Documentation Verification Report

Exponent

📁 Source: Mathlib/GroupTheory/Exponent.lean

Statistics

MetricCount
DefinitionsExponentExists, exponent, ExponentExists, exponent, addCommMonoidOfExponentTwo, commMonoidOfExponentTwo
6
TheoremsaddOrderOf_add_nsmul_eq_lcm, exists_addOrderOf_eq_lcm, of_addOrderOf_dvd_two, exponent_dvd_card, exponent_dvd_iff_forall_zsmul_eq_zero, exponent_dvd_nat_card, exponent_dvd_sub_iff_zsmul_eq_zsmul, exponent_quotient_dvd, addOrderOf_pos, exponent_ne_zero, exponent_pos, isOfFinAddOrder, of_finite, addOrderOf_le_exponent, addOrder_dvd_exponent, exists_addOrderOf_eq_exponent, exp_eq_one_iff, exp_eq_one_of_subsingleton, exponent_addSubmonoid_dvd, exponent_additive, exponent_dvd, exponent_dvd_iff_forall_nsmul_eq_zero, exponent_dvd_of_addMonoidHom, exponent_dvd_of_forall_nsmul_eq_zero, exponent_eq_iSup_addOrderOf, exponent_eq_iSup_addOrderOf', exponent_eq_max'_addOrderOf, exponent_eq_of_addEquiv, exponent_eq_prime_iff, exponent_eq_sInf, exponent_eq_zero_addOrder_zero, exponent_eq_zero_iff, exponent_eq_zero_iff_forall, exponent_eq_zero_iff_range_addOrderOf_infinite, exponent_min, exponent_min', exponent_ne_zero, exponent_ne_zero_iff_range_addOrderOf_finite, exponent_ne_zero_of_finite, exponent_nsmul_eq_zero, exponent_pi, exponent_pi_eq_zero, exponent_pos, exponent_pos_of_exists, exponent_prod, lcm_addOrderOf_dvd_exponent, lcm_addOrderOf_eq_exponent, neZero_exponent_of_finite, nsmul_eq_mod_exponent, one_lt_exponent, exponent_dvd, exponent, exponent_toAddSubmonoid, exponent_top, nsmul_exponent_eq_zero, exponent_top, nsmul_exponent_eq_zero, exists_orderOf_eq_lcm, of_orderOf_dvd_two, orderOf_mul_pow_eq_lcm, exponent_dvd_card, exponent_dvd_iff_forall_zpow_eq_one, exponent_dvd_nat_card, exponent_dvd_sub_iff_zpow_eq_zpow, exponent_quotient_dvd, exponent_ne_zero, exponent_pos, isOfFinOrder, of_finite, orderOf_pos, exists_orderOf_eq_exponent, exp_eq_one_iff, exp_eq_one_of_subsingleton, exponent_dvd, exponent_dvd_iff_forall_pow_eq_one, exponent_dvd_of_forall_pow_eq_one, exponent_dvd_of_monoidHom, exponent_eq_iSup_orderOf, exponent_eq_iSup_orderOf', exponent_eq_max'_orderOf, exponent_eq_of_mulEquiv, exponent_eq_prime_iff, exponent_eq_sInf, exponent_eq_zero_iff, exponent_eq_zero_iff_forall, exponent_eq_zero_iff_range_orderOf_infinite, exponent_eq_zero_of_order_zero, exponent_min, exponent_min', exponent_multiplicative, exponent_ne_zero, exponent_ne_zero_iff_range_orderOf_finite, exponent_ne_zero_of_finite, exponent_pi, exponent_pi_eq_zero, exponent_pos, exponent_pos_of_exists, exponent_prod, exponent_submonoid_dvd, lcm_orderOf_dvd_exponent, lcm_orderOf_eq_exponent, neZero_exponent_of_finite, one_lt_exponent, orderOf_le_exponent, order_dvd_exponent, pow_eq_mod_exponent, pow_exponent_eq_one, exponent_dvd, exponent, exists_addOrderOf_eq_pow_padic_val_nat_add_exponent, exists_orderOf_eq_pow_factorization_exponent, exponent_toSubmonoid, exponent_top, pow_exponent_eq_one, exponent_top, pow_exponent_eq_one, addOrderOf_eq_two_iff, add_comm_of_exponent_two, add_notMem_of_addOrderOf_eq_two, add_notMem_of_exponent_two, inv_eq_self_of_exponent_two, inv_eq_self_of_orderOf_eq_two, mul_comm_of_exponent_two, mul_notMem_of_exponent_two, mul_notMem_of_orderOf_eq_two, neg_eq_self_of_addOrderOf_eq_two, neg_eq_self_of_exponent_two, orderOf_eq_two_iff
128
Total134

AddCommute

Theorems

NameKindAssumesProvesValidatesDepends On
addOrderOf_add_nsmul_eq_lcm 📖mathematicalAddCommute
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
addOrderOf
AddMonoid.toNatSMul
Nat.factorizationLCMLeft
Nat.factorizationLCMRight
addOrderOf_add_eq_mul_addOrderOf_of_coprime
nsmul_nsmul
addOrderOf_nsmul_addOrderOf_sub
Nat.factorizationLCMLeft_dvd_left
Nat.factorizationLCMRight_dvd_right
Nat.coprime_factorizationLCMLeft_factorizationLCMRight
Nat.factorizationLCMLeft_mul_factorizationLCMRight
exists_addOrderOf_eq_lcm 📖mathematicalAddCommute
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddSubmonoid
SetLike.instMembership
AddSubmonoid.instSetLike
AddSubmonoid.closure
Set
Set.instInsert
Set.instSingletonSet
addOrderOf
AddSubmonoid.subset_closure
Set.mem_insert_iff
Set.mem_singleton_iff
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
AddSubmonoid.instAddSubmonoidClass
nsmul_mem
addOrderOf_add_nsmul_eq_lcm
of_addOrderOf_dvd_two 📖mathematicaladdOrderOfAddCommute
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
addCommute_iff_eq
IsCancelAdd.toIsLeftCancelAdd
IsCancelAdd.toIsRightCancelAdd
add_assoc
two_nsmul
addOrderOf_dvd_iff_nsmul_eq_zero
add_zero

AddGroup

Theorems

NameKindAssumesProvesValidatesDepends On
exponent_dvd_card 📖mathematicalAddMonoid.exponent
SubNegMonoid.toAddMonoid
toSubNegMonoid
Fintype.card
AddMonoid.exponent_dvd
addOrderOf_dvd_card
exponent_dvd_iff_forall_zsmul_eq_zero 📖mathematicalAddMonoid.exponent
SubNegMonoid.toAddMonoid
toSubNegMonoid
SubNegMonoid.toZSMul
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
toSubtractionMonoid
Int.natCast_dvd
AddMonoid.exponent_dvd_iff_forall_nsmul_eq_zero
natAbs_nsmul_eq_zero
exponent_dvd_nat_card 📖mathematicalAddMonoid.exponent
SubNegMonoid.toAddMonoid
toSubNegMonoid
Nat.card
AddMonoid.exponent_dvd
addOrderOf_dvd_natCard
exponent_dvd_sub_iff_zsmul_eq_zsmul 📖mathematicalAddMonoid.exponent
SubNegMonoid.toAddMonoid
toSubNegMonoid
SubNegMonoid.toZSMul
exponent_dvd_iff_forall_zsmul_eq_zero
sub_zsmul
add_neg_eq_zero
exponent_quotient_dvd 📖mathematicalAddMonoid.exponent
HasQuotient.Quotient
AddSubgroup
QuotientAddGroup.instHasQuotientAddSubgroup
SubNegMonoid.toAddMonoid
toSubNegMonoid
QuotientAddGroup.Quotient.addGroup
AddMonoidHom.exponent_dvd
AddMonoidHom.instAddMonoidHomClass
QuotientAddGroup.mk'_surjective

AddMonoid

Definitions

NameCategoryTheorems
ExponentExists 📖MathDef
5 mathmath: ExponentExists.of_finite, IsAddTorsion.exponentExists, exponent_ne_zero, exponent_eq_zero_iff, exponent_pos
exponent 📖CompOp
55 mathmath: IsAddKleinFour.exponent_two, exponent_eq_zero_iff_forall, exponent_pos_of_exists, exists_addOrderOf_eq_exponent, exponent_dvd_of_addMonoidHom, AddAction.period_dvd_exponent, AddSubmonoid.nsmul_exponent_eq_zero, exponent_nsmul_eq_zero, exponent_eq_max'_addOrderOf, ExponentExists.exponent_pos, exponent_eq_prime_iff, AddGroup.exponent_quotient_dvd, exponent_min', exponent_dvd_of_forall_nsmul_eq_zero, exponent_dvd, AddSubmonoid.exponent_top, Monoid.exponent_multiplicative, exponent_pi, exponent_prod, exponent_eq_zero_iff, exponent_eq_iSup_addOrderOf, IsAddCyclic.exponent_eq_zero_of_infinite, AddSubgroup.nsmul_exponent_eq_zero, nsmul_eq_mod_exponent, addOrderOf_le_exponent, AddGroup.exponent_dvd_card, exponent_additive, exponent_pos, card_dvd_exponent_nsmul_rank, IsAddCyclic.iff_exponent_eq_card, ZMod.exponent, not_isAddCyclic_iff_exponent_eq_prime, exponent_addSubmonoid_dvd, AddGroup.exponent_dvd_nat_card, exponent_eq_zero_iff_range_addOrderOf_infinite, exponent_eq_sInf, AddOpposite.exponent, AddMonoidHom.exponent_dvd, exponent_eq_iSup_addOrderOf', AddGroup.exponent_dvd_iff_forall_zsmul_eq_zero, AddGroup.exponent_dvd_sub_iff_zsmul_eq_zsmul, lcm_addOrderOf_eq_exponent, AddSubgroup.exponent_top, exponent_dvd_iff_forall_nsmul_eq_zero, exp_eq_one_of_subsingleton, neZero_exponent_of_finite, AddSubgroup.exponent_toAddSubmonoid, exponent_eq_zero_addOrder_zero, Nat.Prime.exists_addOrderOf_eq_pow_padic_val_nat_add_exponent, lcm_addOrderOf_dvd_exponent, exp_eq_one_iff, IsAddCyclic.exponent_eq_card, addOrder_dvd_exponent, one_lt_exponent, exponent_eq_of_addEquiv

Theorems

NameKindAssumesProvesValidatesDepends On
addOrderOf_le_exponent 📖mathematicalExponentExistsaddOrderOf
exponent
ExponentExists.exponent_pos
addOrder_dvd_exponent
addOrder_dvd_exponent 📖mathematicaladdOrderOf
exponent
addOrderOf_dvd_of_nsmul_eq_zero
exponent_nsmul_eq_zero
exists_addOrderOf_eq_exponent 📖mathematicalExponentExists
AddCommMonoid.toAddMonoid
addOrderOf
exponent
ExponentExists.exponent_ne_zero
addOrderOf_zero
exponent_ne_zero_iff_range_addOrderOf_finite
ExponentExists.addOrderOf_pos
Set.Nonempty.csSup_mem
addOrder_dvd_exponent
Nat.dvd_of_primeFactorsList_subperm
Mathlib.Tactic.Push.not_forall_eq
not_le
Nat.prime_of_mem_primeFactorsList
Nat.Prime.exists_addOrderOf_eq_pow_padic_val_nat_add_exponent
Nat.ordProj_dvd
addOrderOf_nsmul'
pow_ne_zero
isReduced_of_noZeroDivisors
IsStrictOrderedRing.noZeroDivisors
Nat.instIsStrictOrderedRing
CanonicallyOrderedAdd.toExistsAddOfLE
Nat.instCanonicallyOrderedAdd
Nat.Prime.ne_zero
Nat.primeFactorsList_count_eq
Nat.coprime_pow_right_iff
pos_of_gt
Nat.coprime_or_dvd_of_prime
pow_one
Nat.instOrderedSub
Nat.factorization_div
Nat.factorization_pow
Nat.Prime.factorization
Finsupp.smul_single
mul_one
Finsupp.single_eq_same
tsub_self
zero_add
Nat.pow_succ_factorization_not_dvd
LT.lt.ne'
AddCommute.addOrderOf_add_eq_mul_addOrderOf_of_coprime
AddCommute.all
pow_add
mul_assoc
lt_mul_iff_one_lt_right
LinearOrderedCommMonoidWithZero.toPosMulStrictMono
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
pow_succ
one_lt_pow₀
Nat.instZeroLEOneClass
IsOrderedRing.toPosMulMono
IsStrictOrderedRing.toIsOrderedRing
Nat.Prime.one_lt
LT.lt.not_ge
le_csSup
Set.Finite.bddAbove
SemilatticeSup.instIsDirectedOrder
Set.mem_range_self
exp_eq_one_iff 📖mathematicalexponentone_nsmul
exponent_nsmul_eq_zero
le_antisymm
exponent_min'
exponent_pos_of_exists
exp_eq_one_of_subsingleton 📖mathematicalexponentexp_eq_one_iff
exponent_addSubmonoid_dvd 📖mathematicalexponent
AddSubmonoid
toAddZeroClass
SetLike.instMembership
AddSubmonoid.instSetLike
AddSubmonoid.toAddMonoid
exponent_dvd_of_addMonoidHom
AddSubmonoid.subtype_injective
exponent_additive 📖mathematicalexponent
Additive
Additive.addMonoid
Monoid.exponent
exponent_dvd 📖mathematicalexponent
addOrderOf
exponent_dvd_iff_forall_nsmul_eq_zero
addOrderOf_dvd_iff_nsmul_eq_zero
exponent_dvd_iff_forall_nsmul_eq_zero 📖mathematicalexponent
toNatSMul
AddZero.toZero
AddZeroClass.toAddZero
toAddZeroClass
dvd_zero
zero_nsmul
nsmul_eq_mod_exponent
exponent_pos_of_exists
exponent_min'
pos_iff_ne_zero
Nat.instCanonicallyOrderedAdd
LT.lt.not_ge
exponent_dvd_of_addMonoidHom 📖mathematicalDFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
toAddZeroClass
AddMonoidHom.instFunLike
exponentexponent_dvd_of_forall_nsmul_eq_zero
map_nsmul
AddMonoidHom.instAddMonoidHomClass
exponent_nsmul_eq_zero
map_zero
AddMonoidHomClass.toZeroHomClass
exponent_dvd_of_forall_nsmul_eq_zero 📖mathematicaltoNatSMul
AddZero.toZero
AddZeroClass.toAddZero
toAddZeroClass
exponentexponent_dvd_iff_forall_nsmul_eq_zero
exponent_eq_iSup_addOrderOf 📖mathematicaladdOrderOf
AddCommMonoid.toAddMonoid
exponent
iSup
Nat.instSupSet
iSup.eq_1
csSup_eq_of_forall_le_of_forall_lt_exists_gt
Set.range_nonempty
Zero.instNonempty
Set.mem_range
addOrderOf_le_exponent
exists_addOrderOf_eq_exponent
exists_exists_eq_and
exponent_eq_zero_iff
Set.Infinite.Nat.sSup_eq_zero
exponent_eq_zero_iff_range_addOrderOf_infinite
exponent_eq_iSup_addOrderOf' 📖mathematicalexponent
AddCommMonoid.toAddMonoid
addOrderOf
iSup
Nat.instSupSet
exponent_eq_zero_addOrder_zero
exponent_eq_iSup_addOrderOf
Ne.bot_lt
exponent_eq_max'_addOrderOf 📖mathematicalexponent
AddCommMonoid.toAddMonoid
AddCancelCommMonoid.toAddCommMonoid
Finset.max'
Nat.instLinearOrder
Finset.image
addOrderOf
Finset.univ
Finset
Finset.instMembership
Finset.Nonempty.csSup_eq_max'
Finset.coe_image
Finset.coe_univ
Set.image_univ
iSup.eq_1
exponent_eq_iSup_addOrderOf
addOrderOf_pos
Finite.of_fintype
exponent_eq_of_addEquiv 📖mathematicalexponentexponent_dvd_of_addMonoidHom
AddEquivClass.instAddMonoidHomClass
AddEquiv.instAddEquivClass
AddEquiv.injective
exponent_eq_prime_iff 📖mathematicalNat.Primeexponent
addOrderOf
Nat.Prime.dvd_iff_eq
addOrderOf_eq_one_iff
addOrder_dvd_exponent
dvd_antisymm
Nat.instIsCancelMulZero
Unique.instSubsingleton
exponent_dvd
addOrderOf_zero
IsUnit.dvd
isUnit_iff_eq_one
dvd_refl
exists_ne
exponent_eq_sInf 📖mathematicalexponent
InfSet.sInf
Nat.instInfSet
setOf
exponent.eq_1
Nat.sInf_def
Set.eq_empty_of_forall_notMem
exponent_eq_zero_iff
Nat.sInf_empty
exponent_eq_zero_addOrder_zero 📖mathematicaladdOrderOfexponentexponent_eq_zero_iff
LT.lt.ne'
ExponentExists.addOrderOf_pos
exponent_eq_zero_iff 📖mathematicalexponent
ExponentExists
Iff.not_right
exponent_ne_zero
exponent_eq_zero_iff_forall 📖mathematicalexponentexponent_eq_zero_iff
ExponentExists.eq_1
Mathlib.Tactic.Push.not_exists
Mathlib.Tactic.Push.not_and_eq
Mathlib.Tactic.Push.not_forall_eq
exponent_eq_zero_iff_range_addOrderOf_infinite 📖mathematicaladdOrderOfexponent
Set.Infinite
Set.range
exponent_ne_zero_iff_range_addOrderOf_finite
not_iff_comm
exponent_min 📖exponentexponent_min'
Mathlib.Tactic.Push.not_exists
exponent_min' 📖mathematicaltoNatSMul
AddZero.toZero
AddZeroClass.toAddZero
toAddZeroClass
exponentexponent.eq_1
Nat.find_min'
exponent_ne_zero 📖mathematicalExponentExistsexponent.eq_1
Nat.find_eq_zero
lt_self_iff_false
zero_nsmul
exponent_ne_zero_iff_range_addOrderOf_finite 📖mathematicaladdOrderOfSet.Finite
Set.range
Set.Infinite.exists_gt
nsmul_ne_zero_of_lt_addOrderOf
exponent_nsmul_eq_zero
CanLift.prf
Set.instCanLiftFinsetCoeFinite
Finset.prod_pos
Nat.instZeroLEOneClass
LinearOrderedCommMonoidWithZero.toPosMulStrictMono
Nat.instNontrivial
Finset.mem_coe
exponent_dvd
Finset.dvd_prod_of_mem
Set.mem_range_self
LT.lt.ne'
zero_dvd_iff
exponent_ne_zero_of_finite 📖ExponentExists.exponent_ne_zero
ExponentExists.of_finite
exponent_nsmul_eq_zero 📖mathematicaltoNatSMul
exponent
AddZero.toZero
AddZeroClass.toAddZero
toAddZeroClass
Nat.find_spec
zero_nsmul
exponent_pi 📖mathematicalexponent
Pi.addMonoid
Finset.lcm
Nat.instCommMonoidWithZero
instNormalizedGCDMonoidNat
Finset.univ
dvd_antisymm
Nat.instIsCancelMulZero
Unique.instSubsingleton
exponent_dvd_of_forall_nsmul_eq_zero
Pi.smul_apply
Pi.zero_apply
addOrderOf_dvd_iff_nsmul_eq_zero
dvd_trans
addOrder_dvd_exponent
Finset.dvd_lcm
Finset.mem_univ
Finset.lcm_dvd
AddMonoidHom.exponent_dvd
AddMonoidHom.instAddMonoidHomClass
Function.surjective_eval
Zero.instNonempty
exponent_pi_eq_zero 📖mathematicalexponentPi.addMonoidexponent_eq_zero_iff
ExponentExists.eq_1
Mathlib.Tactic.Push.not_exists
Mathlib.Tactic.Push.not_and_eq
Mathlib.Tactic.Push.not_forall_eq
Pi.single_eq_same
exponent_pos 📖mathematicalexponent
ExponentExists
pos_iff_ne_zero
Nat.instCanonicallyOrderedAdd
exponent_ne_zero
exponent_pos_of_exists 📖mathematicaltoNatSMul
AddZero.toZero
AddZeroClass.toAddZero
toAddZeroClass
exponentExponentExists.exponent_pos
exponent_prod 📖mathematicalexponent
Prod.instAddMonoid
GCDMonoid.lcm
Nat.instCommMonoidWithZero
instGCDMonoidNat
dvd_antisymm
Nat.instIsCancelMulZero
Unique.instSubsingleton
exponent_dvd_of_forall_nsmul_eq_zero
Prod.smul_fst
Prod.fst_zero
addOrderOf_dvd_iff_nsmul_eq_zero
dvd_trans
addOrder_dvd_exponent
dvd_lcm_left
Prod.smul_snd
Prod.snd_zero
dvd_lcm_right
lcm_dvd
AddMonoidHom.exponent_dvd
AddMonoidHom.instAddMonoidHomClass
Prod.fst_surjective
Zero.instNonempty
Prod.snd_surjective
lcm_addOrderOf_dvd_exponent 📖mathematicalFinset.lcm
Nat.instCommMonoidWithZero
instNormalizedGCDMonoidNat
Finset.univ
addOrderOf
exponent
Finset.lcm_dvd
addOrder_dvd_exponent
lcm_addOrderOf_eq_exponent 📖mathematicalFinset.lcm
Nat.instCommMonoidWithZero
instNormalizedGCDMonoidNat
Finset.univ
addOrderOf
exponent
lcm_addOrderOf_dvd_exponent
exponent_dvd
Finset.dvd_lcm
Finset.mem_univ
neZero_exponent_of_finite 📖mathematicalMulZeroClass.toZero
Nat.instMulZeroClass
exponent
AddLeftCancelMonoid.toAddMonoid
exponent_ne_zero_of_finite
nsmul_eq_mod_exponent 📖mathematicaltoNatSMul
exponent
add_nsmul
mul_nsmul
exponent_nsmul_eq_zero
nsmul_zero
add_zero
one_lt_exponent 📖mathematicalexponent
AddLeftCancelMonoid.toAddMonoid
exponent_ne_zero_of_finite
exp_eq_one_iff
not_subsingleton

AddMonoid.ExponentExists

Theorems

NameKindAssumesProvesValidatesDepends On
addOrderOf_pos 📖mathematicalAddMonoid.ExponentExistsaddOrderOfIsOfFinAddOrder.addOrderOf_pos
isOfFinAddOrder
exponent_ne_zero 📖AddMonoid.ExponentExistsAddMonoid.exponent_ne_zero
exponent_pos 📖mathematicalAddMonoid.ExponentExistsAddMonoid.exponentAddMonoid.exponent_pos
isOfFinAddOrder 📖mathematicalAddMonoid.ExponentExistsIsOfFinAddOrderisOfFinAddOrder_iff_nsmul_eq_zero
of_finite 📖mathematicalAddMonoid.ExponentExists
AddLeftCancelMonoid.toAddMonoid
pos_iff_ne_zero
Nat.instCanonicallyOrderedAdd
Finset.lcm_eq_zero_iff
Nat.instNontrivial
Finset.mem_univ
addOrderOf_eq_zero_iff
LT.lt.ne'
addOrderOf_pos
addOrderOf_dvd_iff_nsmul_eq_zero
AddMonoid.lcm_addOrderOf_eq_exponent
AddMonoid.addOrder_dvd_exponent

AddMonoidHom

Theorems

NameKindAssumesProvesValidatesDepends On
exponent_dvd 📖mathematicalDFunLike.coeAddMonoid.exponentAddMonoid.exponent_dvd_of_forall_nsmul_eq_zero
map_nsmul
AddMonoid.exponent_nsmul_eq_zero
map_zero
AddMonoidHomClass.toZeroHomClass

AddOpposite

Theorems

NameKindAssumesProvesValidatesDepends On
exponent 📖mathematicalAddMonoid.exponent
AddOpposite
instAddMonoid
op_injective
unop_injective
Lean.Meta.FastSubsingleton.elim
Lean.Meta.instFastSubsingletonForall
Lean.Meta.instFastSubsingletonDecidable

AddSubgroup

Theorems

NameKindAssumesProvesValidatesDepends On
exponent_toAddSubmonoid 📖mathematicalAddMonoid.exponent
AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
SetLike.instMembership
AddSubmonoid.instSetLike
toAddSubmonoid
AddSubmonoid.toAddMonoid
AddSubgroup
instSetLike
toAddGroup
AddMonoid.exponent_eq_of_addEquiv
exponent_top 📖mathematicalAddMonoid.exponent
AddSubgroup
SetLike.instMembership
instSetLike
Top.top
instTop
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
toAddGroup
AddMonoid.exponent_eq_of_addEquiv
nsmul_exponent_eq_zero 📖mathematicalAddSubgroup
SetLike.instMembership
instSetLike
AddMonoid.toNatSMul
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddMonoid.exponent
toAddGroup
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
AddSubmonoid.nsmul_exponent_eq_zero
exponent_toAddSubmonoid

AddSubmonoid

Theorems

NameKindAssumesProvesValidatesDepends On
exponent_top 📖mathematicalAddMonoid.exponent
AddSubmonoid
AddMonoid.toAddZeroClass
SetLike.instMembership
instSetLike
Top.top
instTop
toAddMonoid
AddMonoid.exponent_eq_of_addEquiv
nsmul_exponent_eq_zero 📖mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
SetLike.instMembership
instSetLike
AddMonoid.toNatSMul
AddMonoid.exponent
toAddMonoid
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.exponent_nsmul_eq_zero
nsmul_mem
instAddSubmonoidClass
ZeroMemClass.coe_eq_zero
AddSubmonoidClass.mk_nsmul

Commute

Theorems

NameKindAssumesProvesValidatesDepends On
exists_orderOf_eq_lcm 📖mathematicalCommute
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Submonoid
SetLike.instMembership
Submonoid.instSetLike
Submonoid.closure
Set
Set.instInsert
Set.instSingletonSet
orderOf
Submonoid.subset_closure
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
Submonoid.instSubmonoidClass
pow_mem
orderOf_mul_pow_eq_lcm
of_orderOf_dvd_two 📖mathematicalorderOfCommute
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
commute_iff_eq
IsCancelMul.toIsLeftCancelMul
IsCancelMul.toIsRightCancelMul
mul_assoc
pow_two
mul_one
orderOf_mul_pow_eq_lcm 📖mathematicalCommute
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
orderOf
Monoid.toNatPow
Nat.factorizationLCMLeft
Nat.factorizationLCMRight
orderOf_mul_eq_mul_orderOf_of_coprime
pow_pow
orderOf_pow_orderOf_div
Nat.factorizationLCMLeft_mul_factorizationLCMRight

Group

Theorems

NameKindAssumesProvesValidatesDepends On
exponent_dvd_card 📖mathematicalMonoid.exponent
DivInvMonoid.toMonoid
toDivInvMonoid
Fintype.card
Monoid.exponent_dvd
orderOf_dvd_card
exponent_dvd_iff_forall_zpow_eq_one 📖mathematicalMonoid.exponent
DivInvMonoid.toMonoid
toDivInvMonoid
DivInvMonoid.toZPow
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
toDivisionMonoid
exponent_dvd_nat_card 📖mathematicalMonoid.exponent
DivInvMonoid.toMonoid
toDivInvMonoid
Nat.card
Monoid.exponent_dvd
orderOf_dvd_natCard
exponent_dvd_sub_iff_zpow_eq_zpow 📖mathematicalMonoid.exponent
DivInvMonoid.toMonoid
toDivInvMonoid
DivInvMonoid.toZPow
zpow_sub
exponent_quotient_dvd 📖mathematicalMonoid.exponent
HasQuotient.Quotient
Subgroup
QuotientGroup.instHasQuotientSubgroup
DivInvMonoid.toMonoid
toDivInvMonoid
QuotientGroup.Quotient.group
MonoidHom.exponent_dvd
MonoidHom.instMonoidHomClass
QuotientGroup.mk'_surjective

Monoid

Definitions

NameCategoryTheorems
ExponentExists 📖MathDef
5 mathmath: exponent_eq_zero_iff, IsTorsion.exponentExists, exponent_ne_zero, exponent_pos, ExponentExists.of_finite
exponent 📖CompOp
60 mathmath: exists_orderOf_eq_exponent, Nat.Prime.exists_orderOf_eq_pow_factorization_exponent, exponent_eq_zero_iff_forall, ExponentExists.exponent_pos, MulAction.period_dvd_exponent, exponent_dvd_iff_forall_pow_eq_one, neZero_exponent_of_finite, DihedralGroup.exponent, card_dvd_exponent_pow_rank, Group.exponent_dvd_sub_iff_zpow_eq_zpow, Submonoid.pow_exponent_eq_one, exponent_pi, exponent_dvd, MulOpposite.exponent, exp_eq_one_of_subsingleton, pow_eq_mod_exponent, not_isCyclic_iff_exponent_eq_prime, exponent_dvd_of_forall_pow_eq_one, order_dvd_exponent, exponent_min', one_lt_exponent, MonoidHom.exponent_dvd, exponent_multiplicative, exponent_eq_zero_iff, lcm_orderOf_eq_exponent, lcm_orderOf_dvd_exponent, exponent_submonoid_dvd, exponent_eq_of_mulEquiv, Subgroup.exponent_toSubmonoid, Subgroup.exponent_top, AddMonoid.exponent_additive, Submonoid.exponent_top, IsKleinFour.exponent_two, exponent_dvd_of_monoidHom, exponent_eq_iSup_orderOf, IsCyclic.iff_exponent_eq_card, exponent_prod, exponent_eq_max'_orderOf, Group.exponent_dvd_nat_card, IsZGroup.exponent_eq_card, exponent_pos, IsCyclic.exponent_eq_card, exp_eq_one_iff, alternatingGroup.exponent_kleinFour_of_card_eq_four, ArithmeticFunction.carmichael_eq_exponent', IsCyclic.exponent_eq_zero_of_infinite, exponent_eq_sInf, ArithmeticFunction.carmichael_eq_exponent, exponent_eq_prime_iff, Group.exponent_dvd_card, exponent_eq_zero_iff_range_orderOf_infinite, QuaternionGroup.exponent, exponent_pos_of_exists, exponent_eq_iSup_orderOf', pow_exponent_eq_one, Group.exponent_dvd_iff_forall_zpow_eq_one, Subgroup.pow_exponent_eq_one, orderOf_le_exponent, Group.exponent_quotient_dvd, exponent_eq_zero_of_order_zero

Theorems

NameKindAssumesProvesValidatesDepends On
exists_orderOf_eq_exponent 📖mathematicalExponentExists
CommMonoid.toMonoid
orderOf
exponent
ExponentExists.exponent_ne_zero
orderOf_one
exponent_ne_zero_iff_range_orderOf_finite
ExponentExists.orderOf_pos
Set.Nonempty.csSup_mem
order_dvd_exponent
Nat.dvd_of_primeFactorsList_subperm
Mathlib.Tactic.Push.not_forall_eq
Nat.prime_of_mem_primeFactorsList
Nat.Prime.exists_orderOf_eq_pow_factorization_exponent
Nat.ordProj_dvd
orderOf_pow'
pow_ne_zero
isReduced_of_noZeroDivisors
IsStrictOrderedRing.noZeroDivisors
Nat.instIsStrictOrderedRing
CanonicallyOrderedAdd.toExistsAddOfLE
Nat.instCanonicallyOrderedAdd
Nat.Prime.ne_zero
Nat.primeFactorsList_count_eq
Nat.coprime_pow_right_iff
pos_of_gt
Nat.coprime_or_dvd_of_prime
pow_one
Nat.instOrderedSub
Nat.factorization_div
Nat.factorization_pow
Nat.Prime.factorization
Finsupp.smul_single
mul_one
Finsupp.single_eq_same
tsub_self
zero_add
Nat.pow_succ_factorization_not_dvd
LT.lt.ne'
Commute.orderOf_mul_eq_mul_orderOf_of_coprime
Commute.all
pow_add
mul_assoc
lt_mul_iff_one_lt_right
LinearOrderedCommMonoidWithZero.toPosMulStrictMono
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
pow_succ
one_lt_pow₀
Nat.instZeroLEOneClass
IsOrderedRing.toPosMulMono
IsStrictOrderedRing.toIsOrderedRing
Nat.Prime.one_lt
LT.lt.not_ge
le_csSup
Set.Finite.bddAbove
SemilatticeSup.instIsDirectedOrder
Set.mem_range_self
exp_eq_one_iff 📖mathematicalexponentpow_one
pow_exponent_eq_one
le_antisymm
exponent_min'
exponent_pos_of_exists
exp_eq_one_of_subsingleton 📖mathematicalexponentexp_eq_one_iff
exponent_dvd 📖mathematicalexponent
orderOf
exponent_dvd_iff_forall_pow_eq_one 📖mathematicalexponent
toNatPow
MulOne.toOne
MulOneClass.toMulOne
toMulOneClass
pow_zero
pow_eq_mod_exponent
exponent_pos_of_exists
exponent_min'
pos_iff_ne_zero
Nat.instCanonicallyOrderedAdd
LT.lt.not_ge
exponent_dvd_of_forall_pow_eq_one 📖mathematicaltoNatPow
MulOne.toOne
MulOneClass.toMulOne
toMulOneClass
exponentexponent_dvd_iff_forall_pow_eq_one
exponent_dvd_of_monoidHom 📖mathematicalDFunLike.coe
MonoidHom
MulOneClass.toMulOne
toMulOneClass
MonoidHom.instFunLike
exponentexponent_dvd_of_forall_pow_eq_one
map_pow
MonoidHom.instMonoidHomClass
pow_exponent_eq_one
map_one
MonoidHomClass.toOneHomClass
exponent_eq_iSup_orderOf 📖mathematicalorderOf
CommMonoid.toMonoid
exponent
iSup
Nat.instSupSet
iSup.eq_1
csSup_eq_of_forall_le_of_forall_lt_exists_gt
Set.range_nonempty
One.instNonempty
orderOf_le_exponent
exists_orderOf_eq_exponent
exponent_eq_zero_iff
Set.Infinite.Nat.sSup_eq_zero
exponent_eq_zero_iff_range_orderOf_infinite
exponent_eq_iSup_orderOf' 📖mathematicalexponent
CommMonoid.toMonoid
orderOf
iSup
Nat.instSupSet
exponent_eq_zero_of_order_zero
exponent_eq_iSup_orderOf
Ne.bot_lt
exponent_eq_max'_orderOf 📖mathematicalexponent
CommMonoid.toMonoid
CancelCommMonoid.toCommMonoid
Finset.max'
Nat.instLinearOrder
Finset.image
orderOf
Finset.univ
Finset
Finset.instMembership
Finset.Nonempty.csSup_eq_max'
Finset.coe_image
Finset.coe_univ
Set.image_univ
iSup.eq_1
exponent_eq_iSup_orderOf
orderOf_pos
Finite.of_fintype
exponent_eq_of_mulEquiv 📖mathematicalexponentexponent_dvd_of_monoidHom
MulEquivClass.instMonoidHomClass
MulEquiv.instMulEquivClass
MulEquiv.injective
exponent_eq_prime_iff 📖mathematicalNat.Primeexponent
orderOf
Nat.Prime.dvd_iff_eq
orderOf_eq_one_iff
order_dvd_exponent
dvd_antisymm
Nat.instIsCancelMulZero
Unique.instSubsingleton
exponent_dvd
orderOf_one
dvd_refl
exists_ne
exponent_eq_sInf 📖mathematicalexponent
InfSet.sInf
Nat.instInfSet
setOf
exponent.eq_1
Nat.sInf_def
Set.eq_empty_of_forall_notMem
exponent_eq_zero_iff
Nat.sInf_empty
exponent_eq_zero_iff 📖mathematicalexponent
ExponentExists
Iff.not_right
exponent_ne_zero
exponent_eq_zero_iff_forall 📖mathematicalexponentexponent_eq_zero_iff
ExponentExists.eq_1
Mathlib.Tactic.Push.not_and_eq
Mathlib.Tactic.Push.not_forall_eq
exponent_eq_zero_iff_range_orderOf_infinite 📖mathematicalorderOfexponent
Set.Infinite
Set.range
exponent_ne_zero_iff_range_orderOf_finite
not_iff_comm
exponent_eq_zero_of_order_zero 📖mathematicalorderOfexponentexponent_eq_zero_iff
LT.lt.ne'
ExponentExists.orderOf_pos
exponent_min 📖exponentexponent_min'
exponent_min' 📖mathematicaltoNatPow
MulOne.toOne
MulOneClass.toMulOne
toMulOneClass
exponentexponent.eq_1
Nat.find_min'
exponent_multiplicative 📖mathematicalexponent
Multiplicative
Multiplicative.monoid
AddMonoid.exponent
exponent_ne_zero 📖mathematicalExponentExistsexponent.eq_1
pow_zero
exponent_ne_zero_iff_range_orderOf_finite 📖mathematicalorderOfSet.Finite
Set.range
Set.Infinite.exists_gt
pow_ne_one_of_lt_orderOf
pow_exponent_eq_one
CanLift.prf
Set.instCanLiftFinsetCoeFinite
Finset.prod_pos
Nat.instZeroLEOneClass
LinearOrderedCommMonoidWithZero.toPosMulStrictMono
Nat.instNontrivial
Finset.mem_coe
exponent_dvd
Finset.dvd_prod_of_mem
Set.mem_range_self
LT.lt.ne'
zero_dvd_iff
exponent_ne_zero_of_finite 📖ExponentExists.exponent_ne_zero
ExponentExists.of_finite
exponent_pi 📖mathematicalexponent
Pi.monoid
Finset.lcm
Nat.instCommMonoidWithZero
instNormalizedGCDMonoidNat
Finset.univ
dvd_antisymm
Nat.instIsCancelMulZero
Unique.instSubsingleton
exponent_dvd_of_forall_pow_eq_one
Pi.pow_apply
Pi.one_apply
orderOf_dvd_iff_pow_eq_one
dvd_trans
order_dvd_exponent
Finset.dvd_lcm
Finset.mem_univ
Finset.lcm_dvd
MonoidHom.exponent_dvd
MonoidHom.instMonoidHomClass
Function.surjective_eval
One.instNonempty
exponent_pi_eq_zero 📖mathematicalexponentPi.monoidexponent_eq_zero_iff
ExponentExists.eq_1
Mathlib.Tactic.Push.not_and_eq
Mathlib.Tactic.Push.not_forall_eq
Pi.mulSingle_eq_same
exponent_pos 📖mathematicalexponent
ExponentExists
pos_iff_ne_zero
Nat.instCanonicallyOrderedAdd
exponent_ne_zero
exponent_pos_of_exists 📖mathematicaltoNatPow
MulOne.toOne
MulOneClass.toMulOne
toMulOneClass
exponentExponentExists.exponent_pos
exponent_prod 📖mathematicalexponent
Prod.instMonoid
GCDMonoid.lcm
Nat.instCommMonoidWithZero
instGCDMonoidNat
dvd_antisymm
Nat.instIsCancelMulZero
Unique.instSubsingleton
exponent_dvd_of_forall_pow_eq_one
Prod.pow_fst
Prod.fst_one
orderOf_dvd_iff_pow_eq_one
dvd_trans
order_dvd_exponent
dvd_lcm_left
Prod.pow_snd
Prod.snd_one
dvd_lcm_right
lcm_dvd
MonoidHom.exponent_dvd
MonoidHom.instMonoidHomClass
Prod.fst_surjective
One.instNonempty
Prod.snd_surjective
exponent_submonoid_dvd 📖mathematicalexponent
Submonoid
toMulOneClass
SetLike.instMembership
Submonoid.instSetLike
Submonoid.toMonoid
exponent_dvd_of_monoidHom
Submonoid.subtype_injective
lcm_orderOf_dvd_exponent 📖mathematicalFinset.lcm
Nat.instCommMonoidWithZero
instNormalizedGCDMonoidNat
Finset.univ
orderOf
exponent
Finset.lcm_dvd
order_dvd_exponent
lcm_orderOf_eq_exponent 📖mathematicalFinset.lcm
Nat.instCommMonoidWithZero
instNormalizedGCDMonoidNat
Finset.univ
orderOf
exponent
lcm_orderOf_dvd_exponent
exponent_dvd
Finset.dvd_lcm
Finset.mem_univ
neZero_exponent_of_finite 📖mathematicalMulZeroClass.toZero
Nat.instMulZeroClass
exponent
LeftCancelMonoid.toMonoid
exponent_ne_zero_of_finite
one_lt_exponent 📖mathematicalexponent
LeftCancelMonoid.toMonoid
exponent_ne_zero_of_finite
exp_eq_one_iff
not_subsingleton
orderOf_le_exponent 📖mathematicalExponentExistsorderOf
exponent
ExponentExists.exponent_pos
order_dvd_exponent
order_dvd_exponent 📖mathematicalorderOf
exponent
orderOf_dvd_of_pow_eq_one
pow_exponent_eq_one
pow_eq_mod_exponent 📖mathematicaltoNatPow
exponent
pow_add
pow_mul
pow_exponent_eq_one
one_pow
mul_one
pow_exponent_eq_one 📖mathematicaltoNatPow
exponent
MulOne.toOne
MulOneClass.toMulOne
toMulOneClass
Nat.find_spec
pow_zero

Monoid.ExponentExists

Theorems

NameKindAssumesProvesValidatesDepends On
exponent_ne_zero 📖Monoid.ExponentExistsMonoid.exponent_ne_zero
exponent_pos 📖mathematicalMonoid.ExponentExistsMonoid.exponentMonoid.exponent_pos
isOfFinOrder 📖mathematicalMonoid.ExponentExistsIsOfFinOrderisOfFinOrder_iff_pow_eq_one
of_finite 📖mathematicalMonoid.ExponentExists
LeftCancelMonoid.toMonoid
Nat.instCanonicallyOrderedAdd
Nat.instNontrivial
LT.lt.ne'
orderOf_pos
orderOf_dvd_iff_pow_eq_one
Monoid.lcm_orderOf_eq_exponent
Monoid.order_dvd_exponent
orderOf_pos 📖mathematicalMonoid.ExponentExistsorderOfIsOfFinOrder.orderOf_pos
isOfFinOrder

MonoidHom

Theorems

NameKindAssumesProvesValidatesDepends On
exponent_dvd 📖mathematicalDFunLike.coeMonoid.exponentMonoid.exponent_dvd_of_forall_pow_eq_one
map_pow
Monoid.pow_exponent_eq_one
map_one
MonoidHomClass.toOneHomClass

MulOpposite

Theorems

NameKindAssumesProvesValidatesDepends On
exponent 📖mathematicalMonoid.exponent
MulOpposite
instMonoid
op_injective
unop_injective
Lean.Meta.FastSubsingleton.elim
Lean.Meta.instFastSubsingletonForall
Lean.Meta.instFastSubsingletonDecidable

Nat.Prime

Theorems

NameKindAssumesProvesValidatesDepends On
exists_addOrderOf_eq_pow_padic_val_nat_add_exponent 📖mathematicalNat.PrimeaddOrderOf
Monoid.toNatPow
Nat.instMonoid
DFunLike.coe
Finsupp
MulZeroClass.toZero
Nat.instMulZeroClass
Finsupp.instFunLike
Nat.factorization
AddMonoid.exponent
eq_or_ne
pow_zero
addOrderOf_zero
Ne.bot_lt
bot_eq_zero
Nat.instCanonicallyOrderedAdd
Nat.factorization_zero
Finsupp.zero_apply
LT.lt.not_ge
one_lt
mul_le_iff_le_one_left
IsOrderedRing.toMulPosMono
IsStrictOrderedRing.toIsOrderedRing
Nat.instIsStrictOrderedRing
MulPosStrictMono.toMulPosReflectLE
IsStrictOrderedRing.toMulPosStrictMono
Nat.dvd_of_mem_primeFactors
Finsupp.mem_support_iff
AddMonoid.exponent_dvd_iff_forall_nsmul_eq_zero
Nat.ordProj_dvd
addOrderOf_eq_prime_pow
mul_nsmul
pos
mul_assoc
pow_succ
mul_comm
AddMonoid.exponent_nsmul_eq_zero
exists_orderOf_eq_pow_factorization_exponent 📖mathematicalNat.PrimeorderOf
Monoid.toNatPow
Nat.instMonoid
DFunLike.coe
Finsupp
MulZeroClass.toZero
Nat.instMulZeroClass
Finsupp.instFunLike
Nat.factorization
Monoid.exponent
eq_or_ne
pow_zero
orderOf_one
Ne.bot_lt
bot_eq_zero
Nat.instCanonicallyOrderedAdd
Nat.factorization_zero
Finsupp.zero_apply
LT.lt.not_ge
one_lt
mul_le_iff_le_one_left
IsOrderedRing.toMulPosMono
IsStrictOrderedRing.toIsOrderedRing
Nat.instIsStrictOrderedRing
MulPosStrictMono.toMulPosReflectLE
IsStrictOrderedRing.toMulPosStrictMono
Nat.dvd_of_mem_primeFactors
Finsupp.mem_support_iff
Monoid.exponent_dvd_iff_forall_pow_eq_one
Nat.ordProj_dvd
orderOf_eq_prime_pow
pow_mul
pos
mul_assoc
pow_succ
mul_comm
Monoid.pow_exponent_eq_one

Subgroup

Theorems

NameKindAssumesProvesValidatesDepends On
exponent_toSubmonoid 📖mathematicalMonoid.exponent
Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
SetLike.instMembership
Submonoid.instSetLike
toSubmonoid
Submonoid.toMonoid
Subgroup
instSetLike
toGroup
Monoid.exponent_eq_of_mulEquiv
exponent_top 📖mathematicalMonoid.exponent
Subgroup
SetLike.instMembership
instSetLike
Top.top
instTop
DivInvMonoid.toMonoid
Group.toDivInvMonoid
toGroup
Monoid.exponent_eq_of_mulEquiv
pow_exponent_eq_one 📖mathematicalSubgroup
SetLike.instMembership
instSetLike
Monoid.toNatPow
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Monoid.exponent
toGroup
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
Submonoid.pow_exponent_eq_one
exponent_toSubmonoid

Submonoid

Theorems

NameKindAssumesProvesValidatesDepends On
exponent_top 📖mathematicalMonoid.exponent
Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
Top.top
instTop
toMonoid
Monoid.exponent_eq_of_mulEquiv
pow_exponent_eq_one 📖mathematicalSubmonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
Monoid.toNatPow
Monoid.exponent
toMonoid
MulOne.toOne
MulOneClass.toMulOne
Monoid.pow_exponent_eq_one
pow_mem
instSubmonoidClass
OneMemClass.coe_eq_one
SubmonoidClass.mk_pow

(root)

Definitions

NameCategoryTheorems
addCommMonoidOfExponentTwo 📖CompOp
commMonoidOfExponentTwo 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
addOrderOf_eq_two_iff 📖mathematicalAddMonoid.exponentaddOrderOfaddOrderOf_zero
Nat.instAtLeastTwoHAddOfNat
OfNat.one_ne_ofNat
Nat.instCharZero
addOrderOf_eq_prime
Nat.fact_prime_two
AddMonoid.exponent_nsmul_eq_zero
add_comm_of_exponent_two 📖mathematicalAddMonoid.exponentAddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommute.of_addOrderOf_dvd_two
AddMonoid.addOrder_dvd_exponent
add_notMem_of_addOrderOf_eq_two 📖mathematicaladdOrderOf
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Set
Set.instMembership
Set.instInsert
Set.instSingletonSet
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
Set.mem_insert_iff
add_eq_left
AddLeftCancelSemigroup.toIsLeftCancelAdd
add_eq_right
AddRightCancelSemigroup.toIsRightCancelAdd
Set.mem_singleton_iff
add_eq_zero_iff_eq_neg
neg_eq_self_of_addOrderOf_eq_two
addOrderOf_zero
Nat.instAtLeastTwoHAddOfNat
OfNat.one_ne_ofNat
Nat.instCharZero
add_notMem_of_exponent_two 📖mathematicalAddMonoid.exponent
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Set
Set.instMembership
Set.instInsert
Set.instSingletonSet
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
add_notMem_of_addOrderOf_eq_two
addOrderOf_eq_prime
Nat.fact_prime_two
AddMonoid.exponent_nsmul_eq_zero
inv_eq_self_of_exponent_two 📖mathematicalMonoid.exponent
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
inv_eq_of_mul_eq_one_left
Monoid.pow_exponent_eq_one
pow_two
inv_eq_self_of_orderOf_eq_two 📖mathematicalorderOf
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
inv_eq_of_mul_eq_one_left
pow_orderOf_eq_one
pow_two
mul_comm_of_exponent_two 📖mathematicalMonoid.exponentMulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Commute.of_orderOf_dvd_two
Monoid.order_dvd_exponent
mul_notMem_of_exponent_two 📖mathematicalMonoid.exponent
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Set
Set.instMembership
Set.instInsert
Set.instSingletonSet
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
mul_notMem_of_orderOf_eq_two
orderOf_eq_prime
Nat.fact_prime_two
Monoid.pow_exponent_eq_one
mul_notMem_of_orderOf_eq_two 📖mathematicalorderOf
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Set
Set.instMembership
Set.instInsert
Set.instSingletonSet
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
LeftCancelSemigroup.toIsLeftCancelMul
RightCancelSemigroup.toIsRightCancelMul
inv_eq_self_of_orderOf_eq_two
orderOf_one
Nat.instCharZero
Nat.instAtLeastTwoHAddOfNat
neg_eq_self_of_addOrderOf_eq_two 📖mathematicaladdOrderOf
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
neg_eq_of_add_eq_zero_left
addOrderOf_nsmul_eq_zero
two_nsmul
neg_eq_self_of_exponent_two 📖mathematicalAddMonoid.exponent
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
neg_eq_of_add_eq_zero_left
AddMonoid.exponent_nsmul_eq_zero
two_nsmul
orderOf_eq_two_iff 📖mathematicalMonoid.exponentorderOforderOf_one
Nat.instCharZero
Nat.instAtLeastTwoHAddOfNat
orderOf_eq_prime
Nat.fact_prime_two
Monoid.pow_exponent_eq_one

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