Documentation Verification Report

Index

πŸ“ Source: Mathlib/GroupTheory/Index.lean

Statistics

MetricCount
DefinitionsIsFiniteRelIndex, fintypeOfIndexNeZero, fintypeQuotientOfFiniteIndex, index, relIndex, IsFiniteRelIndex, fintypeOfIndexNeZero, fintypeQuotientOfFiniteIndex, index, relIndex
10
Theoremsindex_stabilizer, index_stabilizer_of_transitive, card_fiber_eq_of_mem_range, finite_iff_finite_ker_range, surjective_of_card_ker_le_div, index_ne_zero, relIndex_ne_zero, to_finiteIndex_addSubgroupOf, add_mem_iff_of_index_two, add_self_mem_of_index_two, card_dvd_of_surjective, card_eq_one, card_map_dvd, card_mul_index, card_range_dvd, dvd_index_map, exists_nsmul_mem_of_index_ne_zero, exists_nsmul_mem_of_relIndex_ne_zero, finiteIndex_iInf, finiteIndex_iInf', finiteIndex_iff, finiteIndex_iff_finite_quotient, finiteIndex_ker, finiteIndex_of_finite, finiteIndex_of_finite_quotient, finiteIndex_of_le, finite_iff_finite_and_finiteIndex, finite_quotient_of_finiteIndex, finite_quotient_of_finite_quotient_of_index_ne_zero, finite_quotient_of_pretransitive_of_index_ne_zero, index_antitone, index_bot, index_comap, index_comap_of_surjective, index_dvd_card, index_dvd_of_le, index_dvd_two_iff, index_dvd_two_iff', index_eq_card, index_eq_one, index_eq_two_iff, index_eq_two_iff', index_eq_two_iff_exists_notMem_and, index_eq_two_iff_exists_notMem_and', index_eq_zero_iff_infinite, index_eq_zero_of_relIndex_eq_zero, index_iInf_le, index_iInf_ne_zero, index_inf_le, index_inf_ne_zero, index_ker, index_map, index_map_dvd, index_map_eq, index_map_equiv, index_map_of_bijective, index_map_of_injective, index_map_subtype, index_mul_card, index_ne_zero_iff_finite, index_ne_zero_of_finite, index_pi, index_prod, index_range, index_smul, index_strictAnti, index_toSubgroup, index_top, inf_eq_bot_of_coprime, inf_relIndex_left, inf_relIndex_right, instFiniteIndexMin, instFiniteIndexTop, instFiniteIndex_addSubgroupOf, not_finiteIndex_iff, nsmul_mem_of_index_ne_zero_of_dvd, nsmul_mem_of_relIndex_ne_zero_of_dvd, one_lt_index_of_ne_top, relIindex_dvd_two_iff', relIindex_eq_two_iff', relIndex_addSubgroupOf, relIndex_bot_left, relIndex_bot_right, relIndex_comap, relIndex_comap_ne_zero, relIndex_dvd_card, relIndex_dvd_index_of_le, relIndex_dvd_index_of_normal, relIndex_dvd_of_le_left, relIndex_dvd_two_iff, relIndex_eq_one, relIndex_eq_two_iff, relIndex_eq_two_iff_exists_notMem_and, relIndex_eq_two_iff_exists_notMem_and', relIndex_eq_zero_of_le_left, relIndex_eq_zero_of_le_right, relIndex_iInf_le, relIndex_iInf_ne_zero, relIndex_inf_le, relIndex_inf_mul_relIndex, relIndex_inf_ne_zero, relIndex_inter_ne_zero, relIndex_ker, relIndex_le_of_le_left, relIndex_le_of_le_right, relIndex_map_map, relIndex_map_map_of_injective, relIndex_mul_index, relIndex_mul_relIndex, relIndex_ne_zero, relIndex_ne_zero_trans, relIndex_pointwise_smul, relIndex_self, relIndex_sup_left, relIndex_sup_right, relIndex_toSubgroup, relIndex_top_left, relIndex_top_right, two_smul_mem_of_index_two, card_fiber_eq_of_mem_range, finite_iff_finite_ker_range, surjective_of_card_ker_le_div, index_stabilizer, index_stabilizer_of_transitive, index_ne_zero, relIndex_ne_zero, to_finiteIndex_subgroupOf, card_dvd_of_surjective, card_eq_one, card_map_dvd, card_mul_index, card_range_dvd, dvd_index_map, exists_pow_mem_of_index_ne_zero, exists_pow_mem_of_relIndex_ne_zero, finiteIndex_iInf, finiteIndex_iInf', finiteIndex_iff, finiteIndex_iff_finite_quotient, finiteIndex_ker, finiteIndex_normalCore, finiteIndex_of_finite, finiteIndex_of_finite_quotient, finiteIndex_of_le, finite_iff_finite_and_finiteIndex, finite_quotient_of_finiteIndex, finite_quotient_of_finite_quotient_of_index_ne_zero, finite_quotient_of_pretransitive_of_index_ne_zero, index_antitone, index_bot, index_comap, index_comap_of_surjective, index_dvd_card, index_dvd_of_le, index_dvd_two_iff, index_dvd_two_iff', index_eq_card, index_eq_one, index_eq_two_iff, index_eq_two_iff', index_eq_two_iff_exists_notMem_and, index_eq_two_iff_exists_notMem_and', index_eq_zero_iff_infinite, index_eq_zero_of_relIndex_eq_zero, index_iInf_le, index_iInf_ne_zero, index_inf_le, index_inf_ne_zero, index_ker, index_map, index_map_dvd, index_map_eq, index_map_equiv, index_map_of_bijective, index_map_of_injective, index_map_subtype, index_mul_card, index_ne_zero_iff_finite, index_ne_zero_of_finite, index_pi, index_prod, index_range, index_strictAnti, index_toAddSubgroup, index_top, inf_eq_bot_of_coprime, inf_relIndex_left, inf_relIndex_right, instFiniteIndexMin, instFiniteIndexTop, instFiniteIndex_subgroupOf, mul_mem_iff_of_index_two, mul_self_mem_of_index_two, not_finiteIndex_iff, one_lt_index_of_ne_top, pow_mem_of_index_ne_zero_of_dvd, pow_mem_of_relIndex_ne_zero_of_dvd, relIindex_dvd_two_iff', relIindex_eq_two_iff', relIndex_bot_left, relIndex_bot_right, relIndex_comap, relIndex_comap_ne_zero, relIndex_dvd_card, relIndex_dvd_index_of_le, relIndex_dvd_index_of_normal, relIndex_dvd_of_le_left, relIndex_dvd_two_iff, relIndex_eq_one, relIndex_eq_two_iff, relIndex_eq_two_iff_exists_notMem_and, relIndex_eq_two_iff_exists_notMem_and', relIndex_eq_zero_of_le_left, relIndex_eq_zero_of_le_right, relIndex_iInf_le, relIndex_iInf_ne_zero, relIndex_inf_le, relIndex_inf_mul_relIndex, relIndex_inf_ne_zero, relIndex_inter_ne_zero, relIndex_ker, relIndex_le_of_le_left, relIndex_le_of_le_right, relIndex_map_map, relIndex_map_map_of_injective, relIndex_mul_index, relIndex_mul_relIndex, relIndex_ne_zero, relIndex_ne_zero_trans, relIndex_pointwise_smul, relIndex_self, relIndex_subgroupOf, relIndex_sup_left, relIndex_sup_right, relIndex_toAddSubgroup, relIndex_top_left, relIndex_top_right, sq_mem_of_index_two
238
Total248

AddAction

Theorems

NameKindAssumesProvesValidatesDepends On
index_stabilizer πŸ“–mathematicalβ€”AddSubgroup.index
stabilizer
Set.ncard
orbit
AddSemigroupAction.toVAdd
AddMonoid.toAddSemigroup
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
toAddSemigroupAction
β€”Nat.card_congr
Nat.card_coe_set_eq
index_stabilizer_of_transitive πŸ“–mathematicalβ€”AddSubgroup.index
stabilizer
Nat.card
β€”index_stabilizer
orbit_eq_univ
Set.ncard_univ

AddMonoidHom

Theorems

NameKindAssumesProvesValidatesDepends On
card_fiber_eq_of_mem_range πŸ“–mathematicalSet
Set.instMembership
Set.range
DFunLike.coe
Finset.card
Finset.filter
Finset.univ
β€”add_left_surjective
Finset.map_univ_equiv
Finset.filter_map
Finset.card_map
map_add
AddMonoidHomClass.toAddHomClass
coe_toHomAddUnits
map_neg
instAddMonoidHomClass
AddUnits.add_neg_eq_iff_eq_add
finite_iff_finite_ker_range πŸ“–mathematicalβ€”Finite
AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
ker
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
range
β€”AddSubgroup.finite_iff_finite_and_finiteIndex
Equiv.finite_iff
normal_ker
AddSubgroup.finiteIndex_iff_finite_quotient
surjective_of_card_ker_le_div πŸ“–mathematicalNat.card
AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
ker
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
instFunLike
β€”range_eq_top
SetLike.ext'
Set.eq_of_subset_of_ncard_le
Set.subset_univ
AddSubgroup.coe_top
Set.ncard_univ
Nat.card_coe_set_eq
SetLike.coe_sort_coe
Nat.card_congr
normal_ker
AddSubgroup.card_mul_index
Nat.card_pos
Zero.instNonempty
Subtype.finite
Set.toFinite

AddSubgroup

Definitions

NameCategoryTheorems
IsFiniteRelIndex πŸ“–CompDataβ€”
fintypeOfIndexNeZero πŸ“–CompOpβ€”
fintypeQuotientOfFiniteIndex πŸ“–CompOpβ€”
index πŸ“–CompOp
67 mathmath: index_map_equiv, index_comap_of_surjective, index_eq_zero_iff_infinite, IsComplement.ncard_right, index_prod, IsComplement.card_right, IsAddCyclic.index_nsmulAddMonoidHom_range, index_iInf_le, index_map_eq, index_inf_le, index_eq_natAbs_det, index_eq_two_iff_exists_notMem_and, index_ker, index_pi, index_dvd_card, Ideal.index_pow_le, IsComplement.encard_right, IsComplement.encard_left, dvd_index_map, index_antitone, Ideal.Quotient.index_eq_zero, index_map, index_eq_two_iff_exists_notMem_and', card_mul_index, index_map_dvd, relIndex_mul_index, index_mul_card, exists_index_le_card_of_leftCoset_cover, index_eq_two_iff', index_comap, relIndex_top_right, index_map_of_injective, index_map_subtype, AddAction.index_stabilizer_of_transitive, MeasureTheory.AddSubgroup.index_mul_measure, index_strictAnti, Subgroup.index_toAddSubgroup, exists_nsmul_mem_of_index_ne_zero, one_le_sum_inv_index_of_leftCoset_cover, Int.index_zmultiples, nsmul_index_mem, index_eq_zero_of_relIndex_eq_zero, Module.Basis.SmithNormalForm.toAddSubgroup_index_eq_pow_mul_prod, index_dvd_of_le, index_map_of_bijective, index_eq_two_iff, index_range, IsComplement.ncard_left, one_lt_index_of_ne_top, index_eq_card, leftCoset_cover_filter_FiniteIndex_aux, index_dvd_two_iff', index_toSubgroup, index_top, Submodule.index_smul_le, IsComplement.card_left, IsAddCyclic.index_nsmulAddMonoidHom_ker, index_le_of_leftCoset_cover_const, Module.Basis.SmithNormalForm.toAddSubgroup_index_eq_ite, relIndex_dvd_index_of_le, AddAction.index_stabilizer, index_smul, index_eq_one, index_dvd_two_iff, index_bot, not_finiteIndex_iff, relIndex_dvd_index_of_normal
relIndex πŸ“–CompOp
43 mathmath: relIndex_mul_relIndex, relIndex_eq_two_iff_exists_notMem_and, relIndex_inf_mul_relIndex, exists_nsmul_mem_of_relIndex_ne_zero, relIndex_map_map, relIndex_le_of_le_right, relIndex_sup_left, ZLattice.covolume_div_covolume_eq_relIndex, relIndex_inf_le, relIndex_mul_index, relIndex_eq_abs_det, relIndex_eq_one, relIndex_eq_two_iff_exists_notMem_and', index_comap, relIndex_pointwise_smul, relIndex_top_right, relIndex_map_map_of_injective, relIndex_le_of_le_left, Subgroup.relIndex_strictPeriods, relIndex_self, relIndex_addSubgroupOf, inf_relIndex_right, relIindex_eq_two_iff', relIndex_bot_right, relIndex_dvd_two_iff, relIndex_dvd_of_le_left, inf_relIndex_left, relIndex_eq_natAbs_det, relIindex_dvd_two_iff', relIndex_top_left, relIndex_comap, relIndex_ker, relIndex_dvd_index_of_le, relIndex_iInf_le, relIndex_dvd_card, AddAction.IsBlock.ncard_block_eq_relIndex, relIndex_sup_right, ZLattice.covolume_div_covolume_eq_relIndex', Subgroup.relIndex_toAddSubgroup, relIndex_bot_left, relIndex_toSubgroup, relIndex_dvd_index_of_normal, relIndex_eq_two_iff

Theorems

NameKindAssumesProvesValidatesDepends On
add_mem_iff_of_index_two πŸ“–mathematicalindexAddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”add_mem_cancel_left
instAddSubgroupClass
add_mem_cancel_right
index_eq_two_iff
Xor'.or
add_assoc
add_self_mem_of_index_two πŸ“–mathematicalindexAddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”add_mem_iff_of_index_two
card_dvd_of_surjective πŸ“–mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
Nat.cardβ€”Nat.card_congr
AddMonoidHom.normal_ker
Dvd.intro_left
card_mul_index
card_eq_one πŸ“–mathematicalβ€”Nat.card
AddSubgroup
SetLike.instMembership
instSetLike
Bot.bot
instBot
β€”relIndex_eq_one
le_bot_iff
relIndex_bot_left
card_map_dvd πŸ“–mathematicalβ€”Nat.card
AddSubgroup
SetLike.instMembership
instSetLike
map
β€”card_dvd_of_surjective
AddMonoidHom.addSubgroupMap_surjective
card_mul_index πŸ“–mathematicalβ€”Nat.card
AddSubgroup
SetLike.instMembership
instSetLike
index
β€”relIndex_bot_left
index_bot
relIndex_mul_index
bot_le
card_range_dvd πŸ“–mathematicalβ€”Nat.card
AddSubgroup
SetLike.instMembership
instSetLike
AddMonoidHom.range
β€”card_dvd_of_surjective
AddMonoidHom.rangeRestrict_surjective
dvd_index_map πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddMonoidHom.ker
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
index
map
β€”index_map
sup_of_le_left
dvd_mul_right
exists_nsmul_mem_of_index_ne_zero πŸ“–mathematicalβ€”index
AddSubgroup
SetLike.instMembership
instSetLike
AddMonoid.toNatSMul
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”Fintype.finite
Nat.card_le_card_of_injective
Nat.card_eq_fintype_card
Finset.mem_Icc
Fintype.card_ofFinset
Nat.card_Icc
tsub_zero
Nat.instOrderedSub
add_le_iff_nonpos_right
IsOrderedAddMonoid.toAddLeftMono
Nat.instIsOrderedAddMonoid
IsLeftCancelAdd.addLeftReflectLE_of_addLeftReflectLT
AddLeftCancelSemigroup.toIsLeftCancelAdd
contravariant_lt_of_covariant_le
nonpos_iff_eq_zero
Nat.instCanonicallyOrderedAdd
one_ne_zero
Ne.lt_or_gt
natCast_zsmul
add_comm
add_zsmul
neg_zsmul
QuotientAddGroup.eq
exists_nsmul_mem_of_relIndex_ne_zero πŸ“–mathematicalAddSubgroup
SetLike.instMembership
instSetLike
relIndex
instMin
AddMonoid.toNatSMul
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”exists_nsmul_mem_of_index_ne_zero
mem_inf
nsmul_mem
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
mem_addSubgroupOf
finiteIndex_iInf πŸ“–mathematicalFiniteIndexiInf
AddSubgroup
instInfSet
β€”index_iInf_ne_zero
FiniteIndex.index_ne_zero
finiteIndex_iInf' πŸ“–mathematicalFiniteIndexiInf
AddSubgroup
instInfSet
Finset
Finset.instMembership
β€”iInf_subtype'
finiteIndex_iInf
Finite.of_fintype
finiteIndex_iff πŸ“–mathematicalβ€”FiniteIndexβ€”FiniteIndex.index_ne_zero
finiteIndex_iff_finite_quotient πŸ“–mathematicalβ€”FiniteIndex
Finite
HasQuotient.Quotient
AddSubgroup
QuotientAddGroup.instHasQuotientAddSubgroup
β€”finite_quotient_of_finiteIndex
finiteIndex_of_finite_quotient
finiteIndex_ker πŸ“–mathematicalβ€”FiniteIndex
AddMonoidHom.ker
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”finiteIndex_of_finite_quotient
Finite.of_equiv
AddMonoidHom.normal_ker
finiteIndex_of_finite πŸ“–mathematicalβ€”FiniteIndexβ€”finiteIndex_of_finite_quotient
QuotientAddGroup.finite
finiteIndex_of_finite_quotient πŸ“–mathematicalβ€”FiniteIndexβ€”index_ne_zero_of_finite
finiteIndex_of_le πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
FiniteIndexβ€”ne_zero_of_dvd_ne_zero
FiniteIndex.index_ne_zero
index_dvd_of_le
finite_iff_finite_and_finiteIndex πŸ“–mathematicalβ€”Finite
AddSubgroup
SetLike.instMembership
instSetLike
FiniteIndex
β€”Subtype.finite
finiteIndex_of_finite
Nat.finite_of_card_ne_zero
mul_ne_zero
IsDomain.to_noZeroDivisors
Nat.instIsDomain
LT.lt.ne'
Nat.card_pos
Zero.instNonempty
FiniteIndex.index_ne_zero
card_mul_index
finite_quotient_of_finiteIndex πŸ“–mathematicalβ€”Finite
HasQuotient.Quotient
AddSubgroup
QuotientAddGroup.instHasQuotientAddSubgroup
β€”Fintype.finite
finite_quotient_of_finite_quotient_of_index_ne_zero πŸ“–mathematicalβ€”Finite
AddAction.orbitRel.Quotient
AddSubgroup
SetLike.instMembership
instSetLike
toAddGroup
instAddAction
β€”AddAction.finite_quotient_of_finite_quotient_of_finite_quotient
Finite.of_fintype
finite_quotient_of_pretransitive_of_index_ne_zero πŸ“–mathematicalβ€”Finite
AddAction.orbitRel.Quotient
AddSubgroup
SetLike.instMembership
instSetLike
toAddGroup
instAddAction
β€”AddAction.pretransitive_iff_subsingleton_quotient
finite_quotient_of_finite_quotient_of_index_ne_zero
Finite.of_subsingleton
index_antitone πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
indexβ€”FiniteIndex.index_ne_zero
index_dvd_of_le
index_bot πŸ“–mathematicalβ€”index
Bot.bot
AddSubgroup
instBot
Nat.card
β€”Nat.card_congr
normal_bot
index_comap πŸ“–mathematicalβ€”index
comap
relIndex
AddMonoidHom.range
β€”index_comap_of_surjective
AddMonoidHom.rangeRestrict_surjective
index_comap_of_surjective πŸ“–mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
index
comap
β€”QuotientAddGroup.leftRel_apply
AddMonoidHom.map_add
AddMonoidHom.map_neg
Nat.card_congr
Quotient.ind'
Quotient.eq''
Quotient.map'_mk''
index_dvd_card πŸ“–mathematicalβ€”index
Nat.card
β€”index_mul_card
index_dvd_of_le πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
indexβ€”dvd_of_mul_left_eq
relIndex_mul_index
index_dvd_two_iff πŸ“–mathematicalβ€”index
AddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”two_pos
Nat.instZeroLEOneClass
IsOrderedAddMonoid.toAddLeftMono
Nat.instIsOrderedAddMonoid
two_ne_zero
index_eq_one
mem_top
Zero.instNonempty
index_eq_two_iff
Xor'.or
add_neg_cancel_right
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
NegMemClass.neg_mem
AddSubgroupClass.toNegMemClass
eq_top_iff'
index_top
IsUnit.dvd
isUnit_iff_eq_one
Unique.instSubsingleton
dvd_of_eq
xor_iff_or_and_not_and
neg_add_cancel_left
index_dvd_two_iff' πŸ“–mathematicalβ€”index
AddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”index_dvd_two_iff
Equiv.exists_congr
Equiv.forall_congr
neg_add_rev
neg_mem_iff
AddSubgroupClass.toNegMemClass
instAddSubgroupClass
index_eq_card πŸ“–mathematicalβ€”index
Nat.card
HasQuotient.Quotient
AddSubgroup
QuotientAddGroup.instHasQuotientAddSubgroup
β€”β€”
index_eq_one πŸ“–mathematicalβ€”index
Top.top
AddSubgroup
instTop
β€”QuotientAddGroup.addSubgroup_eq_top_of_subsingleton
Nat.card_eq_one_iff_unique
index_top
index_eq_two_iff πŸ“–mathematicalβ€”index
Xor'
AddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”Nat.card_eq_two_iff'
QuotientAddGroup.eq
add_zero
neg_mem_iff
AddSubgroupClass.toNegMemClass
instAddSubgroupClass
xor_iff_iff_not
add_mem_cancel_left
neg_neg
neg_add_cancel
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
AddSubgroupClass.toAddSubmonoidClass
index_eq_two_iff' πŸ“–mathematicalβ€”index
Xor'
AddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”index_eq_two_iff
Equiv.exists_congr
Equiv.forall_congr
neg_add_rev
neg_mem_iff
AddSubgroupClass.toNegMemClass
instAddSubgroupClass
index_eq_two_iff_exists_notMem_and πŸ“–mathematicalβ€”index
AddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”index_eq_two_iff
xor_iff_or_and_not_and
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
neg_add_cancel_left
NegMemClass.neg_mem
AddSubgroupClass.toNegMemClass
index_eq_two_iff_exists_notMem_and' πŸ“–mathematicalβ€”index
AddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”index_eq_two_iff'
xor_iff_or_and_not_and
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
add_neg_cancel_right
NegMemClass.neg_mem
AddSubgroupClass.toNegMemClass
index_eq_zero_iff_infinite πŸ“–mathematicalβ€”index
Infinite
HasQuotient.Quotient
AddSubgroup
QuotientAddGroup.instHasQuotientAddSubgroup
β€”Nat.card_eq_zero
not_isEmpty_of_nonempty
index_eq_zero_of_relIndex_eq_zero πŸ“–mathematicalrelIndexindexβ€”relIndex_top_right
relIndex_eq_zero_of_le_right
le_top
index_iInf_le πŸ“–mathematicalβ€”index
iInf
AddSubgroup
instInfSet
Finset.prod
Nat.instCommMonoid
Finset.univ
β€”relIndex_top_right
Finset.prod_congr
relIndex_iInf_le
index_iInf_ne_zero πŸ“–β€”β€”β€”β€”relIndex_top_right
relIndex_iInf_ne_zero
index_inf_le πŸ“–mathematicalβ€”index
AddSubgroup
instMin
β€”relIndex_top_right
relIndex_inf_le
index_inf_ne_zero πŸ“–β€”β€”β€”β€”relIndex_top_right
relIndex_inf_ne_zero
index_ker πŸ“–mathematicalβ€”index
AddMonoidHom.ker
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Nat.card
AddSubgroup
SetLike.instMembership
instSetLike
AddMonoidHom.range
β€”AddMonoidHom.comap_bot
index_comap
relIndex_bot_left
index_map πŸ“–mathematicalβ€”index
map
AddSubgroup
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
AddMonoidHom.ker
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.range
β€”comap_map_eq
index_comap
relIndex_mul_index
map_le_range
index_map_dvd πŸ“–mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
index
map
β€”index_map
AddMonoidHom.range_eq_top_of_surjective
index_top
mul_one
index_dvd_of_le
le_sup_left
index_map_eq πŸ“–mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddMonoidHom.ker
index
map
β€”index_map_dvd
dvd_index_map
index_map_equiv πŸ“–mathematicalβ€”index
map
AddMonoidHomClass.toAddMonoidHom
AddEquiv
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
EquivLike.toFunLike
AddEquiv.instEquivLike
AddEquivClass.instAddMonoidHomClass
AddEquiv.instAddEquivClass
β€”index_map_of_bijective
AddEquivClass.instAddMonoidHomClass
AddEquiv.instAddEquivClass
AddEquiv.bijective
index_map_of_bijective πŸ“–mathematicalFunction.Bijective
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
index
map
β€”index_map_eq
AddMonoidHom.ker_eq_bot_iff
bot_le
index_map_of_injective πŸ“–mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
index
map
AddMonoidHom.range
β€”index_map
AddMonoidHom.ker_eq_bot_iff
sup_bot_eq
index_map_subtype πŸ“–mathematicalβ€”index
map
AddSubgroup
SetLike.instMembership
instSetLike
toAddGroup
subtype
β€”index_map_of_injective
subtype_injective
range_subtype
index_mul_card πŸ“–mathematicalβ€”index
Nat.card
AddSubgroup
SetLike.instMembership
instSetLike
β€”mul_comm
card_mul_index
index_ne_zero_iff_finite πŸ“–mathematicalβ€”Finite
HasQuotient.Quotient
AddSubgroup
QuotientAddGroup.instHasQuotientAddSubgroup
β€”index_eq_zero_iff_infinite
not_infinite_iff_finite
index_ne_zero_of_finite πŸ“–β€”β€”β€”β€”nonempty_fintype
index_eq_card
LT.lt.ne'
Nat.card_pos
index_pi πŸ“–mathematicalβ€”index
Pi.addGroup
pi
Set.univ
Finset.prod
Nat.instCommMonoid
Finset.univ
β€”Finset.prod_congr
Nat.card_pi
Nat.card_congr
QuotientAddGroup.leftRel_pi
index_prod πŸ“–mathematicalβ€”index
Prod.instAddGroup
prod
β€”Nat.card_prod
Nat.card_congr
QuotientAddGroup.leftRel_prod
index_range πŸ“–mathematicalβ€”index
AddMonoidHom.range
Nat.card
AddSubgroup
SetLike.instMembership
instSetLike
AddMonoidHom.ker
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”mul_left_inj'
IsCancelMulZero.toIsRightCancelMulZero
Nat.instIsCancelMulZero
FiniteIndex.index_ne_zero
card_mul_index
index_ker
index_mul_card
index_smul πŸ“–mathematicalβ€”index
AddSubgroup
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
pointwiseMulAction
β€”index_map_of_bijective
MulAction.bijective
index_strictAnti πŸ“–mathematicalAddSubgroup
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
indexβ€”FiniteIndex.index_ne_zero
finiteIndex_of_le
LT.lt.le
lt_of_le_of_ne
index_antitone
relIndex_mul_index
mul_eq_rightβ‚€
IsCancelMulZero.toIsRightCancelMulZero
Nat.instIsCancelMulZero
relIndex_eq_one
LT.lt.not_ge
index_toSubgroup πŸ“–mathematicalβ€”Subgroup.index
Multiplicative
Multiplicative.group
DFunLike.coe
OrderIso
AddSubgroup
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Subgroup.instPartialOrder
instFunLikeOrderIso
toSubgroup
index
β€”β€”
index_top πŸ“–mathematicalβ€”index
Top.top
AddSubgroup
instTop
β€”Nat.card_eq_one_iff_unique
QuotientAddGroup.subsingleton_quotient_top
normal_top
inf_eq_bot_of_coprime πŸ“–mathematicalNat.card
AddSubgroup
SetLike.instMembership
instSetLike
instMin
Bot.bot
instBot
β€”card_eq_one
Nat.eq_one_of_dvd_coprimes
card_dvd_of_le
inf_le_left
inf_le_right
inf_relIndex_left πŸ“–mathematicalβ€”relIndex
AddSubgroup
instMin
β€”inf_comm
inf_relIndex_right
inf_relIndex_right πŸ“–mathematicalβ€”relIndex
AddSubgroup
instMin
β€”relIndex.eq_1
inf_addSubgroupOf_right
instFiniteIndexMin πŸ“–mathematicalβ€”FiniteIndex
AddSubgroup
instMin
β€”index_inf_ne_zero
FiniteIndex.index_ne_zero
instFiniteIndexTop πŸ“–mathematicalβ€”FiniteIndex
Top.top
AddSubgroup
instTop
β€”ne_of_eq_of_ne
index_top
one_ne_zero
instFiniteIndex_addSubgroupOf πŸ“–mathematicalβ€”FiniteIndex
AddSubgroup
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
β€”index_ne_zero_of_finite
finite_quotient_of_finiteIndex
index_eq_zero_of_relIndex_eq_zero
not_finiteIndex_iff πŸ“–mathematicalβ€”FiniteIndex
index
β€”finiteIndex_iff
nsmul_mem_of_index_ne_zero_of_dvd πŸ“–mathematicalβ€”AddSubgroup
SetLike.instMembership
instSetLike
AddMonoid.toNatSMul
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”exists_nsmul_mem_of_index_ne_zero
mul_nsmul
nsmul_mem
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
nsmul_mem_of_relIndex_ne_zero_of_dvd πŸ“–mathematicalAddSubgroup
SetLike.instMembership
instSetLike
instMin
AddMonoid.toNatSMul
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”mem_inf
nsmul_mem
AddSubgroupClass.toAddSubmonoidClass
instAddSubgroupClass
mem_addSubgroupOf
nsmul_mem_of_index_ne_zero_of_dvd
one_lt_index_of_ne_top πŸ“–mathematicalβ€”indexβ€”index_ne_zero_of_finite
index_eq_one
relIindex_dvd_two_iff' πŸ“–mathematicalβ€”relIndex
AddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”index_dvd_two_iff'
mem_addSubgroupOf
relIindex_eq_two_iff' πŸ“–mathematicalβ€”relIndex
AddSubgroup
SetLike.instMembership
instSetLike
Xor'
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”index_eq_two_iff'
mem_addSubgroupOf
relIndex_addSubgroupOf πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndex
SetLike.instMembership
instSetLike
toAddGroup
addSubgroupOf
β€”index_comap
inclusion_range
relIndex_bot_left πŸ“–mathematicalβ€”relIndex
Bot.bot
AddSubgroup
instBot
Nat.card
SetLike.instMembership
instSetLike
β€”relIndex.eq_1
bot_addSubgroupOf
index_bot
relIndex_bot_right πŸ“–mathematicalβ€”relIndex
Bot.bot
AddSubgroup
instBot
β€”relIndex.eq_1
addSubgroupOf_bot_eq_top
index_top
relIndex_comap πŸ“–mathematicalβ€”relIndex
comap
map
β€”relIndex.eq_1
addSubgroupOf.eq_1
comap_comap
index_comap
AddMonoidHom.map_range
range_subtype
relIndex_comap_ne_zero πŸ“–β€”β€”β€”β€”relIndex_comap
relIndex_eq_zero_of_le_right
map_comap_le
relIndex_dvd_card πŸ“–mathematicalβ€”relIndex
Nat.card
AddSubgroup
SetLike.instMembership
instSetLike
β€”index_dvd_card
relIndex_dvd_index_of_le πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndex
index
β€”dvd_of_mul_right_eq
relIndex_mul_index
relIndex_dvd_index_of_normal πŸ“–mathematicalβ€”relIndex
index
β€”relIndex_dvd_index_of_le
le_sup_right
relIndex_sup_right
relIndex_dvd_of_le_left πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndexβ€”dvd_of_mul_left_eq
relIndex_inf_mul_relIndex
inf_of_le_left
relIndex_dvd_two_iff πŸ“–mathematicalβ€”relIndex
AddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”index_dvd_two_iff
mem_addSubgroupOf
relIndex_eq_one πŸ“–mathematicalβ€”relIndex
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
β€”index_eq_one
addSubgroupOf_eq_top
relIndex_eq_two_iff πŸ“–mathematicalβ€”relIndex
AddSubgroup
SetLike.instMembership
instSetLike
Xor'
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”index_eq_two_iff
mem_addSubgroupOf
relIndex_eq_two_iff_exists_notMem_and πŸ“–mathematicalβ€”relIndex
AddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”relIndex.eq_1
index_eq_two_iff_exists_notMem_and
mem_addSubgroupOf
relIndex_eq_two_iff_exists_notMem_and' πŸ“–mathematicalβ€”relIndex
AddSubgroup
SetLike.instMembership
instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”relIndex.eq_1
index_eq_two_iff_exists_notMem_and'
mem_addSubgroupOf
relIndex_eq_zero_of_le_left πŸ“–β€”AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndex
β€”β€”eq_zero_of_zero_dvd
relIndex_dvd_of_le_left
relIndex_eq_zero_of_le_right πŸ“–β€”AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndex
β€”β€”Finite.card_eq_zero_of_embedding
relIndex_iInf_le πŸ“–mathematicalβ€”relIndex
iInf
AddSubgroup
instInfSet
Finset.prod
Nat.instCommMonoid
Finset.univ
β€”Finite.card_le_of_embedding'
Finset.prod_eq_zero_iff
Nat.instNontrivial
IsDomain.to_noZeroDivisors
Nat.instIsDomain
Nat.card_pi
relIndex_eq_zero_of_le_left
iInf_le
relIndex_iInf_ne_zero πŸ“–β€”β€”β€”β€”Finset.prod_ne_zero_iff
Nat.instNontrivial
IsDomain.to_noZeroDivisors
Nat.instIsDomain
Nat.card_pi
Finite.card_eq_zero_of_embedding
relIndex_inf_le πŸ“–mathematicalβ€”relIndex
AddSubgroup
instMin
β€”LE.le.trans
le_of_eq
relIndex_eq_zero_of_le_left
inf_le_left
zero_le
Nat.instCanonicallyOrderedAdd
inf_relIndex_right
inf_assoc
relIndex_mul_relIndex
inf_le_right
le_imp_le_of_le_of_le
mul_le_mul_left
covariant_swap_mul_of_covariant_mul
Nat.instMulLeftMono
relIndex_le_of_le_right
le_refl
relIndex_inf_mul_relIndex πŸ“–mathematicalβ€”relIndex
AddSubgroup
instMin
β€”inf_relIndex_right
inf_assoc
relIndex_mul_relIndex
inf_le_right
relIndex_inf_ne_zero πŸ“–β€”β€”β€”β€”relIndex_eq_zero_of_le_right
inf_le_right
inf_relIndex_right
inf_assoc
relIndex_ne_zero_trans
relIndex_inter_ne_zero πŸ“–β€”β€”β€”β€”range_subtype
inf_comm
map_comap_eq
relIndex_comap
comap_map_eq_self_of_injective
subtype_injective
relIndex_comap_ne_zero
relIndex_ker πŸ“–mathematicalβ€”relIndex
AddMonoidHom.ker
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Nat.card
AddSubgroup
SetLike.instMembership
instSetLike
map
β€”AddMonoidHom.comap_bot
relIndex_comap
relIndex_bot_left
relIndex_le_of_le_left πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndexβ€”relIndex_dvd_of_le_left
relIndex_le_of_le_right πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndexβ€”Finite.card_le_of_embedding'
relIndex_map_map πŸ“–mathematicalβ€”relIndex
map
AddSubgroup
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
AddMonoidHom.ker
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”comap_map_eq
relIndex_comap
GaloisConnection.l_u_l_eq_l
gc_map_comap
relIndex_map_map_of_injective πŸ“–mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoidHom.instFunLike
relIndex
map
β€”relIndex_comap
comap_map_eq_self_of_injective
relIndex_mul_index πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndex
index
β€”mul_comm
Nat.card_prod
Nat.card_congr
relIndex_mul_relIndex πŸ“–mathematicalAddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndexβ€”relIndex_addSubgroupOf
relIndex_mul_index
relIndex_ne_zero πŸ“–β€”β€”β€”β€”IsFiniteRelIndex.relIndex_ne_zero
relIndex_ne_zero_trans πŸ“–β€”β€”β€”β€”mul_ne_zero
IsDomain.to_noZeroDivisors
Nat.instIsDomain
relIndex_eq_zero_of_le_right
inf_le_left
relIndex_inf_mul_relIndex
relIndex_eq_zero_of_le_left
relIndex_pointwise_smul πŸ“–mathematicalβ€”relIndex
AddSubgroup
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
pointwiseMulAction
β€”pointwise_smul_def
relIndex_comap
comap_map_eq_self_of_injective
relIndex_self πŸ“–mathematicalβ€”relIndexβ€”relIndex.eq_1
addSubgroupOf_self
index_top
relIndex_sup_left πŸ“–mathematicalβ€”relIndex
AddSubgroup
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
β€”sup_comm
relIndex_sup_right
relIndex_sup_right πŸ“–mathematicalβ€”relIndex
AddSubgroup
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
β€”Nat.card_congr
normal_addSubgroupOf
relIndex_toSubgroup πŸ“–mathematicalβ€”Subgroup.relIndex
Multiplicative
Multiplicative.group
DFunLike.coe
OrderIso
AddSubgroup
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Subgroup.instPartialOrder
instFunLikeOrderIso
toSubgroup
relIndex
β€”β€”
relIndex_top_left πŸ“–mathematicalβ€”relIndex
Top.top
AddSubgroup
instTop
β€”index_top
relIndex_top_right πŸ“–mathematicalβ€”relIndex
Top.top
AddSubgroup
instTop
index
β€”relIndex_mul_index
le_top
index_top
mul_one
two_smul_mem_of_index_two πŸ“–mathematicalindexAddSubgroup
SetLike.instMembership
instSetLike
AddMonoid.toNatSMul
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”add_self_mem_of_index_two
two_nsmul

AddSubgroup.FiniteIndex

Theorems

NameKindAssumesProvesValidatesDepends On
index_ne_zero πŸ“–β€”β€”β€”β€”β€”

AddSubgroup.IsFiniteRelIndex

Theorems

NameKindAssumesProvesValidatesDepends On
relIndex_ne_zero πŸ“–β€”β€”β€”β€”β€”
to_finiteIndex_addSubgroupOf πŸ“–mathematicalβ€”AddSubgroup.FiniteIndex
AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
AddSubgroup.toAddGroup
AddSubgroup.addSubgroupOf
β€”AddSubgroup.relIndex_ne_zero

MonoidHom

Theorems

NameKindAssumesProvesValidatesDepends On
card_fiber_eq_of_mem_range πŸ“–mathematicalSet
Set.instMembership
Set.range
DFunLike.coe
Finset.card
Finset.filter
Finset.univ
β€”mul_left_surjective
Finset.map_univ_equiv
Finset.filter_map
Finset.card_map
map_mul
MonoidHomClass.toMulHomClass
coe_toHomUnits
map_inv
instMonoidHomClass
Units.mul_inv_eq_iff_eq_mul
finite_iff_finite_ker_range πŸ“–mathematicalβ€”Finite
Subgroup
SetLike.instMembership
Subgroup.instSetLike
ker
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
range
β€”Subgroup.finite_iff_finite_and_finiteIndex
Equiv.finite_iff
normal_ker
Subgroup.finiteIndex_iff_finite_quotient
surjective_of_card_ker_le_div πŸ“–mathematicalNat.card
Subgroup
SetLike.instMembership
Subgroup.instSetLike
ker
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
instFunLike
β€”range_eq_top
SetLike.ext'
Set.eq_of_subset_of_ncard_le
Set.subset_univ
Subgroup.coe_top
Set.ncard_univ
Nat.card_coe_set_eq
SetLike.coe_sort_coe
Nat.card_congr
normal_ker
Subgroup.card_mul_index
Nat.card_pos
One.instNonempty
Subtype.finite
Set.toFinite

MulAction

Theorems

NameKindAssumesProvesValidatesDepends On
index_stabilizer πŸ“–mathematicalβ€”Subgroup.index
stabilizer
Set.ncard
orbit
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
toSemigroupAction
β€”Nat.card_congr
Nat.card_coe_set_eq
index_stabilizer_of_transitive πŸ“–mathematicalβ€”Subgroup.index
stabilizer
Nat.card
β€”index_stabilizer
orbit_eq_univ
Set.ncard_univ

Subgroup

Definitions

NameCategoryTheorems
IsFiniteRelIndex πŸ“–CompData
2 mathmath: IsArithmetic.isFiniteRelIndexSL, instIsFiniteRelIndexGeneralLinearGroupAdjoinNegOne
fintypeOfIndexNeZero πŸ“–CompOpβ€”
fintypeQuotientOfFiniteIndex πŸ“–CompOp
1 mathmath: Rep.coindToInd_apply
index πŸ“–CompOp
90 mathmath: relIndex_dvd_index_of_normal, Sylow.not_dvd_index', index_center_le_pow, MonoidHom.transferSylow_eq_pow, index_dvd_two_iff', index_map_of_bijective, not_finiteIndex_iff, relIndex_mul_index, IsZGroup.coprime_commutator_index, Sylow.card_dvd_index, IsComplement.encard_left, relIndex_dvd_index_of_le, IsComplement'.index_eq_card, Units.index_posSubgroup, MonoidHom.transfer_eq_pow_aux, MulAction.index_stabilizer_of_transitive, MulAction.index_stabilizer, Sylow.not_dvd_index, MonoidHom.transferSylow_restrict_eq_pow, IsComplement.encard_right, index_ker, leftCoset_cover_filter_FiniteIndex_aux, rank_le_index_mul_rank, index_mul_card, Sylow.not_dvd_card_commutator_or_not_dvd_index_commutator, NumberField.IsCMField.indexRealUnits_mul_eq, index_inf_le, index_eq_zero_iff_infinite, index_iInf_le, IntermediateField.finrank_eq_fixingSubgroup_index, index_map_of_injective, index_le_of_leftCoset_cover_const, pow_index_mem, index_eq_two_iff, NumberField.Units.regOfFamily_div_regulator, index_dvd_two_iff, exists_index_le_card_of_leftCoset_cover, card_commutator_dvd_index_center_pow, IsComplement.ncard_right, MonoidHom.transfer_center_eq_pow, index_bot, alternatingGroup.index_eq_one, index_pi, exists_finset_card_le_mul, IsComplement.card_right, IsCyclic.index_powMonoidHom_ker, dvd_index_map, index_toAddSubgroup, one_le_sum_inv_index_of_leftCoset_cover, MonoidHom.transferCenterPow_apply, index_map_equiv, MulAction.IsMultiplyPretransitive.index_of_fixingSubgroup_eq, IntermediateField.map_fixingSubgroup_index, IsComplement.card_left, IsCyclic.index_powMonoidHom_range, index_eq_two_iff_exists_notMem_and', index_comap, IsPGroup.index, index_eq_two_iff_exists_notMem_and, index_eq_card, index_eq_two_iff', index_prod, index_eq_one, commProb_subgroup_le, index_top, index_map_eq, index_eq_zero_of_relIndex_eq_zero, index_dvd_of_le, index_map_dvd, index_strictAnti, index_comap_of_surjective, exists_pow_mem_of_index_ne_zero, IsComplement.ncard_left, AddSubgroup.index_toSubgroup, index_map, card_mul_index, MonoidHom.transfer_eq_pow, index_range, index_map_subtype, alternatingGroup.index_eq_two, index_antitone, relIndex_top_right, smul_diff', Sylow.card_eq_index_normalizer, NumberField.IsCMField.index_unitsMulComplexConjInv_range_dvd, MeasureTheory.Subgroup.index_mul_measure, Sylow.card_coprime_index, MulAction.IsMultiplyPretransitive.index_of_fixingSubgroup_mul, one_lt_index_of_ne_top, index_dvd_card
relIndex πŸ“–CompOp
40 mathmath: relIndex_le_of_le_left, relIndex_dvd_index_of_normal, inf_relIndex_left, relIndex_eq_two_iff_exists_notMem_and', relIndex_mul_index, relIndex_dvd_index_of_le, relIndex_self, exists_pow_mem_of_relIndex_ne_zero, relIndex_subgroupOf, relIndex_map_map, relIndex_dvd_of_le_left, relIndex_eq_two_iff_exists_notMem_and, relIndex_eq_two_iff, relIndex_mul_relIndex, relIndex_le_of_le_right, relIndex_sup_right, relIndex_bot_left, relIindex_dvd_two_iff', inf_relIndex_right, relIndex_inf_mul_relIndex, NumberField.Units.regOfFamily_div_regOfFamily, relIndex_iInf_le, relIindex_eq_two_iff', MulAction.IsBlock.ncard_block_eq_relIndex, relindex_adjoinNegOne_eq_two, relIndex_inf_le, relIndex_sup_left, relIndex_eq_one, index_comap, relIndex_map_map_of_injective, relIndex_dvd_two_iff, relIndex_comap, relIndex_pointwise_smul, relIndex_ker, relIndex_bot_right, relIndex_dvd_card, relIndex_top_right, relIndex_toAddSubgroup, AddSubgroup.relIndex_toSubgroup, relIndex_top_left

Theorems

NameKindAssumesProvesValidatesDepends On
card_dvd_of_surjective πŸ“–mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
Nat.cardβ€”Nat.card_congr
MonoidHom.normal_ker
Dvd.intro_left
card_mul_index
card_eq_one πŸ“–mathematicalβ€”Nat.card
Subgroup
SetLike.instMembership
instSetLike
Bot.bot
instBot
β€”relIndex_eq_one
le_bot_iff
relIndex_bot_left
card_map_dvd πŸ“–mathematicalβ€”Nat.card
Subgroup
SetLike.instMembership
instSetLike
map
β€”card_dvd_of_surjective
MonoidHom.subgroupMap_surjective
card_mul_index πŸ“–mathematicalβ€”Nat.card
Subgroup
SetLike.instMembership
instSetLike
index
β€”relIndex_bot_left
index_bot
relIndex_mul_index
bot_le
card_range_dvd πŸ“–mathematicalβ€”Nat.card
Subgroup
SetLike.instMembership
instSetLike
MonoidHom.range
β€”card_dvd_of_surjective
MonoidHom.rangeRestrict_surjective
dvd_index_map πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
MonoidHom.ker
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
index
map
β€”index_map
sup_of_le_left
dvd_mul_right
exists_pow_mem_of_index_ne_zero πŸ“–mathematicalβ€”index
Subgroup
SetLike.instMembership
instSetLike
Monoid.toNatPow
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Fintype.finite
Nat.card_le_card_of_injective
Nat.card_eq_fintype_card
Finset.mem_Icc
Fintype.card_ofFinset
Nat.card_Icc
tsub_zero
Nat.instOrderedSub
IsOrderedAddMonoid.toAddLeftMono
Nat.instIsOrderedAddMonoid
IsLeftCancelAdd.addLeftReflectLE_of_addLeftReflectLT
AddLeftCancelSemigroup.toIsLeftCancelAdd
contravariant_lt_of_covariant_le
Nat.instCanonicallyOrderedAdd
Ne.lt_or_gt
zpow_natCast
add_comm
zpow_add
zpow_neg
QuotientGroup.eq
exists_pow_mem_of_relIndex_ne_zero πŸ“–mathematicalSubgroup
SetLike.instMembership
instSetLike
relIndex
instMin
Monoid.toNatPow
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”exists_pow_mem_of_index_ne_zero
pow_mem
SubgroupClass.toSubmonoidClass
instSubgroupClass
finiteIndex_iInf πŸ“–mathematicalFiniteIndexiInf
Subgroup
instInfSet
β€”index_iInf_ne_zero
FiniteIndex.index_ne_zero
finiteIndex_iInf' πŸ“–mathematicalFiniteIndexiInf
Subgroup
instInfSet
Finset
Finset.instMembership
β€”iInf_subtype'
finiteIndex_iInf
Finite.of_fintype
finiteIndex_iff πŸ“–mathematicalβ€”FiniteIndexβ€”FiniteIndex.index_ne_zero
finiteIndex_iff_finite_quotient πŸ“–mathematicalβ€”FiniteIndex
Finite
HasQuotient.Quotient
Subgroup
QuotientGroup.instHasQuotientSubgroup
β€”finite_quotient_of_finiteIndex
finiteIndex_of_finite_quotient
finiteIndex_ker πŸ“–mathematicalβ€”FiniteIndex
MonoidHom.ker
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”finiteIndex_of_finite_quotient
Finite.of_equiv
MonoidHom.normal_ker
finiteIndex_normalCore πŸ“–mathematicalβ€”FiniteIndex
normalCore
β€”MulAction.left_quotientAction
normalCore_eq_ker
finiteIndex_ker
Subtype.finite
Equiv.finite_left
finite_quotient_of_finiteIndex
finiteIndex_of_finite πŸ“–mathematicalβ€”FiniteIndexβ€”finiteIndex_of_finite_quotient
QuotientGroup.finite
finiteIndex_of_finite_quotient πŸ“–mathematicalβ€”FiniteIndexβ€”index_ne_zero_of_finite
finiteIndex_of_le πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
FiniteIndexβ€”ne_zero_of_dvd_ne_zero
FiniteIndex.index_ne_zero
index_dvd_of_le
finite_iff_finite_and_finiteIndex πŸ“–mathematicalβ€”Finite
Subgroup
SetLike.instMembership
instSetLike
FiniteIndex
β€”Subtype.finite
finiteIndex_of_finite
Nat.finite_of_card_ne_zero
mul_ne_zero
IsDomain.to_noZeroDivisors
Nat.instIsDomain
LT.lt.ne'
Nat.card_pos
One.instNonempty
FiniteIndex.index_ne_zero
card_mul_index
finite_quotient_of_finiteIndex πŸ“–mathematicalβ€”Finite
HasQuotient.Quotient
Subgroup
QuotientGroup.instHasQuotientSubgroup
β€”Fintype.finite
finite_quotient_of_finite_quotient_of_index_ne_zero πŸ“–mathematicalβ€”Finite
MulAction.orbitRel.Quotient
Subgroup
SetLike.instMembership
instSetLike
toGroup
instMulAction
β€”MulAction.finite_quotient_of_finite_quotient_of_finite_quotient
Finite.of_fintype
finite_quotient_of_pretransitive_of_index_ne_zero πŸ“–mathematicalβ€”Finite
MulAction.orbitRel.Quotient
Subgroup
SetLike.instMembership
instSetLike
toGroup
instMulAction
β€”MulAction.pretransitive_iff_subsingleton_quotient
finite_quotient_of_finite_quotient_of_index_ne_zero
Finite.of_subsingleton
index_antitone πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
indexβ€”FiniteIndex.index_ne_zero
index_dvd_of_le
index_bot πŸ“–mathematicalβ€”index
Bot.bot
Subgroup
instBot
Nat.card
β€”Nat.card_congr
normal_bot
index_comap πŸ“–mathematicalβ€”index
comap
relIndex
MonoidHom.range
β€”index_comap_of_surjective
MonoidHom.rangeRestrict_surjective
index_comap_of_surjective πŸ“–mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
index
comap
β€”MonoidHom.map_mul
MonoidHom.map_inv
Nat.card_congr
Quotient.ind'
Quotient.map'_mk''
index_dvd_card πŸ“–mathematicalβ€”index
Nat.card
β€”index_mul_card
index_dvd_of_le πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
indexβ€”dvd_of_mul_left_eq
relIndex_mul_index
index_dvd_two_iff πŸ“–mathematicalβ€”index
Subgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”two_pos
Nat.instZeroLEOneClass
IsOrderedAddMonoid.toAddLeftMono
Nat.instIsOrderedAddMonoid
two_ne_zero
index_eq_one
One.instNonempty
index_eq_two_iff
Xor'.or
mul_inv_cancel_right
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
SubgroupClass.toSubmonoidClass
instSubgroupClass
InvMemClass.inv_mem
SubgroupClass.toInvMemClass
eq_top_iff'
index_top
Unique.instSubsingleton
dvd_of_eq
xor_iff_or_and_not_and
inv_mul_cancel_left
index_dvd_two_iff' πŸ“–mathematicalβ€”index
Subgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”index_dvd_two_iff
Equiv.exists_congr
Equiv.forall_congr
SubgroupClass.toInvMemClass
instSubgroupClass
index_eq_card πŸ“–mathematicalβ€”index
Nat.card
HasQuotient.Quotient
Subgroup
QuotientGroup.instHasQuotientSubgroup
β€”β€”
index_eq_one πŸ“–mathematicalβ€”index
Top.top
Subgroup
instTop
β€”QuotientGroup.subgroup_eq_top_of_subsingleton
Nat.card_eq_one_iff_unique
index_top
index_eq_two_iff πŸ“–mathematicalβ€”index
Xor'
Subgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Nat.card_eq_two_iff'
mul_one
SubgroupClass.toInvMemClass
instSubgroupClass
mul_mem_cancel_left
inv_mem_iff
inv_inv
inv_mul_cancel
OneMemClass.one_mem
SubmonoidClass.toOneMemClass
SubgroupClass.toSubmonoidClass
index_eq_two_iff' πŸ“–mathematicalβ€”index
Xor'
Subgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”index_eq_two_iff
Equiv.exists_congr
Equiv.forall_congr
SubgroupClass.toInvMemClass
instSubgroupClass
index_eq_two_iff_exists_notMem_and πŸ“–mathematicalβ€”index
Subgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
SubgroupClass.toSubmonoidClass
instSubgroupClass
inv_mul_cancel_left
InvMemClass.inv_mem
SubgroupClass.toInvMemClass
index_eq_two_iff_exists_notMem_and' πŸ“–mathematicalβ€”index
Subgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
SubgroupClass.toSubmonoidClass
instSubgroupClass
mul_inv_cancel_right
InvMemClass.inv_mem
SubgroupClass.toInvMemClass
index_eq_zero_iff_infinite πŸ“–mathematicalβ€”index
Infinite
HasQuotient.Quotient
Subgroup
QuotientGroup.instHasQuotientSubgroup
β€”β€”
index_eq_zero_of_relIndex_eq_zero πŸ“–mathematicalrelIndexindexβ€”relIndex_top_right
relIndex_eq_zero_of_le_right
le_top
index_iInf_le πŸ“–mathematicalβ€”index
iInf
Subgroup
instInfSet
Finset.prod
Nat.instCommMonoid
Finset.univ
β€”Finset.prod_congr
index_iInf_ne_zero πŸ“–β€”β€”β€”β€”relIndex_iInf_ne_zero
index_inf_le πŸ“–mathematicalβ€”index
Subgroup
instMin
β€”β€”
index_inf_ne_zero πŸ“–β€”β€”β€”β€”relIndex_top_right
relIndex_inf_ne_zero
index_ker πŸ“–mathematicalβ€”index
MonoidHom.ker
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Nat.card
Subgroup
SetLike.instMembership
instSetLike
MonoidHom.range
β€”MonoidHom.comap_bot
index_comap
relIndex_bot_left
index_map πŸ“–mathematicalβ€”index
map
Subgroup
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
MonoidHom.ker
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.range
β€”comap_map_eq
index_comap
relIndex_mul_index
map_le_range
index_map_dvd πŸ“–mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
index
map
β€”index_map
MonoidHom.range_eq_top_of_surjective
index_top
mul_one
index_dvd_of_le
le_sup_left
index_map_eq πŸ“–mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
MonoidHom.ker
index
map
β€”index_map_dvd
dvd_index_map
index_map_equiv πŸ“–mathematicalβ€”index
map
MonoidHomClass.toMonoidHom
MulEquiv
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
EquivLike.toFunLike
MulEquiv.instEquivLike
MulEquivClass.instMonoidHomClass
MulEquiv.instMulEquivClass
β€”index_map_of_bijective
MulEquivClass.instMonoidHomClass
MulEquiv.instMulEquivClass
MulEquiv.bijective
index_map_of_bijective πŸ“–mathematicalFunction.Bijective
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
index
map
β€”index_map_eq
MonoidHom.ker_eq_bot_iff
bot_le
index_map_of_injective πŸ“–mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
index
map
MonoidHom.range
β€”index_map
MonoidHom.ker_eq_bot_iff
sup_bot_eq
index_map_subtype πŸ“–mathematicalβ€”index
map
Subgroup
SetLike.instMembership
instSetLike
toGroup
subtype
β€”index_map_of_injective
subtype_injective
range_subtype
index_mul_card πŸ“–mathematicalβ€”index
Nat.card
Subgroup
SetLike.instMembership
instSetLike
β€”mul_comm
card_mul_index
index_ne_zero_iff_finite πŸ“–mathematicalβ€”Finite
HasQuotient.Quotient
Subgroup
QuotientGroup.instHasQuotientSubgroup
β€”β€”
index_ne_zero_of_finite πŸ“–β€”β€”β€”β€”nonempty_fintype
index_eq_card
LT.lt.ne'
Nat.card_pos
index_pi πŸ“–mathematicalβ€”index
Pi.group
pi
Set.univ
Finset.prod
Nat.instCommMonoid
Finset.univ
β€”Finset.prod_congr
Nat.card_congr
QuotientGroup.leftRel_pi
index_prod πŸ“–mathematicalβ€”index
Prod.instGroup
prod
β€”Nat.card_congr
QuotientGroup.leftRel_prod
index_range πŸ“–mathematicalβ€”index
MonoidHom.range
Nat.card
Subgroup
SetLike.instMembership
instSetLike
MonoidHom.ker
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”mul_left_inj'
IsCancelMulZero.toIsRightCancelMulZero
Nat.instIsCancelMulZero
FiniteIndex.index_ne_zero
card_mul_index
index_ker
index_mul_card
index_strictAnti πŸ“–mathematicalSubgroup
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
indexβ€”FiniteIndex.index_ne_zero
finiteIndex_of_le
LT.lt.le
lt_of_le_of_ne
index_antitone
relIndex_mul_index
mul_eq_rightβ‚€
IsCancelMulZero.toIsRightCancelMulZero
Nat.instIsCancelMulZero
relIndex_eq_one
LT.lt.not_ge
index_toAddSubgroup πŸ“–mathematicalβ€”AddSubgroup.index
Additive
Additive.addGroup
DFunLike.coe
OrderIso
Subgroup
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddSubgroup.instPartialOrder
instFunLikeOrderIso
toAddSubgroup
index
β€”β€”
index_top πŸ“–mathematicalβ€”index
Top.top
Subgroup
instTop
β€”Nat.card_eq_one_iff_unique
QuotientGroup.subsingleton_quotient_top
normal_top
inf_eq_bot_of_coprime πŸ“–mathematicalNat.card
Subgroup
SetLike.instMembership
instSetLike
instMin
Bot.bot
instBot
β€”card_eq_one
Nat.eq_one_of_dvd_coprimes
card_dvd_of_le
inf_le_left
inf_le_right
inf_relIndex_left πŸ“–mathematicalβ€”relIndex
Subgroup
instMin
β€”inf_comm
inf_relIndex_right
inf_relIndex_right πŸ“–mathematicalβ€”relIndex
Subgroup
instMin
β€”relIndex.eq_1
inf_subgroupOf_right
instFiniteIndexMin πŸ“–mathematicalβ€”FiniteIndex
Subgroup
instMin
β€”index_inf_ne_zero
FiniteIndex.index_ne_zero
instFiniteIndexTop πŸ“–mathematicalβ€”FiniteIndex
Top.top
Subgroup
instTop
β€”ne_of_eq_of_ne
index_top
one_ne_zero
instFiniteIndex_subgroupOf πŸ“–mathematicalβ€”FiniteIndex
Subgroup
SetLike.instMembership
instSetLike
toGroup
subgroupOf
β€”index_ne_zero_of_finite
finite_quotient_of_finiteIndex
index_eq_zero_of_relIndex_eq_zero
mul_mem_iff_of_index_two πŸ“–mathematicalindexSubgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”mul_mem_cancel_left
instSubgroupClass
mul_mem_cancel_right
index_eq_two_iff
Xor'.or
mul_assoc
mul_self_mem_of_index_two πŸ“–mathematicalindexSubgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”mul_mem_iff_of_index_two
not_finiteIndex_iff πŸ“–mathematicalβ€”FiniteIndex
index
β€”β€”
one_lt_index_of_ne_top πŸ“–mathematicalβ€”indexβ€”index_ne_zero_of_finite
index_eq_one
pow_mem_of_index_ne_zero_of_dvd πŸ“–mathematicalβ€”Subgroup
SetLike.instMembership
instSetLike
Monoid.toNatPow
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”exists_pow_mem_of_index_ne_zero
pow_mul
pow_mem
SubgroupClass.toSubmonoidClass
instSubgroupClass
pow_mem_of_relIndex_ne_zero_of_dvd πŸ“–mathematicalSubgroup
SetLike.instMembership
instSetLike
instMin
Monoid.toNatPow
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”pow_mem
SubgroupClass.toSubmonoidClass
instSubgroupClass
pow_mem_of_index_ne_zero_of_dvd
relIindex_dvd_two_iff' πŸ“–mathematicalβ€”relIndex
Subgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”β€”
relIindex_eq_two_iff' πŸ“–mathematicalβ€”relIndex
Subgroup
SetLike.instMembership
instSetLike
Xor'
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”β€”
relIndex_bot_left πŸ“–mathematicalβ€”relIndex
Bot.bot
Subgroup
instBot
Nat.card
SetLike.instMembership
instSetLike
β€”relIndex.eq_1
bot_subgroupOf
index_bot
relIndex_bot_right πŸ“–mathematicalβ€”relIndex
Bot.bot
Subgroup
instBot
β€”relIndex.eq_1
subgroupOf_bot_eq_top
index_top
relIndex_comap πŸ“–mathematicalβ€”relIndex
comap
map
β€”relIndex.eq_1
subgroupOf.eq_1
comap_comap
index_comap
MonoidHom.map_range
range_subtype
relIndex_comap_ne_zero πŸ“–β€”β€”β€”β€”relIndex_comap
relIndex_eq_zero_of_le_right
map_comap_le
relIndex_dvd_card πŸ“–mathematicalβ€”relIndex
Nat.card
Subgroup
SetLike.instMembership
instSetLike
β€”index_dvd_card
relIndex_dvd_index_of_le πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndex
index
β€”dvd_of_mul_right_eq
relIndex_mul_index
relIndex_dvd_index_of_normal πŸ“–mathematicalβ€”relIndex
index
β€”relIndex_dvd_index_of_le
le_sup_right
relIndex_sup_right
relIndex_dvd_of_le_left πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndexβ€”dvd_of_mul_left_eq
relIndex_inf_mul_relIndex
inf_of_le_left
relIndex_dvd_two_iff πŸ“–mathematicalβ€”relIndex
Subgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”β€”
relIndex_eq_one πŸ“–mathematicalβ€”relIndex
Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
β€”index_eq_one
subgroupOf_eq_top
relIndex_eq_two_iff πŸ“–mathematicalβ€”relIndex
Subgroup
SetLike.instMembership
instSetLike
Xor'
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”β€”
relIndex_eq_two_iff_exists_notMem_and πŸ“–mathematicalβ€”relIndex
Subgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”relIndex.eq_1
index_eq_two_iff_exists_notMem_and
relIndex_eq_two_iff_exists_notMem_and' πŸ“–mathematicalβ€”relIndex
Subgroup
SetLike.instMembership
instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”relIndex.eq_1
index_eq_two_iff_exists_notMem_and'
relIndex_eq_zero_of_le_left πŸ“–β€”Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndex
β€”β€”eq_zero_of_zero_dvd
relIndex_dvd_of_le_left
relIndex_eq_zero_of_le_right πŸ“–β€”Subgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndex
β€”β€”Finite.card_eq_zero_of_embedding
relIndex_iInf_le πŸ“–mathematicalβ€”relIndex
iInf
Subgroup
instInfSet
Finset.prod
Nat.instCommMonoid
Finset.univ
β€”Finite.card_le_of_embedding'
Finset.prod_eq_zero_iff
Nat.instNontrivial
IsDomain.to_noZeroDivisors
Nat.instIsDomain
Nat.card_pi
relIndex_eq_zero_of_le_left
iInf_le
relIndex_iInf_ne_zero πŸ“–β€”β€”β€”β€”Finset.prod_ne_zero_iff
Nat.instNontrivial
IsDomain.to_noZeroDivisors
Nat.instIsDomain
Nat.card_pi
Finite.card_eq_zero_of_embedding
relIndex_inf_le πŸ“–mathematicalβ€”relIndex
Subgroup
instMin
β€”LE.le.trans
le_of_eq
relIndex_eq_zero_of_le_left
inf_le_left
zero_le
Nat.instCanonicallyOrderedAdd
inf_relIndex_right
inf_assoc
relIndex_mul_relIndex
inf_le_right
le_imp_le_of_le_of_le
mul_le_mul_left
covariant_swap_mul_of_covariant_mul
Nat.instMulLeftMono
relIndex_le_of_le_right
le_refl
relIndex_inf_mul_relIndex πŸ“–mathematicalβ€”relIndex
Subgroup
instMin
β€”inf_relIndex_right
inf_assoc
relIndex_mul_relIndex
inf_le_right
relIndex_inf_ne_zero πŸ“–β€”β€”β€”β€”relIndex_eq_zero_of_le_right
inf_le_right
inf_relIndex_right
inf_assoc
relIndex_ne_zero_trans
relIndex_inter_ne_zero πŸ“–β€”β€”β€”β€”range_subtype
inf_comm
map_comap_eq
relIndex_comap
comap_map_eq_self_of_injective
subtype_injective
relIndex_comap_ne_zero
relIndex_ker πŸ“–mathematicalβ€”relIndex
MonoidHom.ker
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Nat.card
Subgroup
SetLike.instMembership
instSetLike
map
β€”MonoidHom.comap_bot
relIndex_comap
relIndex_bot_left
relIndex_le_of_le_left πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndexβ€”relIndex_dvd_of_le_left
relIndex_le_of_le_right πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndexβ€”Finite.card_le_of_embedding'
relIndex_map_map πŸ“–mathematicalβ€”relIndex
map
Subgroup
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
MonoidHom.ker
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”comap_map_eq
relIndex_comap
GaloisConnection.l_u_l_eq_l
gc_map_comap
relIndex_map_map_of_injective πŸ“–mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
relIndex
map
β€”relIndex_comap
comap_map_eq_self_of_injective
relIndex_mul_index πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndex
index
β€”mul_comm
Nat.card_congr
relIndex_mul_relIndex πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndexβ€”relIndex_subgroupOf
relIndex_mul_index
relIndex_ne_zero πŸ“–β€”β€”β€”β€”IsFiniteRelIndex.relIndex_ne_zero
relIndex_ne_zero_trans πŸ“–β€”β€”β€”β€”mul_ne_zero
IsDomain.to_noZeroDivisors
Nat.instIsDomain
relIndex_eq_zero_of_le_right
inf_le_left
relIndex_inf_mul_relIndex
relIndex_eq_zero_of_le_left
relIndex_pointwise_smul πŸ“–mathematicalβ€”relIndex
Subgroup
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
pointwiseMulAction
β€”pointwise_smul_def
relIndex_comap
comap_map_eq_self_of_injective
relIndex_self πŸ“–mathematicalβ€”relIndexβ€”relIndex.eq_1
subgroupOf_self
index_top
relIndex_subgroupOf πŸ“–mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
relIndex
SetLike.instMembership
instSetLike
toGroup
subgroupOf
β€”index_comap
inclusion_range
relIndex_sup_left πŸ“–mathematicalβ€”relIndex
Subgroup
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
β€”sup_comm
relIndex_sup_right
relIndex_sup_right πŸ“–mathematicalβ€”relIndex
Subgroup
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
β€”Nat.card_congr
normal_subgroupOf
relIndex_toAddSubgroup πŸ“–mathematicalβ€”AddSubgroup.relIndex
Additive
Additive.addGroup
DFunLike.coe
OrderIso
Subgroup
AddSubgroup
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddSubgroup.instPartialOrder
instFunLikeOrderIso
toAddSubgroup
relIndex
β€”β€”
relIndex_top_left πŸ“–mathematicalβ€”relIndex
Top.top
Subgroup
instTop
β€”index_top
relIndex_top_right πŸ“–mathematicalβ€”relIndex
Top.top
Subgroup
instTop
index
β€”relIndex_mul_index
le_top
index_top
mul_one
sq_mem_of_index_two πŸ“–mathematicalindexSubgroup
SetLike.instMembership
instSetLike
Monoid.toNatPow
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”mul_self_mem_of_index_two
pow_two

Subgroup.FiniteIndex

Theorems

NameKindAssumesProvesValidatesDepends On
index_ne_zero πŸ“–β€”β€”β€”β€”β€”

Subgroup.IsFiniteRelIndex

Theorems

NameKindAssumesProvesValidatesDepends On
relIndex_ne_zero πŸ“–β€”β€”β€”β€”β€”
to_finiteIndex_subgroupOf πŸ“–mathematicalβ€”Subgroup.FiniteIndex
Subgroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.toGroup
Subgroup.subgroupOf
β€”Subgroup.relIndex_ne_zero

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