Documentation Verification Report

PrincipalIdealDomain

πŸ“ Source: Mathlib/RingTheory/PrincipalIdealDomain.lean

Statistics

MetricCount
DefinitionsnonPrincipals, IsBezout, gcd, toGCDDomain, nonPrincipals
5
TheoremsisPrincipalIdealRing, to_principal_ideal_domain, of_comap, exists_maximal_not_isPrincipal, nonPrincipals_eq_empty_iff, nonPrincipals_zorn, span_singleton_generator, coprime_iff_not_dvd, coprime_pow_of_not_dvd, dvd_iff_not_isCoprime, isCoprime_or_dvd, associated_gcd_gcd, dvd_gcd, gcd_dvd_left, gcd_dvd_right, gcd_eq_sum, isPrincipal_of_FG, nonemptyGCDMonoid, of_isPrincipalIdealRing, span_gcd, span_gcd_eq_span_gcd, span_pair_isPrincipal, isPrincipalIdealRing, to_maximal_ideal, of_isNoetherianRing_of_isBezout, of_surjective, isCoprime, coprime_iff_not_dvd, factors_spec, isMaximal_of_irreducible, isNoetherianRing, mem_submonoid_of_factors_subset_of_units_subset, ne_zero_of_mem_factors, ringHom_mem_submonoid_of_factors_subset_of_units_subset, to_uniqueFactorizationMonoid, associated_generator_span_self, dvd_generator_span_iff, eq_bot_iff_generator_eq_zero, fg, generator_map_dvd_of_mem, generator_mem, generator_submoduleImage_dvd_of_mem, map, map_ringHom, mem_iff_eq_smul_generator, mem_iff_generator_dvd, of_comap, prime_generator_of_isPrime, bot_isPrincipal, dvd_or_isCoprime, exists_associated_pow_of_associated_pow_mul, exists_associated_pow_of_mul_eq_pow', exists_gcd_eq_mul_add_mul, gcd_dvd_iff_exists, gcd_isUnit_iff, isCoprime_of_dvd, isCoprime_of_irreducible_dvd, isCoprime_of_prime_dvd, isPrincipalIdealRing_pi_iff, isPrincipalIdealRing_prod_iff, isRelPrime_iff_isCoprime, mod_mem_iff, nonPrincipals_def, nonPrincipals_eq_empty_iff, nonPrincipals_zorn, span_gcd, span_singleton_inf_span_singleton, top_isPrincipal
68
Total73

DivisionSemiring

Theorems

NameKindAssumesProvesValidatesDepends On
isPrincipalIdealRing πŸ“–mathematicalβ€”IsPrincipalIdealRing
toSemiring
β€”Ideal.eq_bot_or_top
bot_isPrincipal
top_isPrincipal

EuclideanDomain

Theorems

NameKindAssumesProvesValidatesDepends On
to_principal_ideal_domain πŸ“–mathematicalβ€”IsPrincipalIdealRing
CommSemiring.toSemiring
CommRing.toCommSemiring
toCommRing
β€”r_wellFounded
WellFounded.min_mem
Submodule.ext
Ideal.mem_span_singleton
dvd_add
Distrib.leftDistribClass
dvd_mul_right
WellFounded.not_lt_min
mod_lt
mod_mem_iff
div_add_mod
Ideal.mul_mem_right
Ideal.instIsTwoSided_1
Submodule.bot_coe
Submodule.span_eq
Submodule.mem_bot
by_contra
Ideal.zero_mem

Ideal

Definitions

NameCategoryTheorems
nonPrincipals πŸ“–CompOp
1 mathmath: nonPrincipals_eq_empty_iff

Theorems

NameKindAssumesProvesValidatesDepends On
exists_maximal_not_isPrincipal πŸ“–mathematicalIsPrincipalIdealRingMaximal
Ideal
Preorder.toLE
PartialOrder.toPreorder
Submodule.instPartialOrder
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
Submodule.IsPrincipal
β€”zorn_leβ‚€
nonPrincipals_zorn
nonPrincipals_eq_empty_iff πŸ“–mathematicalβ€”nonPrincipals
Set
Ideal
Set.instEmptyCollection
IsPrincipalIdealRing
β€”β€”
nonPrincipals_zorn πŸ“–mathematicalIsPrincipalIdealRing
Set
Ideal
Set.instHasSubset
nonPrincipals
IsChain
Preorder.toLE
PartialOrder.toPreorder
Submodule.instPartialOrder
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
Set.instMembershipβ€”Set.nonempty_def
Submodule.mem_span_singleton_self
Submodule.mem_sSup_of_directed
IsChain.directedOn
instReflLe
le_antisymm
le_sSup
Set.not_nonempty_iff_eq_empty
instIsEmptyFalse
span_singleton_generator πŸ“–mathematicalβ€”span
Set
Set.instSingletonSet
Submodule.IsPrincipal.generator
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
β€”Submodule.IsPrincipal.principal

Ideal.IsPrincipal

Theorems

NameKindAssumesProvesValidatesDepends On
of_comap πŸ“–mathematicalDFunLike.coeSubmodule.IsPrincipal
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
β€”Ideal.map_comap_of_surjective
Submodule.IsPrincipal.map_ringHom

Irreducible

Theorems

NameKindAssumesProvesValidatesDepends On
coprime_iff_not_dvd πŸ“–mathematicalIrreducible
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
IsCoprime
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
CommRing.toNonUnitalCommRing
β€”isRelPrime_iff_isCoprime
IsBezout.span_pair_isPrincipal
isRelPrime_iff_not_dvd
coprime_pow_of_not_dvd πŸ“–mathematicalIrreducible
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
CommRing.toNonUnitalCommRing
IsCoprime
Monoid.toNatPow
β€”IsCoprime.pow_right
IsCoprime.symm
coprime_iff_not_dvd
dvd_iff_not_isCoprime πŸ“–mathematicalIrreducible
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
CommRing.toNonUnitalCommRing
IsCoprime
β€”iff_not_comm
coprime_iff_not_dvd
isCoprime_or_dvd πŸ“–mathematicalIrreducible
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
IsCoprime
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
CommRing.toNonUnitalCommRing
β€”dvd_iff_not_isCoprime
em

IsBezout

Definitions

NameCategoryTheorems
gcd πŸ“–CompOp
7 mathmath: gcd_dvd_left, dvd_gcd, associated_gcd_gcd, span_gcd_eq_span_gcd, gcd_dvd_right, span_gcd, gcd_eq_sum
toGCDDomain πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
associated_gcd_gcd πŸ“–mathematicalβ€”Associated
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
gcd
CommRing.toRing
GCDMonoid.gcd
CommSemiring.toCommMonoidWithZero
β€”gcd_greatest_associated
gcd_dvd_left
gcd_dvd_right
dvd_gcd
dvd_gcd πŸ“–mathematicalsemigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
CommRing.toNonUnitalCommRing
gcd
CommRing.toRing
β€”Ideal.span_singleton_le_span_singleton
span_gcd
Ideal.span_insert
sup_le_iff
gcd_dvd_left πŸ“–mathematicalβ€”semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
CommRing.toNonUnitalCommRing
gcd
CommRing.toRing
β€”Submodule.IsPrincipal.mem_iff_generator_dvd
Ideal.subset_span
gcd_dvd_right πŸ“–mathematicalβ€”semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
CommRing.toNonUnitalCommRing
gcd
CommRing.toRing
β€”Submodule.IsPrincipal.mem_iff_generator_dvd
Ideal.subset_span
gcd_eq_sum πŸ“–mathematicalβ€”Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Distrib.toMul
gcd
CommRing.toRing
β€”Ideal.mem_span_pair
span_gcd
Ideal.subset_span
isPrincipal_of_FG πŸ“–mathematicalIdeal.FGSubmodule.IsPrincipal
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
β€”β€”
nonemptyGCDMonoid πŸ“–mathematicalβ€”GCDMonoid
CommSemiring.toCommMonoidWithZero
CommRing.toCommSemiring
β€”β€”
of_isPrincipalIdealRing πŸ“–mathematicalβ€”IsBezoutβ€”IsPrincipalIdealRing.principal
span_gcd πŸ“–mathematicalβ€”Ideal.span
Ring.toSemiring
Set
Set.instSingletonSet
gcd
Set.instInsert
β€”Ideal.span_singleton_generator
span_gcd_eq_span_gcd πŸ“–mathematicalβ€”Ideal.span
CommSemiring.toSemiring
CommRing.toCommSemiring
Set
Set.instSingletonSet
GCDMonoid.gcd
CommSemiring.toCommMonoidWithZero
gcd
CommRing.toRing
span_pair_isPrincipal
β€”span_pair_isPrincipal
Ideal.span_singleton_eq_span_singleton
associated_of_dvd_dvd
IsCancelMulZero.toIsLeftCancelMulZero
IsDomain.toIsCancelMulZero
dvd_gcd
GCDMonoid.gcd_dvd_left
GCDMonoid.gcd_dvd_right
GCDMonoid.dvd_gcd
gcd_dvd_left
gcd_dvd_right
span_pair_isPrincipal πŸ“–mathematicalβ€”Submodule.IsPrincipal
Ring.toSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
Ideal.span
Set
Set.instInsert
Set.instSingletonSet
β€”isPrincipal_of_FG
Finset.coe_insert
Finset.coe_singleton

IsField

Theorems

NameKindAssumesProvesValidatesDepends On
isPrincipalIdealRing πŸ“–mathematicalIsField
Ring.toSemiring
IsPrincipalIdealRingβ€”EuclideanDomain.to_principal_ideal_domain

IsPrime

Theorems

NameKindAssumesProvesValidatesDepends On
to_maximal_ideal πŸ“–mathematicalβ€”Ideal.IsMaximal
CommSemiring.toSemiring
CommRing.toCommSemiring
β€”Ideal.isMaximal_iff
Ideal.ne_top_iff_one
Ideal.IsPrime.ne_top'
IsPrincipalIdealRing.principal
Submodule.IsPrincipal.mem_iff_generator_dvd
Submodule.IsPrincipal.generator_mem
Ideal.IsPrime.mem_or_mem
Ideal.span_singleton_generator
Ideal.span_le
Set.singleton_subset_iff
Submodule.IsPrincipal.eq_bot_iff_generator_eq_zero
Ideal.mul_mem_right
Ideal.instIsTwoSided_1
mul_right_inj'
IsCancelMulZero.toIsLeftCancelMulZero
IsDomain.toIsCancelMulZero
mul_left_comm
mul_one

IsPrincipalIdealRing

Theorems

NameKindAssumesProvesValidatesDepends On
of_isNoetherianRing_of_isBezout πŸ“–mathematicalβ€”IsPrincipalIdealRingβ€”IsBezout.isPrincipal_of_FG
IsNoetherian.noetherian
of_surjective πŸ“–mathematicalDFunLike.coeIsPrincipalIdealRingβ€”Ideal.IsPrincipal.of_comap
principal

IsRelPrime

Theorems

NameKindAssumesProvesValidatesDepends On
isCoprime πŸ“–mathematicalIsRelPrime
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
IsCoprimeβ€”Ideal.isCoprime_span_singleton_iff
Ideal.isCoprime_iff_sup_eq
Ideal.span_union
Set.singleton_union
IsBezout.span_gcd
Ideal.span_singleton_eq_top
IsBezout.gcd_dvd_left
IsBezout.gcd_dvd_right

Prime

Theorems

NameKindAssumesProvesValidatesDepends On
coprime_iff_not_dvd πŸ“–mathematicalPrime
CommSemiring.toCommMonoidWithZero
CommRing.toCommSemiring
IsCoprime
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
CommRing.toNonUnitalCommRing
β€”Irreducible.coprime_iff_not_dvd
irreducible
IsDomain.toIsCancelMulZero

PrincipalIdealRing

Theorems

NameKindAssumesProvesValidatesDepends On
factors_spec πŸ“–mathematicalβ€”Irreducible
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
Associated
Multiset.prod
CommRing.toCommMonoid
factors
β€”WfDvdMonoid.exists_factors
UniqueFactorizationMonoid.toIsWellFounded
ufm_of_decomposition_of_wfDvdMonoid
IsDomain.toIsCancelMulZero
IsNoetherianRing.wfDvdMonoid
isNoetherianRing
instDecompositionMonoidOfNonemptyGCDMonoid
IsBezout.nonemptyGCDMonoid
IsBezout.of_isPrincipalIdealRing
isMaximal_of_irreducible πŸ“–mathematicalIrreducible
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
Ideal.IsMaximal
Submodule.span
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
Set
Set.instSingletonSet
β€”Ideal.span_singleton_eq_top
Irreducible.not_isUnit
IsPrincipalIdealRing.principal
Ideal.submodule_span_eq
Ideal.span_singleton_le_span_singleton
le_of_lt
of_irreducible_mul
IsUnit.mul_right_dvd
dvd_refl
not_le_of_gt
isNoetherianRing πŸ“–mathematicalβ€”IsNoetherianRingβ€”Submodule.IsPrincipal.fg
IsPrincipalIdealRing.principal
mem_submonoid_of_factors_subset_of_units_subset πŸ“–β€”Submonoid
MulZeroOneClass.toMulOneClass
NonAssocSemiring.toMulZeroOneClass
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
SetLike.instMembership
Submonoid.instSetLike
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
β€”β€”factors_spec
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
Submonoid.instSubmonoidClass
multiset_prod_mem
ne_zero_of_mem_factors πŸ“–β€”Multiset
Multiset.instMembership
factors
β€”β€”Irreducible.ne_zero
factors_spec
ringHom_mem_submonoid_of_factors_subset_of_units_subset πŸ“–β€”Submonoid
MulZeroOneClass.toMulOneClass
NonAssocSemiring.toMulZeroOneClass
SetLike.instMembership
Submonoid.instSetLike
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
β€”β€”mem_submonoid_of_factors_subset_of_units_subset
MonoidHom.instMonoidHomClass
to_uniqueFactorizationMonoid πŸ“–mathematicalβ€”UniqueFactorizationMonoid
CommSemiring.toCommMonoidWithZero
CommRing.toCommSemiring
β€”IsNoetherianRing.wfDvdMonoid
isNoetherianRing
IsDomain.toIsCancelMulZero
irreducible_iff_prime
instDecompositionMonoidOfNonemptyGCDMonoid
IsBezout.nonemptyGCDMonoid
IsBezout.of_isPrincipalIdealRing

Submodule.IsPrincipal

Theorems

NameKindAssumesProvesValidatesDepends On
associated_generator_span_self πŸ“–mathematicalβ€”Associated
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
generator
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
Ideal.span
Set
Set.instSingletonSet
instIsPrincipalSpanSingletonSet
β€”instIsPrincipalSpanSingletonSet
Ideal.span_singleton_eq_span_singleton
Ideal.span_singleton_generator
dvd_generator_span_iff πŸ“–mathematicalβ€”semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
CommSemiring.toNonUnitalCommSemiring
generator
CommSemiring.toSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
Ideal.span
β€”Dvd.dvd.trans
mem_iff_generator_dvd
Ideal.subset_span
Submodule.span_induction
dvd_zero
dvd_add
Distrib.leftDistribClass
Dvd.dvd.mul_left
generator_mem
eq_bot_iff_generator_eq_zero πŸ“–mathematicalβ€”Bot.bot
Submodule
Submodule.instBot
generator
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
β€”Submodule.span_singleton_eq_bot
span_singleton_generator
fg πŸ“–mathematicalβ€”Submodule.FGβ€”Finset.coe_singleton
span_singleton_generator
generator_map_dvd_of_mem πŸ“–mathematicalSubmodule
CommSemiring.toSemiring
SetLike.instMembership
Submodule.setLike
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
CommSemiring.toNonUnitalCommSemiring
generator
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
Submodule.map
RingHom.id
RingHomSurjective.ids
DFunLike.coe
LinearMap
LinearMap.instFunLike
β€”RingHomSurjective.ids
mem_iff_generator_dvd
Submodule.mem_map
generator_mem πŸ“–mathematicalβ€”Submodule
SetLike.instMembership
Submodule.setLike
generator
β€”Submodule.subset_span
Set.mem_singleton
span_singleton_generator
generator_submoduleImage_dvd_of_mem πŸ“–mathematicalSubmodule
CommSemiring.toSemiring
Preorder.toLE
PartialOrder.toPreorder
Submodule.instPartialOrder
SetLike.instMembership
Submodule.setLike
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
CommSemiring.toNonUnitalCommSemiring
generator
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
LinearMap.submoduleImage
DFunLike.coe
LinearMap
RingHom.id
Submodule.addCommMonoid
Submodule.module
LinearMap.instFunLike
β€”mem_iff_generator_dvd
LinearMap.mem_submoduleImage_of_le
map πŸ“–mathematicalβ€”Submodule.IsPrincipal
Submodule.map
RingHom.id
Semiring.toNonAssocSemiring
RingHomSurjective.ids
β€”RingHomSurjective.ids
Set.image_singleton
Submodule.map_span
span_singleton_generator
map_ringHom πŸ“–mathematicalβ€”Submodule.IsPrincipal
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
Ideal.map
β€”Ideal.submodule_span_eq
Set.image_singleton
Ideal.map_span
Ideal.span_singleton_generator
mem_iff_eq_smul_generator πŸ“–mathematicalβ€”Submodule
SetLike.instMembership
Submodule.setLike
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Module.toDistribMulAction
generator
β€”span_singleton_generator
mem_iff_generator_dvd πŸ“–mathematicalβ€”Ideal
CommSemiring.toSemiring
SetLike.instMembership
Submodule.setLike
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
CommSemiring.toNonUnitalCommSemiring
generator
β€”mem_iff_eq_smul_generator
mul_comm
of_comap πŸ“–mathematicalDFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
LinearMap.instFunLike
Submodule.IsPrincipalβ€”RingHomSurjective.ids
Submodule.map_comap_eq_of_surjective
map
prime_generator_of_isPrime πŸ“–mathematicalβ€”Prime
CommSemiring.toCommMonoidWithZero
generator
CommSemiring.toSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
β€”eq_bot_iff_generator_eq_zero
Ideal.IsPrime.ne_top
Ideal.eq_top_of_isUnit_mem
generator_mem
mem_iff_generator_dvd
Ideal.IsPrime.mem_or_mem'

(root)

Definitions

NameCategoryTheorems
IsBezout πŸ“–CompData
6 mathmath: ValuationRing.TFAE, ValuationRing.instIsBezout, IsBezout.iff_span_pair_isPrincipal, Function.Surjective.isBezout, ValuationRing.iff_local_bezout_domain, IsBezout.of_isPrincipalIdealRing
nonPrincipals πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
bot_isPrincipal πŸ“–mathematicalβ€”Submodule.IsPrincipal
Bot.bot
Submodule
Submodule.instBot
β€”Submodule.span_zero_singleton
dvd_or_isCoprime πŸ“–mathematicalIrreducible
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
CommRing.toNonUnitalCommRing
IsCoprime
β€”IsRelPrime.isCoprime
IsBezout.span_pair_isPrincipal
Irreducible.dvd_or_isRelPrime
exists_associated_pow_of_associated_pow_mul πŸ“–β€”IsCoprime
CommRing.toCommSemiring
Associated
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
Monoid.toNatPow
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
β€”β€”Associated.symm
exists_associated_pow_of_mul_eq_pow'
isCoprime_mul_unit_right_right
Units.isUnit
mul_assoc
exists_associated_pow_of_mul_eq_pow' πŸ“–mathematicalIsCoprime
CommRing.toCommSemiring
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
Associatedβ€”exists_associated_pow_of_mul_eq_pow
gcd_isUnit_iff
exists_gcd_eq_mul_add_mul πŸ“–mathematicalβ€”GCDMonoid.gcd
CommSemiring.toCommMonoidWithZero
CommRing.toCommSemiring
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Distrib.toMul
β€”gcd_dvd_iff_exists
dvd_refl
gcd_dvd_iff_exists πŸ“–mathematicalβ€”semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
CommRing.toNonUnitalCommRing
GCDMonoid.gcd
CommSemiring.toCommMonoidWithZero
CommRing.toCommSemiring
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
Distrib.toMul
β€”mul_comm
gcd_isUnit_iff πŸ“–mathematicalβ€”IsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
GCDMonoid.gcd
CommSemiring.toCommMonoidWithZero
IsCoprime
β€”IsCoprime.eq_1
Ideal.mem_span_pair
span_gcd
Ideal.span_singleton_eq_top
Ideal.eq_top_iff_one
isCoprime_of_dvd πŸ“–mathematicalMulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
IsCoprime
CommRing.toCommSemiring
β€”IsRelPrime.isCoprime
IsBezout.span_pair_isPrincipal
isRelPrime_of_no_nonunits_factors
isCoprime_of_irreducible_dvd πŸ“–mathematicalMulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
IsCoprime
CommRing.toCommSemiring
β€”IsRelPrime.isCoprime
IsBezout.span_pair_isPrincipal
IsBezout.of_isPrincipalIdealRing
WfDvdMonoid.isRelPrime_of_no_irreducible_factors
UniqueFactorizationMonoid.toIsWellFounded
PrincipalIdealRing.to_uniqueFactorizationMonoid
isCoprime_of_prime_dvd πŸ“–mathematicalMulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
NonUnitalCommRing.toNonUnitalCommSemiring
IsCoprime
CommRing.toCommSemiring
β€”isCoprime_of_irreducible_dvd
Irreducible.prime
UniqueFactorizationMonoid.instDecompositionMonoid
PrincipalIdealRing.to_uniqueFactorizationMonoid
isPrincipalIdealRing_pi_iff πŸ“–mathematicalβ€”IsPrincipalIdealRing
Pi.semiring
β€”IsPrincipalIdealRing.of_surjective
RingHom.instRingHomClass
Function.surjective_eval
Zero.instNonempty
Ideal.instIsPrincipalIdealRingForallOfFinite
isPrincipalIdealRing_prod_iff πŸ“–mathematicalβ€”IsPrincipalIdealRing
Prod.instSemiring
β€”IsPrincipalIdealRing.of_surjective
RingHom.instRingHomClass
Prod.fst_surjective
Zero.instNonempty
Prod.snd_surjective
instIsPrincipalIdealRingProd
isRelPrime_iff_isCoprime πŸ“–mathematicalβ€”IsRelPrime
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
IsCoprime
β€”IsRelPrime.isCoprime
IsCoprime.isRelPrime
mod_mem_iff πŸ“–mathematicalIdeal
CommSemiring.toSemiring
CommRing.toCommSemiring
EuclideanDomain.toCommRing
SetLike.instMembership
Submodule.setLike
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
EuclideanDomain.instModβ€”Ideal.add_mem
Ideal.mul_mem_right
Ideal.instIsTwoSided_1
EuclideanDomain.div_add_mod
Ideal.sub_mem
EuclideanDomain.mod_eq_sub_mul_div
nonPrincipals_def πŸ“–mathematicalβ€”Ideal
CommSemiring.toSemiring
CommRing.toCommSemiring
Set
Set.instMembership
setOf
Submodule.IsPrincipal
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
β€”β€”
nonPrincipals_eq_empty_iff πŸ“–mathematicalβ€”setOf
Ideal
CommSemiring.toSemiring
CommRing.toCommSemiring
Submodule.IsPrincipal
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
Set
Set.instEmptyCollection
IsPrincipalIdealRing
β€”β€”
nonPrincipals_zorn πŸ“–β€”Set
Ideal
CommSemiring.toSemiring
CommRing.toCommSemiring
Set.instHasSubset
setOf
Submodule.IsPrincipal
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
IsChain
Preorder.toLE
PartialOrder.toPreorder
Submodule.instPartialOrder
Set.instMembership
β€”β€”Submodule.mem_span_singleton_self
Submodule.mem_sSup_of_directed
IsChain.directedOn
instReflLe
le_antisymm
le_sSup
span_gcd πŸ“–mathematicalβ€”Ideal.span
CommSemiring.toSemiring
CommRing.toCommSemiring
Set
Set.instSingletonSet
GCDMonoid.gcd
CommSemiring.toCommMonoidWithZero
Set.instInsert
β€”IsBezout.span_pair_isPrincipal
IsBezout.span_gcd
IsBezout.span_gcd_eq_span_gcd
span_singleton_inf_span_singleton πŸ“–mathematicalβ€”Ideal
CommSemiring.toSemiring
CommRing.toCommSemiring
EuclideanDomain.toCommRing
Submodule.instMin
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
Ideal.span
Set
Set.instSingletonSet
GCDMonoid.lcm
CommSemiring.toCommMonoidWithZero
β€”Ideal.ext_iff
Ideal.mem_inf
lcm_dvd_iff
top_isPrincipal πŸ“–mathematicalβ€”Submodule.IsPrincipal
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
Top.top
Submodule
Submodule.instTop
β€”Ideal.span_singleton_one

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