathematicalCharacteristic
Top.top
Subgroup
instTop
characteristic_iff_map_le
le_top
top_prod 📖mathematicalprod
Top.top
Subgroup
instTop
comap
Prod.instGroup
MonoidHom.snd
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
ext
top_prod_top 📖mathematicalprod
Top.top
Subgroup
instTop
Prod.instGroup
top_prod
comap_top

Subgroup.Characteristic

Theorems

NameKindAssumesProvesValidatesDepends On
fixed 📖mathematicalSubgroup.comap
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid

Subgroup.Normal

Theorems

NameKindAssumesProvesValidatesDepends On
comap 📖mathematicalSubgroup.Normal
Subgroup.comap
map_mul
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
map_inv
conj_mem
map 📖mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
Subgroup.Normal
Subgroup.map
Subgroup.normalizer_eq_top_iff
top_le_iff
MonoidHom.range_eq_top_of_surjective
MonoidHom.range_eq_map
Subgroup.normalizer_eq_top
Subgroup.le_normalizer_map
of_map_injective 📖mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
Subgroup.Normalcomap
Subgroup.comap_map_eq_self_of_injective
of_map_subtype 📖mathematicalSubgroup.Normal
Subgroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.toGroup
of_map_injective
Subgroup.subtype_injective
subgroupOf 📖mathematicalSubgroup.Normal
Subgroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.toGroup
Subgroup.subgroupOf
comap

Subgroup.SubgroupNormal

Theorems

NameKindAssumesProvesValidatesDepends On
mem_comm 📖Subgroup
Preorder.toLE
PartialOrder.toPreorder
Subgroup.instPartialOrder
SetLike.instMembership
Subgroup.instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Subgroup.normal_subgroupOf_iff
mul_one
mul_inv_cancel
mul_assoc

(root)

Definitions

NameCategoryTheorems
NormalizerCondition 📖MathDef
3 mathmath: isNilpotent_of_finite_tfae, normalizerCondition_iff_only_full_group_self_normalizing, normalizerCondition_of_isNilpotent

Theorems

NameKindAssumesProvesValidatesDepends On
div_mem_comm_iff 📖mathematicalSetLike.instMembership
DivInvMonoid.toDiv
Group.toDivInvMonoid
inv_mem_iff
SubgroupClass.toInvMemClass
inv_div
normalizerCondition_iff_only_full_group_self_normalizing 📖mathematicalNormalizerCondition
Top.top
Subgroup
Subgroup.instTop
sub_mem_comm_iff 📖mathematicalSetLike.instMembership
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
neg_mem_iff
AddSubgroupClass.toNegMemClass
neg_sub

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