Documentation Verification Report

Basic

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

Statistics

MetricCount
DefinitionsaddConjugatesOfSet, liftOfRightInverse, liftOfRightInverseAux, liftOfSurjective, Characteristic, inertia, normalClosure, normalCore, pi, prod, prodEquiv, noncenter, conjugatesOfSet, liftOfRightInverse, liftOfRightInverseAux, liftOfSurjective, NormalizerCondition, Characteristic, normalClosure, normalCore, pi, prod, prodEquiv
23
TheoremsaddConj_mem_addConjugatesOfSet, addConjugatesOfSet_mono, addConjugatesOfSet_subset, addConjugates_subset_normal, mem_addConjugatesOfSet_iff, subset_addConjugatesOfSet, eq_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, addConjugatesOfSet_subset_normalClosure, 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_le_normalClosure, 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_normalClosure, 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, normalClosure_closure_eq_normalClosure, normalClosure_empty, 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_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_le_normalCore, 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_normalClosure, 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_empty, 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
214
Total237

AddGroup

Definitions

NameCategoryTheorems
addConjugatesOfSet πŸ“–CompOp
6 mathmath: addConj_mem_addConjugatesOfSet, subset_addConjugatesOfSet, addConjugatesOfSet_mono, mem_addConjugatesOfSet_iff, AddSubgroup.addConjugatesOfSet_subset_normalClosure, addConjugatesOfSet_subset

Theorems

NameKindAssumesProvesValidatesDepends On
addConj_mem_addConjugatesOfSet πŸ“–mathematicalSet
Set.instMembership
addConjugatesOfSet
Set
Set.instMembership
addConjugatesOfSet
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
toSubtractionMonoid
β€”mem_addConjugatesOfSet_iff
IsAddConj.trans
isAddConj_iff
addConjugatesOfSet_mono πŸ“–mathematicalSet
Set.instHasSubset
Set
Set.instHasSubset
addConjugatesOfSet
β€”Set.biUnion_subset_biUnion_left
addConjugatesOfSet_subset πŸ“–mathematicalSet
Set.instHasSubset
SetLike.coe
AddSubgroup
AddSubgroup.instSetLike
Set
Set.instHasSubset
addConjugatesOfSet
SetLike.coe
AddSubgroup
AddSubgroup.instSetLike
β€”Set.iUnionβ‚‚_subset
addConjugates_subset_normal
addConjugates_subset_normal πŸ“–mathematicalAddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
Set
Set.instHasSubset
addConjugatesOf
SubNegMonoid.toAddMonoid
toSubNegMonoid
SetLike.coe
AddSubgroup
AddSubgroup.instSetLike
β€”isAddConj_iff
AddSubgroup.Normal.conj_mem
mem_addConjugatesOfSet_iff πŸ“–mathematicalβ€”Set
Set.instMembership
addConjugatesOfSet
IsAddConj
SubNegMonoid.toAddMonoid
toSubNegMonoid
β€”addConjugatesOfSet.eq_1
Set.mem_iUnionβ‚‚
isAddConj_iff
subset_addConjugatesOfSet πŸ“–mathematicalβ€”Set
Set.instHasSubset
addConjugatesOfSet
β€”mem_addConjugatesOfSet_iff
IsAddConj.refl

AddMonoidHom

Definitions

NameCategoryTheorems
liftOfRightInverse πŸ“–CompOp
3 mathmath: liftOfRightInverse_comp_apply, liftOfRightInverse_comp, eq_liftOfRightInverse
liftOfRightInverseAux πŸ“–CompOp
1 mathmath: liftOfRightInverseAux_comp_apply
liftOfSurjective πŸ“–CompOpβ€”

Theorems

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

AddSubgroup

Definitions

NameCategoryTheorems
Characteristic πŸ“–CompData
12 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, addCommutator_characteristic, characteristic_iff_comap_eq, centerCharacteristic, characteristic_iff_map_le
inertia πŸ“–CompOp
3 mathmath: subgroupOf_inertia, mem_inertia, inertia_map_subtype
normalClosure πŸ“–CompOp
15 mathmath: normalClosure_closure_eq_normalClosure, closure_le_normalClosure, normalClosure_le_normal, normalClosure_empty, addCommutator_eq_normalClosure, normalClosure_eq_iInf, normalClosure_subset_iff, normalClosure_mono, normalClosure_eq_self, normalClosure_normal, le_normalClosure, normalClosure_idempotent, addConjugatesOfSet_subset_normalClosure, addCommutator_def', subset_normalClosure
normalCore πŸ“–CompOp
8 mathmath: normalCore_eq_self, normal_le_normalCore, normalCore_le, normalCore_mono, normalCore_idempotent, normalCore_normal, finiteIndex_normalCore, normalCore_eq_iSup
pi πŸ“–CompOp
19 mathmath: DFinsupp.range_mapRangeAddMonoidHom, addCommutator_pi_pi_le, 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 πŸ“–CompOp
26 mathmath: coe_prod, le_prod_iff, OpenAddSubgroup.toAddSubgroup_prod, prod_mono_left, bot_prod_bot, mem_prod, index_prod, AddMonoidHom.ker_fst, addCommutator_sum_sum, 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 πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
addCommute_of_normal_of_disjoint πŸ“–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
β€”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
addConjugatesOfSet_subset_normalClosure πŸ“–mathematicalβ€”Set
Set.instHasSubset
AddGroup.addConjugatesOfSet
SetLike.coe
AddSubgroup
instSetLike
normalClosure
β€”subset_closure
addSubgroupOf_normalizer_eq πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
addSubgroupOf
normalizer
SetLike.coe
AddSubgroup
instSetLike
SetLike.instMembership
toAddGroup
β€”comap_normalizer_eq_of_le_range
LE.le.trans_eq
range_subtype
botCharacteristic πŸ“–mathematicalβ€”Characteristic
Bot.bot
AddSubgroup
instBot
β€”characteristic_iff_le_map
bot_le
bot_prod_bot πŸ“–mathematicalβ€”prod
Bot.bot
AddSubgroup
instBot
Prod.instAddGroup
β€”SetLike.coe_injective
Set.singleton_prod_singleton
Set.singleton_eq_singleton_iff
Prod.mk_eq_zero
characteristic_iff_comap_eq πŸ“–mathematicalβ€”Characteristic
comap
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”Characteristic.fixed
characteristic_iff_comap_le πŸ“–mathematicalβ€”Characteristic
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”characteristic_iff_comap_eq
le_of_eq
le_antisymm
AddEquiv.symm_apply_apply
characteristic_iff_le_comap πŸ“–mathematicalβ€”Characteristic
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”characteristic_iff_comap_eq
ge_of_eq
le_antisymm
AddEquiv.symm_apply_apply
characteristic_iff_le_map πŸ“–mathematicalβ€”Characteristic
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”map_equiv_eq_comap_symm'
characteristic_iff_le_comap
characteristic_iff_map_eq πŸ“–mathematicalβ€”Characteristic
map
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”map_equiv_eq_comap_symm'
characteristic_iff_comap_eq
characteristic_iff_map_le πŸ“–mathematicalβ€”Characteristic
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”map_equiv_eq_comap_symm'
characteristic_iff_comap_le
closure_le_normalClosure πŸ“–mathematicalβ€”AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
closure
normalClosure
β€”closure_le
subset_normalClosure
closure_prod πŸ“–mathematicalSet
Set.instMembership
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
closure
Prod.instAddGroup
SProd.sprod
Set
Set.instSProd
prod
β€”le_antisymm
closure_le
Set.prod_subset_prod_iff
subset_closure
prod_le_iff
map_le_iff_le_comap
coe_pi πŸ“–mathematicalβ€”SetLike.coe
AddSubgroup
Pi.addGroup
instSetLike
pi
Set.pi
β€”β€”
coe_prod πŸ“–mathematicalβ€”SetLike.coe
AddSubgroup
Prod.instAddGroup
instSetLike
prod
SProd.sprod
Set
Set.instSProd
β€”β€”
comap_normalizer_eq_of_le_range πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddMonoidHom.range
comap
normalizer
SetLike.coe
AddSubgroup
instSetLike
β€”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 πŸ“–mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
comap
normalizer
SetLike.coe
AddSubgroup
instSetLike
β€”comap_normalizer_eq_of_le_range
inertia_map_subtype πŸ“–mathematicalβ€”Subgroup.map
Subgroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.toGroup
Subgroup.subtype
inertia
Submonoid.instMulActionSubtypeMem
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Subgroup.instSubgroupClass
Subgroup.instMin
β€”Subgroup.instSubgroupClass
subgroupOf_inertia
Subgroup.subgroupOf_map_subtype
inf_addSubgroupOf_inf_normal_of_left πŸ“–mathematicalβ€”Normal
AddSubgroup
SetLike.instMembership
instSetLike
instMin
toAddGroup
addSubgroupOf
β€”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 πŸ“–mathematicalβ€”Normal
AddSubgroup
SetLike.instMembership
instSetLike
instMin
toAddGroup
addSubgroupOf
β€”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 πŸ“–mathematicalβ€”AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
instMin
normalizer
SetLike.coe
instSetLike
β€”β€”
instIsAddTorsionFree πŸ“–mathematicalβ€”IsAddTorsionFree
AddSubgroup
SetLike.instMembership
instSetLike
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
toAddGroup
β€”nsmul_right_injective
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
SetLike.coe_eq_coe
le_normalClosure πŸ“–mathematicalβ€”AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
SetLike.coe
instSetLike
β€”subset_normalClosure
le_normalizer_comap πŸ“–mathematicalβ€”AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
normalizer
SetLike.coe
instSetLike
β€”mem_comap
mem_normalizer_iff
map_add
AddMonoidHomClass.toAddHomClass
AddMonoidHom.instAddMonoidHomClass
map_neg
le_normalizer_map πŸ“–mathematicalβ€”AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
normalizer
SetLike.coe
instSetLike
β€”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 πŸ“–mathematicalβ€”AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
instSetLike
β€”subset_normalizer_of_normal
le_normalizer_of_normal_addSubgroupOf πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
instSetLike
β€”normal_addSubgroupOf_iff_le_normalizer
le_pi_iff πŸ“–mathematicalβ€”AddSubgroup
Pi.addGroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
pi
comap
Pi.evalAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”Set.subset_pi_iff
le_prod_iff πŸ“–mathematicalβ€”AddSubgroup
Prod.instAddGroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
prod
map
AddMonoidHom.fst
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.snd
β€”toAddSubmonoid_le
AddSubmonoid.le_prod_iff
map_equiv_normalizer_eq πŸ“–mathematicalβ€”map
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
normalizer
SetLike.coe
AddSubgroup
instSetLike
β€”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 πŸ“–mathematicalFunction.Bijective
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
map
normalizer
SetLike.coe
AddSubgroup
instSetLike
β€”map_equiv_normalizer_eq
AddMonoidHomClass.toAddHomClass
AddMonoidHom.instAddMonoidHomClass
mem_inertia πŸ“–mathematicalβ€”Subgroup
SetLike.instMembership
Subgroup.instSetLike
inertia
AddSubgroup
instSetLike
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
β€”β€”
mem_pi πŸ“–mathematicalβ€”AddSubgroup
Pi.addGroup
SetLike.instMembership
instSetLike
pi
β€”β€”
mem_prod πŸ“–mathematicalβ€”AddSubgroup
Prod.instAddGroup
SetLike.instMembership
instSetLike
prod
β€”β€”
normalClosure_closure_eq_normalClosure πŸ“–mathematicalβ€”normalClosure
SetLike.coe
AddSubgroup
instSetLike
closure
β€”le_antisymm
normalClosure_le_normal
normalClosure_normal
closure_le_normalClosure
normalClosure_mono
subset_closure
normalClosure_empty πŸ“–mathematicalβ€”normalClosure
Set
Set.instEmptyCollection
Bot.bot
AddSubgroup
instBot
β€”normalClosure_closure_eq_normalClosure
closure_empty
normalClosure_eq_self
normal_bot
normalClosure_eq_iInf πŸ“–mathematicalβ€”normalClosure
iInf
AddSubgroup
instInfSet
Normal
Set
Set.instHasSubset
SetLike.coe
instSetLike
β€”le_antisymm
le_iInf
normalClosure_le_normal
iInf_le_of_le
normalClosure_normal
subset_normalClosure
le_rfl
normalClosure_eq_self πŸ“–mathematicalβ€”normalClosure
SetLike.coe
AddSubgroup
instSetLike
β€”le_antisymm
normalClosure_le_normal
Eq.subset
Set.instReflSubset
le_normalClosure
normalClosure_idempotent πŸ“–mathematicalβ€”normalClosure
SetLike.coe
AddSubgroup
instSetLike
β€”normalClosure_eq_self
normalClosure_normal
normalClosure_le_normal πŸ“–mathematicalSet
Set.instHasSubset
SetLike.coe
AddSubgroup
instSetLike
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
β€”closure_induction
AddGroup.addConjugatesOfSet_subset
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
NegMemClass.neg_mem
AddSubgroupClass.toNegMemClass
normalClosure_mono πŸ“–mathematicalSet
Set.instHasSubset
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
β€”normalClosure_le_normal
normalClosure_normal
Set.Subset.trans
subset_normalClosure
normalClosure_normal πŸ“–mathematicalβ€”Normal
normalClosure
β€”closure_induction
addConjugatesOfSet_subset_normalClosure
AddGroup.addConj_mem_addConjugatesOfSet
add_zero
add_neg_cancel
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
addConj_add
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
addConj_neg
NegMemClass.neg_mem
AddSubgroupClass.toNegMemClass
normalClosure_subset_iff πŸ“–mathematicalβ€”Set
Set.instHasSubset
SetLike.coe
AddSubgroup
instSetLike
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
β€”normalClosure_le_normal
Set.Subset.trans
subset_normalClosure
normalCore_eq_iSup πŸ“–mathematicalβ€”normalCore
iSup
AddSubgroup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
Normal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
β€”le_antisymm
le_iSup_of_le
normalCore_normal
normalCore_le
le_rfl
iSup_le
normal_le_normalCore
normalCore_eq_self πŸ“–mathematicalβ€”normalCoreβ€”le_antisymm
normalCore_le
normal_le_normalCore
le_rfl
normalCore_idempotent πŸ“–mathematicalβ€”normalCoreβ€”normalCore_eq_self
normalCore_normal
normalCore_le πŸ“–mathematicalβ€”AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalCore
β€”add_zero
neg_zero
zero_add
normalCore_mono πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalCore
β€”normal_le_normalCore
normalCore_normal
LE.le.trans
normalCore_le
normalCore_normal πŸ“–mathematicalβ€”Normal
normalCore
β€”add_assoc
neg_add_rev
normal_addSubgroupOf πŸ“–mathematicalβ€”Normal
AddSubgroup
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
β€”normal_comap
normal_addSubgroupOf_iff πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Normal
AddSubgroup
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”Normal.conj_mem
normal_addSubgroupOf_iff_le_normalizer πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Normal
AddSubgroup
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
β€”addSubgroupOf_eq_top
addSubgroupOf_normalizer_eq
normalizer_eq_top_iff
normal_addSubgroupOf_iff_le_normalizer_inf πŸ“–mathematicalβ€”Normal
AddSubgroup
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
instMin
β€”normal_addSubgroupOf_iff_le_normalizer
inf_le_right
inf_addSubgroupOf_right
normal_addSubgroupOf_of_le_normalizer πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
instSetLike
Normal
AddSubgroup
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
β€”normal_addSubgroupOf_iff_le_normalizer_inf
LE.le.trans
le_inf
le_normalizer
inf_normalizer_le_normalizer_inf
normal_addSubgroupOf_sup_of_le_normalizer πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
instSetLike
Normal
AddSubgroup
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
toAddGroup
addSubgroupOf
β€”normal_addSubgroupOf_iff_le_normalizer
le_sup_right
sup_le
le_normalizer
normal_bot πŸ“–mathematicalβ€”Normal
Bot.bot
AddSubgroup
instBot
β€”mem_bot
addConj_eq_zero_iff
Zero.instNonempty
normal_comap πŸ“–mathematicalβ€”Normal
comap
β€”Normal.comap
normal_iInf_normal πŸ“–mathematicalNormalNormal
iInf
AddSubgroup
instInfSet
β€”mem_iInf
Normal.conj_mem
normal_in_normalizer πŸ“–mathematicalβ€”Normal
AddSubgroup
SetLike.instMembership
instSetLike
normalizer
SetLike.coe
toAddGroup
addSubgroupOf
β€”normal_addSubgroupOf_iff_le_normalizer
le_normalizer
le_rfl
normal_inf_normal πŸ“–mathematicalβ€”Normal
AddSubgroup
instMin
β€”Normal.conj_mem
normal_le_normalCore πŸ“–mathematicalβ€”AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalCore
β€”ge_trans
normalCore_le
Normal.conj_mem
normal_of_characteristic πŸ“–mathematicalβ€”Normalβ€”SetLike.ext_iff
Characteristic.fixed
normal_top πŸ“–mathematicalβ€”Normal
Top.top
AddSubgroup
instTop
β€”β€”
normalizer_eq_top πŸ“–mathematicalβ€”normalizer
SetLike.coe
AddSubgroup
instSetLike
Top.top
instTop
β€”normalizer_eq_top_iff
normalizer_eq_top_iff πŸ“–mathematicalβ€”normalizer
SetLike.coe
AddSubgroup
instSetLike
Top.top
instTop
Normal
β€”eq_top_iff
mem_top
Normal.conj_mem
Normal.mem_comm_iff
neg_add_cancel_left
pi_bot πŸ“–mathematicalβ€”pi
Set.univ
Bot.bot
AddSubgroup
instBot
Pi.addGroup
β€”ext
mem_pi
Set.mem_univ
mem_bot
pi_empty πŸ“–mathematicalβ€”pi
Set
Set.instEmptyCollection
Top.top
AddSubgroup
Pi.addGroup
instTop
β€”ext
mem_pi
Set.mem_empty_iff_false
IsEmpty.forall_iff
instIsEmptyFalse
mem_top
pi_eq_bot_iff πŸ“–mathematicalβ€”pi
Set.univ
Bot.bot
AddSubgroup
Pi.addGroup
instBot
β€”SetLike.ext'_iff
Set.univ_pi_eq_singleton_iff
pi_top πŸ“–mathematicalβ€”pi
Top.top
AddSubgroup
instTop
Pi.addGroup
β€”ext
mem_pi
mem_top
prod_addSubgroupOf_prod_normal πŸ“–mathematicalβ€”Normal
AddSubgroup
Prod.instAddGroup
SetLike.instMembership
instSetLike
prod
toAddGroup
addSubgroupOf
β€”Normal.conj_mem
mem_prod
prod_eq_bot_iff πŸ“–mathematicalβ€”prod
Bot.bot
AddSubgroup
Prod.instAddGroup
instBot
β€”AddSubmonoid.prod_eq_bot_iff
prod_le_iff πŸ“–mathematicalβ€”AddSubgroup
Prod.instAddGroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
prod
map
AddMonoidHom.inl
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.inr
β€”toAddSubmonoid_le
AddSubmonoid.prod_le_iff
prod_mono πŸ“–mathematicalβ€”Relator.LiftFun
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Prod.instAddGroup
prod
β€”Set.prod_mono
prod_mono_left πŸ“–mathematicalβ€”Monotone
AddSubgroup
Prod.instAddGroup
PartialOrder.toPreorder
instPartialOrder
prod
β€”prod_mono
le_refl
prod_mono_right πŸ“–mathematicalβ€”Monotone
AddSubgroup
Prod.instAddGroup
PartialOrder.toPreorder
instPartialOrder
prod
β€”prod_mono
le_refl
prod_normal πŸ“–mathematicalβ€”Normal
Prod.instAddGroup
prod
β€”Normal.conj_mem
mem_prod
prod_top πŸ“–mathematicalβ€”prod
Top.top
AddSubgroup
instTop
comap
Prod.instAddGroup
AddMonoidHom.fst
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”ext
mem_prod
mem_top
mem_comap
single_mem_pi πŸ“–mathematicalβ€”AddSubgroup
Pi.addGroup
SetLike.instMembership
instSetLike
pi
Pi.single
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”Set.update_mem_pi_iff_of_mem
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
sub_mem_comm_iff πŸ“–mathematicalβ€”AddSubgroup
SetLike.instMembership
instSetLike
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
β€”sub_mem_comm_iff
instAddSubgroupClass
subgroupOf_inertia πŸ“–mathematicalβ€”Subgroup.subgroupOf
inertia
Subgroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.toGroup
Submonoid.instMulActionSubtypeMem
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Subgroup.instSubgroupClass
β€”β€”
subset_normalClosure πŸ“–mathematicalβ€”Set
Set.instHasSubset
SetLike.coe
AddSubgroup
instSetLike
normalClosure
β€”Set.Subset.trans
AddGroup.subset_addConjugatesOfSet
addConjugatesOfSet_subset_normalClosure
subset_normalizer_of_normal πŸ“–mathematicalβ€”Set
Set.instHasSubset
SetLike.coe
AddSubgroup
instSetLike
normalizer
β€”le_top
normalizer_eq_top
topCharacteristic πŸ“–mathematicalβ€”Characteristic
Top.top
AddSubgroup
instTop
β€”characteristic_iff_map_le
le_top
top_prod πŸ“–mathematicalβ€”prod
Top.top
AddSubgroup
instTop
comap
Prod.instAddGroup
AddMonoidHom.snd
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”ext
mem_prod
mem_top
mem_comap
top_prod_top πŸ“–mathematicalβ€”prod
Top.top
AddSubgroup
instTop
Prod.instAddGroup
β€”top_prod
comap_top

AddSubgroup.Characteristic

Theorems

NameKindAssumesProvesValidatesDepends On
fixed πŸ“–mathematicalβ€”AddSubgroup.comap
AddEquiv.toAddMonoidHom
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”β€”

AddSubgroup.Normal

Theorems

NameKindAssumesProvesValidatesDepends On
addSubgroupOf πŸ“–mathematicalβ€”AddSubgroup.Normal
AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
AddSubgroup.toAddGroup
AddSubgroup.addSubgroupOf
β€”comap
comap πŸ“–mathematicalβ€”AddSubgroup.Normal
AddSubgroup.comap
β€”AddSubgroup.mem_comap
map_add
AddMonoidHomClass.toAddHomClass
AddMonoidHom.instAddMonoidHomClass
map_neg
conj_mem
map πŸ“–mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
AddSubgroup.Normal
AddSubgroup.map
β€”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 πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
AddSubgroup.instPartialOrder
SetLike.instMembership
AddSubgroup.instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”AddSubgroup.normal_addSubgroupOf_iff
add_zero
add_neg_cancel
add_assoc

ConjClasses

Definitions

NameCategoryTheorems
noncenter πŸ“–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 πŸ“–mathematicalβ€”ConjClasses
Set
Set.instMembership
noncenter
Set.Nontrivial
carrier
β€”β€”

Group

Definitions

NameCategoryTheorems
conjugatesOfSet πŸ“–CompOp
6 mathmath: Subgroup.conjugatesOfSet_subset_normalClosure, conj_mem_conjugatesOfSet, conjugatesOfSet_mono, mem_conjugatesOfSet_iff, subset_conjugatesOfSet, conjugatesOfSet_subset

Theorems

NameKindAssumesProvesValidatesDepends On
conj_mem_conjugatesOfSet πŸ“–mathematicalSet
Set.instMembership
conjugatesOfSet
Set
Set.instMembership
conjugatesOfSet
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
toDivisionMonoid
β€”mem_conjugatesOfSet_iff
IsConj.trans
isConj_iff
conjugatesOfSet_mono πŸ“–mathematicalSet
Set.instHasSubset
Set
Set.instHasSubset
conjugatesOfSet
β€”Set.biUnion_subset_biUnion_left
conjugatesOfSet_subset πŸ“–mathematicalSet
Set.instHasSubset
SetLike.coe
Subgroup
Subgroup.instSetLike
Set
Set.instHasSubset
conjugatesOfSet
SetLike.coe
Subgroup
Subgroup.instSetLike
β€”Set.iUnionβ‚‚_subset
conjugates_subset_normal
conjugates_subset_normal πŸ“–mathematicalSubgroup
SetLike.instMembership
Subgroup.instSetLike
Set
Set.instHasSubset
conjugatesOf
DivInvMonoid.toMonoid
toDivInvMonoid
SetLike.coe
Subgroup
Subgroup.instSetLike
β€”isConj_iff
Subgroup.Normal.conj_mem
mem_conjugatesOfSet_iff πŸ“–mathematicalβ€”Set
Set.instMembership
conjugatesOfSet
IsConj
DivInvMonoid.toMonoid
toDivInvMonoid
β€”conjugatesOfSet.eq_1
Set.mem_iUnionβ‚‚
subset_conjugatesOfSet πŸ“–mathematicalβ€”Set
Set.instHasSubset
conjugatesOfSet
β€”mem_conjugatesOfSet_iff
IsConj.refl

IsConj

Theorems

NameKindAssumesProvesValidatesDepends On
normalClosure_eq_top_of πŸ“–mathematicalSubgroup
SetLike.instMembership
Subgroup.instSetLike
IsConj
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Subgroup.normalClosure
Subgroup.toGroup
Set
Set.instSingletonSet
Top.top
Subgroup.instTop
Subgroup.normalClosure
Subgroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.toGroup
Set
Set.instSingletonSet
Top.top
Subgroup.instTop
β€”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 πŸ“–CompOp
3 mathmath: eq_liftOfRightInverse, liftOfRightInverse_comp, liftOfRightInverse_comp_apply
liftOfRightInverseAux πŸ“–CompOp
1 mathmath: liftOfRightInverseAux_comp_apply
liftOfSurjective πŸ“–CompOpβ€”

Theorems

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

Subgroup

Definitions

NameCategoryTheorems
Characteristic πŸ“–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 πŸ“–CompOp
24 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, IsConj.normalClosure_eq_top_of, Equiv.Perm.IsThreeCycle.alternating_normalClosure, normalClosure_normal, normalClosure_le_normal, normalClosure_closure_eq_normalClosure, closure_le_normalClosure, subset_normalClosure, normalClosure_idempotent, normalClosure_empty, normalClosure_eq_self, comap_normalClosure
normalCore πŸ“–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 πŸ“–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 πŸ“–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 πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
botCharacteristic πŸ“–mathematicalβ€”Characteristic
Bot.bot
Subgroup
instBot
β€”characteristic_iff_le_map
bot_le
bot_prod_bot πŸ“–mathematicalβ€”prod
Bot.bot
Subgroup
instBot
Prod.instGroup
β€”SetLike.coe_injective
Set.singleton_prod_singleton
characteristic_iff_comap_eq πŸ“–mathematicalβ€”Characteristic
comap
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Characteristic.fixed
characteristic_iff_comap_le πŸ“–mathematicalβ€”Characteristic
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”characteristic_iff_comap_eq
le_of_eq
le_antisymm
MulEquiv.symm_apply_apply
characteristic_iff_le_comap πŸ“–mathematicalβ€”Characteristic
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”characteristic_iff_comap_eq
ge_of_eq
le_antisymm
MulEquiv.symm_apply_apply
characteristic_iff_le_map πŸ“–mathematicalβ€”Characteristic
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”map_equiv_eq_comap_symm'
characteristic_iff_le_comap
characteristic_iff_map_eq πŸ“–mathematicalβ€”Characteristic
map
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”map_equiv_eq_comap_symm'
characteristic_iff_comap_eq
characteristic_iff_map_le πŸ“–mathematicalβ€”Characteristic
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”map_equiv_eq_comap_symm'
characteristic_iff_comap_le
closure_le_normalClosure πŸ“–mathematicalβ€”Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
closure
normalClosure
β€”β€”
closure_prod πŸ“–mathematicalSet
Set.instMembership
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
closure
Prod.instGroup
SProd.sprod
Set
Set.instSProd
prod
β€”le_antisymm
closure_le
Set.prod_subset_prod_iff
subset_closure
prod_le_iff
map_le_iff_le_comap
coe_pi πŸ“–mathematicalβ€”SetLike.coe
Subgroup
Pi.group
instSetLike
pi
Set.pi
β€”β€”
coe_prod πŸ“–mathematicalβ€”SetLike.coe
Subgroup
Prod.instGroup
instSetLike
prod
SProd.sprod
Set
Set.instSProd
β€”β€”
comap_normalClosure πŸ“–mathematicalβ€”normalClosure
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
β€”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 πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
MonoidHom.range
comap
normalizer
SetLike.coe
Subgroup
instSetLike
β€”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 πŸ“–mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
comap
normalizer
SetLike.coe
Subgroup
instSetLike
β€”comap_normalizer_eq_of_le_range
commute_of_normal_of_disjoint πŸ“–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
β€”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 πŸ“–mathematicalβ€”Set
Set.instHasSubset
Group.conjugatesOfSet
SetLike.coe
Subgroup
instSetLike
normalClosure
β€”subset_closure
div_mem_comm_iff πŸ“–mathematicalβ€”Subgroup
SetLike.instMembership
instSetLike
DivInvMonoid.toDiv
Group.toDivInvMonoid
β€”div_mem_comm_iff
instSubgroupClass
inf_normalizer_le_normalizer_inf πŸ“–mathematicalβ€”Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
instMin
normalizer
SetLike.coe
instSetLike
β€”β€”
inf_subgroupOf_inf_normal_of_left πŸ“–mathematicalβ€”Normal
Subgroup
SetLike.instMembership
instSetLike
instMin
toGroup
subgroupOf
β€”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 πŸ“–mathematicalβ€”Normal
Subgroup
SetLike.instMembership
instSetLike
instMin
toGroup
subgroupOf
β€”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 πŸ“–mathematicalβ€”IsMulTorsionFree
Subgroup
SetLike.instMembership
instSetLike
DivInvMonoid.toMonoid
Group.toDivInvMonoid
toGroup
β€”pow_left_injective
SubgroupClass.toSubmonoidClass
instSubgroupClass
le_normalClosure πŸ“–mathematicalβ€”Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
SetLike.coe
instSetLike
β€”subset_normalClosure
le_normalizer_comap πŸ“–mathematicalβ€”Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
normalizer
SetLike.coe
instSetLike
β€”map_mul
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
map_inv
le_normalizer_map πŸ“–mathematicalβ€”Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
normalizer
SetLike.coe
instSetLike
β€”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 πŸ“–mathematicalβ€”Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
instSetLike
β€”subset_normalizer_of_normal
le_normalizer_of_normal_subgroupOf πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
instSetLike
β€”normal_subgroupOf_iff_le_normalizer
le_pi_iff πŸ“–mathematicalβ€”Subgroup
Pi.group
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
pi
comap
Pi.evalMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Set.subset_pi_iff
le_prod_iff πŸ“–mathematicalβ€”Subgroup
Prod.instGroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
prod
map
MonoidHom.fst
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.snd
β€”Submonoid.le_prod_iff
map_equiv_normalizer_eq πŸ“–mathematicalβ€”map
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
normalizer
SetLike.coe
Subgroup
instSetLike
β€”ext
Equiv.forall_congr
MulEquiv.symm_apply_apply
map_mul
MonoidHomClass.toMulHomClass
MulEquivClass.instMonoidHomClass
MulEquiv.instMulEquivClass
map_inv
map_normalClosure πŸ“–mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
map
normalClosure
Set.image
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
β€”Normal.map
normalClosure_normal
le_antisymm
normal_comap
normalClosure_le_normal
Set.image_mono
subset_normalClosure
map_normalizer_eq_of_bijective πŸ“–mathematicalFunction.Bijective
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
map
normalizer
SetLike.coe
Subgroup
instSetLike
β€”map_equiv_normalizer_eq
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
mem_pi πŸ“–mathematicalβ€”Subgroup
Pi.group
SetLike.instMembership
instSetLike
pi
β€”β€”
mem_prod πŸ“–mathematicalβ€”Subgroup
Prod.instGroup
SetLike.instMembership
instSetLike
prod
β€”β€”
mulSingle_mem_pi πŸ“–mathematicalβ€”Subgroup
Pi.group
SetLike.instMembership
instSetLike
pi
Pi.mulSingle
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”Set.update_mem_pi_iff_of_mem
OneMemClass.one_mem
SubmonoidClass.toOneMemClass
SubgroupClass.toSubmonoidClass
instSubgroupClass
normalClosure_closure_eq_normalClosure πŸ“–mathematicalβ€”normalClosure
SetLike.coe
Subgroup
instSetLike
closure
β€”le_antisymm
normalClosure_le_normal
normalClosure_normal
closure_le_normalClosure
normalClosure_mono
subset_closure
normalClosure_empty πŸ“–mathematicalβ€”normalClosure
Set
Set.instEmptyCollection
Bot.bot
Subgroup
instBot
β€”normalClosure_closure_eq_normalClosure
closure_empty
normalClosure_eq_self
normal_bot
normalClosure_eq_iInf πŸ“–mathematicalβ€”normalClosure
iInf
Subgroup
instInfSet
Normal
Set
Set.instHasSubset
SetLike.coe
instSetLike
β€”le_antisymm
le_iInf
normalClosure_le_normal
iInf_le_of_le
normalClosure_normal
subset_normalClosure
le_rfl
normalClosure_eq_self πŸ“–mathematicalβ€”normalClosure
SetLike.coe
Subgroup
instSetLike
β€”le_antisymm
normalClosure_le_normal
Eq.subset
Set.instReflSubset
le_normalClosure
normalClosure_idempotent πŸ“–mathematicalβ€”normalClosure
SetLike.coe
Subgroup
instSetLike
β€”normalClosure_eq_self
normalClosure_normal
normalClosure_le_normal πŸ“–mathematicalSet
Set.instHasSubset
SetLike.coe
Subgroup
instSetLike
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
β€”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 πŸ“–mathematicalSet
Set.instHasSubset
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
β€”normalClosure_le_normal
normalClosure_normal
Set.Subset.trans
subset_normalClosure
normalClosure_normal πŸ“–mathematicalβ€”Normal
normalClosure
β€”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 πŸ“–mathematicalβ€”Set
Set.instHasSubset
SetLike.coe
Subgroup
instSetLike
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalClosure
β€”normalClosure_le_normal
Set.Subset.trans
subset_normalClosure
normalCore_eq_iSup πŸ“–mathematicalβ€”normalCore
iSup
Subgroup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
instCompleteLattice
Normal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
β€”le_antisymm
le_iSup_of_le
normalCore_normal
normalCore_le
le_rfl
iSup_le
normal_le_normalCore
normalCore_eq_self πŸ“–mathematicalβ€”normalCoreβ€”le_antisymm
normalCore_le
normal_le_normalCore
le_rfl
normalCore_idempotent πŸ“–mathematicalβ€”normalCoreβ€”normalCore_eq_self
normalCore_normal
normalCore_le πŸ“–mathematicalβ€”Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalCore
β€”mul_one
inv_one
one_mul
normalCore_mono πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalCore
β€”normal_le_normalCore
normalCore_normal
LE.le.trans
normalCore_le
normalCore_normal πŸ“–mathematicalβ€”Normal
normalCore
β€”mul_assoc
mul_inv_rev
normal_bot πŸ“–mathematicalβ€”Normal
Bot.bot
Subgroup
instBot
β€”One.instNonempty
normal_comap πŸ“–mathematicalβ€”Normal
comap
β€”Normal.comap
normal_iInf_normal πŸ“–mathematicalNormalNormal
iInf
Subgroup
instInfSet
β€”mem_iInf
Normal.conj_mem
normal_in_normalizer πŸ“–mathematicalβ€”Normal
Subgroup
SetLike.instMembership
instSetLike
normalizer
SetLike.coe
toGroup
subgroupOf
β€”normal_subgroupOf_iff_le_normalizer
le_normalizer
le_rfl
normal_inf_normal πŸ“–mathematicalβ€”Normal
Subgroup
instMin
β€”Normal.conj_mem
normal_le_normalCore πŸ“–mathematicalβ€”Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalCore
β€”ge_trans
normalCore_le
Normal.conj_mem
normal_of_characteristic πŸ“–mathematicalβ€”Normalβ€”SetLike.ext_iff
Characteristic.fixed
normal_subgroupOf πŸ“–mathematicalβ€”Normal
Subgroup
SetLike.instMembership
instSetLike
toGroup
subgroupOf
β€”normal_comap
normal_subgroupOf_iff πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Normal
Subgroup
SetLike.instMembership
instSetLike
toGroup
subgroupOf
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”Normal.conj_mem
normal_subgroupOf_iff_le_normalizer πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Normal
Subgroup
SetLike.instMembership
instSetLike
toGroup
subgroupOf
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
β€”subgroupOf_eq_top
subgroupOf_normalizer_eq
normalizer_eq_top_iff
normal_subgroupOf_iff_le_normalizer_inf πŸ“–mathematicalβ€”Normal
Subgroup
SetLike.instMembership
instSetLike
toGroup
subgroupOf
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
instMin
β€”normal_subgroupOf_iff_le_normalizer
inf_le_right
inf_subgroupOf_right
normal_subgroupOf_of_le_normalizer πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
instSetLike
Normal
Subgroup
SetLike.instMembership
instSetLike
toGroup
subgroupOf
β€”normal_subgroupOf_iff_le_normalizer_inf
LE.le.trans
le_inf
le_normalizer
inf_normalizer_le_normalizer_inf
normal_subgroupOf_sup_of_le_normalizer πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
normalizer
SetLike.coe
instSetLike
Normal
Subgroup
SetLike.instMembership
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
toGroup
subgroupOf
β€”normal_subgroupOf_iff_le_normalizer
le_sup_right
sup_le
le_normalizer
normal_top πŸ“–mathematicalβ€”Normal
Top.top
Subgroup
instTop
β€”β€”
normalizer_eq_top πŸ“–mathematicalβ€”normalizer
SetLike.coe
Subgroup
instSetLike
Top.top
instTop
β€”normalizer_eq_top_iff
normalizer_eq_top_iff πŸ“–mathematicalβ€”normalizer
SetLike.coe
Subgroup
instSetLike
Top.top
instTop
Normal
β€”eq_top_iff
mem_top
Normal.conj_mem
Normal.mem_comm_iff
inv_mul_cancel_left
pi_bot πŸ“–mathematicalβ€”pi
Set.univ
Bot.bot
Subgroup
instBot
Pi.group
β€”ext
pi_empty πŸ“–mathematicalβ€”pi
Set
Set.instEmptyCollection
Top.top
Subgroup
Pi.group
instTop
β€”ext
instIsEmptyFalse
pi_eq_bot_iff πŸ“–mathematicalβ€”pi
Set.univ
Bot.bot
Subgroup
Pi.group
instBot
β€”Set.univ_pi_eq_singleton_iff
pi_top πŸ“–mathematicalβ€”pi
Top.top
Subgroup
instTop
Pi.group
β€”ext
prod_eq_bot_iff πŸ“–mathematicalβ€”prod
Bot.bot
Subgroup
Prod.instGroup
instBot
β€”Submonoid.prod_eq_bot_iff
prod_le_iff πŸ“–mathematicalβ€”Subgroup
Prod.instGroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
prod
map
MonoidHom.inl
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.inr
β€”Submonoid.prod_le_iff
prod_mono πŸ“–mathematicalβ€”Relator.LiftFun
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Prod.instGroup
prod
β€”Set.prod_mono
prod_mono_left πŸ“–mathematicalβ€”Monotone
Subgroup
Prod.instGroup
PartialOrder.toPreorder
instPartialOrder
prod
β€”prod_mono
le_refl
prod_mono_right πŸ“–mathematicalβ€”Monotone
Subgroup
Prod.instGroup
PartialOrder.toPreorder
instPartialOrder
prod
β€”prod_mono
le_refl
prod_normal πŸ“–mathematicalβ€”Normal
Prod.instGroup
prod
β€”Normal.conj_mem
mem_prod
prod_subgroupOf_prod_normal πŸ“–mathematicalβ€”Normal
Subgroup
Prod.instGroup
SetLike.instMembership
instSetLike
prod
toGroup
subgroupOf
β€”Normal.conj_mem
mem_prod
prod_top πŸ“–mathematicalβ€”prod
Top.top
Subgroup
instTop
comap
Prod.instGroup
MonoidHom.fst
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”ext
subgroupOf_normalizer_eq πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
subgroupOf
normalizer
SetLike.coe
Subgroup
instSetLike
SetLike.instMembership
toGroup
β€”comap_normalizer_eq_of_le_range
LE.le.trans_eq
range_subtype
subset_normalClosure πŸ“–mathematicalβ€”Set
Set.instHasSubset
SetLike.coe
Subgroup
instSetLike
normalClosure
β€”Set.Subset.trans
Group.subset_conjugatesOfSet
conjugatesOfSet_subset_normalClosure
subset_normalizer_of_normal πŸ“–mathematicalβ€”Set
Set.instHasSubset
SetLike.coe
Subgroup
instSetLike
normalizer
β€”le_top
normalizer_eq_top
topCharacteristic πŸ“–mathematicalβ€”Characteristic
Top.top
Subgroup
instTop
β€”characteristic_iff_map_le
le_top
top_prod πŸ“–mathematicalβ€”prod
Top.top
Subgroup
instTop
comap
Prod.instGroup
MonoidHom.snd
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”ext
top_prod_top πŸ“–mathematicalβ€”prod
Top.top
Subgroup
instTop
Prod.instGroup
β€”top_prod
comap_top

Subgroup.Characteristic

Theorems

NameKindAssumesProvesValidatesDepends On
fixed πŸ“–mathematicalβ€”Subgroup.comap
MulEquiv.toMonoidHom
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”β€”

Subgroup.Normal

Theorems

NameKindAssumesProvesValidatesDepends On
comap πŸ“–mathematicalβ€”Subgroup.Normal
Subgroup.comap
β€”map_mul
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
map_inv
conj_mem
map πŸ“–mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
Subgroup.Normal
Subgroup.map
β€”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 πŸ“–mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
Subgroup.Normalβ€”comap
Subgroup.comap_map_eq_self_of_injective
of_map_subtype πŸ“–mathematicalβ€”Subgroup.Normal
Subgroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.toGroup
β€”of_map_injective
Subgroup.subtype_injective
subgroupOf πŸ“–mathematicalβ€”Subgroup.Normal
Subgroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.toGroup
Subgroup.subgroupOf
β€”comap

Subgroup.SubgroupNormal

Theorems

NameKindAssumesProvesValidatesDepends On
mem_comm πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
Subgroup.instPartialOrder
SetLike.instMembership
Subgroup.instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Subgroup
SetLike.instMembership
Subgroup.instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Subgroup.normal_subgroupOf_iff
mul_one
mul_inv_cancel
mul_assoc

(root)

Definitions

NameCategoryTheorems
NormalizerCondition πŸ“–MathDef
3 mathmath: isNilpotent_of_finite_tfae, normalizerCondition_iff_only_full_group_self_normalizing, normalizerCondition_of_isNilpotent

Theorems

NameKindAssumesProvesValidatesDepends On
div_mem_comm_iff πŸ“–mathematicalβ€”SetLike.instMembership
DivInvMonoid.toDiv
Group.toDivInvMonoid
β€”inv_mem_iff
SubgroupClass.toInvMemClass
inv_div
normalizerCondition_iff_only_full_group_self_normalizing πŸ“–mathematicalβ€”NormalizerCondition
Top.top
Subgroup
Subgroup.instTop
β€”β€”
sub_mem_comm_iff πŸ“–mathematicalβ€”SetLike.instMembership
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
β€”neg_mem_iff
AddSubgroupClass.toNegMemClass
neg_sub

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