Documentation Verification Report

Basic

πŸ“ Source: Mathlib/RingTheory/Algebraic/Basic.lean

Statistics

MetricCount
DefinitionsalgEquivEquivAlgHom
1
TheoremsisAlgebraic, isAlgebraic_iff, algEquivEquivAlgHom_apply, algEquivEquivAlgHom_symm_apply, algHom_bijective, algHom_bijectiveβ‚‚, bijective_of_isScalarTower, bijective_of_isScalarTower', exists_smul_eq_mul, extendScalars, faithfulSMul_tower_top, injective_tower_top, nontrivial, of_injective, of_ringHom_of_comp_eq, ringHom_of_comp_eq, tower_bot, tower_bot_of_injective, tower_top, infinite, of_ringHom_of_comp_eq, ringHom_of_comp_eq, injective_of_transcendental, isAlgebraic_of_not_injective, isAlgebraic_ringHom_iff_of_comp_eq, transcendental_of_subsingleton, transcendental_ringHom_iff_of_comp_eq, algHom, algebraMap, exists_nonzero_coeff_and_aeval_eq_zero, exists_nonzero_dvd, exists_nonzero_eq_adjoin_mul, exists_smul_eq_mul, extendScalars, inv, invOf, invOf_iff, inv_iff, nontrivial, of_aeval, of_aeval_of_transcendental, of_pow, of_ringHom_of_comp_eq, ringHom_of_comp_eq, tower_top, tower_top_of_subalgebra_le, isAlgebraic, transcendental, transcendental_X, algebra_isAlgebraic_bot_left_iff, algebra_isAlgebraic_bot_right, algebra_isAlgebraic_of_algebra_isAlgebraic_bot_left, inv_mem_of_algebraic, inv_mem_of_root_of_coeff_zero_ne_zero, isAlgebraic_bot_iff, isAlgebraic_iff_isAlgebraic_val, isAlgebraic_of_isAlgebraic_bot, isField_of_algebraic, aeval, aeval_of_transcendental, infinite, of_aeval, of_ringHom_of_comp_eq, of_tower_top, of_tower_top_of_subalgebra_le, pow, restrictScalars, ringHom_of_comp_eq, inv_eq_of_aeval_divX_ne_zero, inv_eq_of_root_of_coeff_zero_ne_zero, isAlgebraic_algHom_iff, isAlgebraic_algebraMap, isAlgebraic_algebraMap_iff, isAlgebraic_iff_not_injective, isAlgebraic_int, isAlgebraic_nat, isAlgebraic_of_mem_rootSet, isAlgebraic_one, isAlgebraic_rat, isAlgebraic_ringHom_iff_of_comp_eq, isAlgebraic_zero, is_transcendental_of_subsingleton, transcendental_algebraMap_iff, transcendental_iff_injective, transcendental_iff_ker_eq_bot, transcendental_ringHom_iff_of_comp_eq
86
Total87

AlgEquiv

Theorems

NameKindAssumesProvesValidatesDepends On
isAlgebraic πŸ“–mathematicalβ€”Algebra.IsAlgebraicβ€”Algebra.IsAlgebraic.of_injective
injective
isAlgebraic_iff πŸ“–mathematicalβ€”Algebra.IsAlgebraicβ€”isAlgebraic

Algebra

Theorems

NameKindAssumesProvesValidatesDepends On
injective_of_transcendental πŸ“–mathematicalβ€”DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
β€”Mathlib.Tactic.Contrapose.contraposeβ‚‚
isAlgebraic_of_not_injective
transcendental_iff_not_isAlgebraic
isAlgebraic_of_not_injective πŸ“–mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
IsAlgebraicβ€”isAlgebraic_iff_not_injective
Polynomial.aeval_C
Polynomial.C_injective
isAlgebraic_ringHom_iff_of_comp_eq πŸ“–mathematicalRingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
EquivLike.toFunLike
RingEquivClass.toRingHomClass
IsAlgebraicβ€”RingEquivClass.toRingHomClass
IsAlgebraic.of_ringHom_of_comp_eq
EquivLike.surjective
EquivLike.injective
IsAlgebraic.ringHom_of_comp_eq
transcendental_of_subsingleton πŸ“–mathematicalβ€”Transcendentalβ€”is_transcendental_of_subsingleton
transcendental_ringHom_iff_of_comp_eq πŸ“–mathematicalRingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
EquivLike.toFunLike
RingEquivClass.toRingHomClass
Transcendentalβ€”RingEquivClass.toRingHomClass
isAlgebraic_ringHom_iff_of_comp_eq

Algebra.IsAlgebraic

Definitions

NameCategoryTheorems
algEquivEquivAlgHom πŸ“–CompOp
2 mathmath: algEquivEquivAlgHom_apply, algEquivEquivAlgHom_symm_apply

Theorems

NameKindAssumesProvesValidatesDepends On
algEquivEquivAlgHom_apply πŸ“–mathematicalβ€”DFunLike.coe
MulEquiv
AlgEquiv
CommRing.toCommSemiring
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
AlgHom
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
AlgEquiv.aut
AlgHom.End
EquivLike.toFunLike
MulEquiv.instEquivLike
algEquivEquivAlgHom
AlgEquiv.toAlgHom
β€”β€”
algEquivEquivAlgHom_symm_apply πŸ“–mathematicalβ€”DFunLike.coe
MulEquiv
AlgHom
CommRing.toCommSemiring
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
AlgEquiv
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
AlgHom.End
DivInvMonoid.toMonoid
Group.toDivInvMonoid
AlgEquiv.aut
EquivLike.toFunLike
MulEquiv.instEquivLike
MulEquiv.symm
algEquivEquivAlgHom
AlgEquiv.ofBijective
algHom_bijective
β€”β€”
algHom_bijective πŸ“–mathematicalβ€”Function.Bijective
DFunLike.coe
AlgHom
CommRing.toCommSemiring
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
AlgHom.funLike
β€”RingHom.injective
DivisionRing.isSimpleRing
IsSimpleRing.instNontrivial
isAlgebraic
instIsDomain
Polynomial.rootSet_maps_to'
Finite.injective_iff_surjective
Finite.of_fintype
Polynomial.mem_rootSet
algHom_bijectiveβ‚‚ πŸ“–mathematicalβ€”Function.Bijective
DFunLike.coe
AlgHom
CommRing.toCommSemiring
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
DivisionRing.toDivisionSemiring
AlgHom.funLike
β€”Function.Injective.bijectiveβ‚‚_of_surjective
RingHom.injective
DivisionRing.isSimpleRing
IsSimpleRing.instNontrivial
algHom_bijective
bijective_of_isScalarTower πŸ“–mathematicalβ€”Function.Bijective
DFunLike.coe
AlgHom
CommRing.toCommSemiring
DivisionSemiring.toSemiring
DivisionRing.toDivisionSemiring
Semifield.toDivisionSemiring
Field.toSemifield
AlgHom.funLike
β€”algHom_bijectiveβ‚‚
bijective_of_isScalarTower' πŸ“–mathematicalβ€”Function.Bijective
DFunLike.coe
AlgHom
CommRing.toCommSemiring
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
AlgHom.funLike
β€”algHom_bijectiveβ‚‚
exists_smul_eq_mul πŸ“–mathematicalβ€”Algebra.toSMul
CommRing.toCommSemiring
Ring.toSemiring
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”IsAlgebraic.exists_smul_eq_mul
isAlgebraic
mem_nonZeroDivisors_of_ne_zero
extendScalars πŸ“–mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
algebraMap
Algebra.IsAlgebraicβ€”IsAlgebraic.extendScalars
isAlgebraic
faithfulSMul_tower_top πŸ“–mathematicalβ€”FaithfulSMul
Algebra.toSMul
CommRing.toCommSemiring
Ring.toSemiring
β€”faithfulSMul_iff_algebraMap_injective
injective_tower_top
injective_tower_top πŸ“–β€”DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
β€”β€”injective_iff_map_eq_zero
RingHomClass.toAddMonoidHomClass
RingHom.instRingHomClass
of_not_not
IsAlgebraic.exists_nonzero_dvd
isAlgebraic
mem_nonZeroDivisors_of_ne_zero
zero_dvd_iff
IsScalarTower.algebraMap_apply
map_dvd
NonUnitalRingHomClass.toMulHomClass
RingHomClass.toNonUnitalRingHomClass
map_zero
MonoidWithZeroHomClass.toZeroHomClass
RingHomClass.toMonoidWithZeroHomClass
nontrivial πŸ“–mathematicalβ€”Nontrivialβ€”IsAlgebraic.nontrivial
isAlgebraic
of_injective πŸ“–mathematicalDFunLike.coe
AlgHom
CommRing.toCommSemiring
Ring.toSemiring
AlgHom.funLike
Algebra.IsAlgebraicβ€”isAlgebraic_algHom_iff
isAlgebraic
of_ringHom_of_comp_eq πŸ“–mathematicalDFunLike.coe
RingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
Algebra.IsAlgebraicβ€”IsAlgebraic.of_ringHom_of_comp_eq
isAlgebraic
ringHom_of_comp_eq πŸ“–mathematicalDFunLike.coe
RingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
Algebra.IsAlgebraicβ€”IsAlgebraic.ringHom_of_comp_eq
isAlgebraic
tower_bot πŸ“–mathematicalβ€”Algebra.IsAlgebraic
DivisionRing.toRing
Field.toDivisionRing
β€”tower_bot_of_injective
RingHom.injective
DivisionRing.isSimpleRing
tower_bot_of_injective πŸ“–mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
Algebra.IsAlgebraic
CommRing.toRing
β€”isAlgebraic_algebraMap_iff
isAlgebraic
tower_top πŸ“–mathematicalβ€”Algebra.IsAlgebraic
Field.toCommRing
β€”extendScalars
RingHom.injective
DivisionRing.isSimpleRing
IsSimpleRing.instNontrivial

Algebra.Transcendental

Theorems

NameKindAssumesProvesValidatesDepends On
infinite πŸ“–mathematicalβ€”Infiniteβ€”Transcendental.infinite
of_ringHom_of_comp_eq πŸ“–mathematicalDFunLike.coe
RingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
Algebra.Transcendentalβ€”Algebra.transcendental_iff_not_isAlgebraic
Algebra.IsAlgebraic.ringHom_of_comp_eq
ringHom_of_comp_eq πŸ“–mathematicalDFunLike.coe
RingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
Algebra.Transcendentalβ€”Algebra.transcendental_iff_not_isAlgebraic
Algebra.IsAlgebraic.of_ringHom_of_comp_eq

IsAlgebraic

Theorems

NameKindAssumesProvesValidatesDepends On
algHom πŸ“–mathematicalIsAlgebraicDFunLike.coe
AlgHom
CommRing.toCommSemiring
Ring.toSemiring
AlgHom.funLike
β€”Polynomial.aeval_algHom
AlgHom.comp_apply
map_zero
MonoidWithZeroHomClass.toZeroHomClass
RingHomClass.toMonoidWithZeroHomClass
AlgHomClass.toRingHomClass
AlgHom.algHomClass
algebraMap πŸ“–mathematicalIsAlgebraic
CommRing.toRing
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
β€”Polynomial.aeval_algebraMap_apply
map_zero
MonoidWithZeroHomClass.toZeroHomClass
RingHomClass.toMonoidWithZeroHomClass
RingHom.instRingHomClass
exists_nonzero_coeff_and_aeval_eq_zero πŸ“–mathematicalIsAlgebraic
Submonoid
MulZeroOneClass.toMulOneClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
Ring.toSemiring
SetLike.instMembership
Submonoid.instSetLike
nonZeroDivisors
DFunLike.coe
AlgHom
Polynomial
CommSemiring.toSemiring
CommRing.toCommSemiring
Polynomial.semiring
Polynomial.algebraOfAlgebra
Algebra.id
AlgHom.funLike
Polynomial.aeval
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”Polynomial.exists_eq_pow_rootMultiplicity_mul_and_not_dvd
Polynomial.C_0
sub_zero
Submonoid.pow_mem
Polynomial.aeval_X_pow
map_mul
NonUnitalAlgSemiHomClass.toMulHomClass
AlgHom.instNonUnitalAlgHomClassOfAlgHomClass
AlgHom.algHomClass
Polynomial.X_pow_mul
exists_nonzero_dvd πŸ“–mathematicalIsAlgebraic
Submonoid
MulZeroOneClass.toMulOneClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
Ring.toSemiring
SetLike.instMembership
Submonoid.instSetLike
nonZeroDivisors
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
Semiring.toNonUnitalSemiring
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
algebraMap
β€”exists_nonzero_coeff_and_aeval_eq_zero
map_dvd
NonUnitalAlgSemiHomClass.toMulHomClass
AlgHom.instNonUnitalAlgHomClassOfAlgHomClass
AlgHom.algHomClass
Polynomial.X_dvd_sub_C
Polynomial.aeval_C
Polynomial.aeval_X
dvd_neg
zero_sub
map_sub
DistribMulActionSemiHomClass.toAddMonoidHomClass
NonUnitalAlgSemiHomClass.toDistribMulActionSemiHomClass
exists_nonzero_eq_adjoin_mul πŸ“–mathematicalIsAlgebraic
Submonoid
MulZeroOneClass.toMulOneClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
Ring.toSemiring
SetLike.instMembership
Submonoid.instSetLike
nonZeroDivisors
Subalgebra
CommRing.toCommSemiring
Subalgebra.instSetLike
Algebra.adjoin
Set
Set.instSingletonSet
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
algebraMap
β€”exists_nonzero_coeff_and_aeval_eq_zero
Polynomial.X_dvd_sub_C
Polynomial.aeval_mem_adjoin_singleton
neg_ne_zero
map_neg
RingHomClass.toAddMonoidHomClass
RingHom.instRingHomClass
Polynomial.aeval_C
Polynomial.aeval_X
map_mul
NonUnitalAlgSemiHomClass.toMulHomClass
AlgHom.instNonUnitalAlgHomClassOfAlgHomClass
AlgHom.algHomClass
zero_sub
map_sub
DistribMulActionSemiHomClass.toAddMonoidHomClass
NonUnitalAlgSemiHomClass.toDistribMulActionSemiHomClass
exists_smul_eq_mul πŸ“–mathematicalIsAlgebraic
Submonoid
MulZeroOneClass.toMulOneClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
Ring.toSemiring
SetLike.instMembership
Submonoid.instSetLike
nonZeroDivisors
Algebra.toSMul
CommRing.toCommSemiring
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”exists_nonzero_dvd
Algebra.smul_def
mul_assoc
extendScalars πŸ“–β€”DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
algebraMap
IsAlgebraic
β€”β€”Polynomial.degree_eq_bot
Polynomial.degree_map_eq_of_injective
Polynomial.aeval_map_algebraMap
inv πŸ“–mathematicalIsAlgebraic
DivisionRing.toRing
Field.toDivisionRing
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
Field.toSemifield
β€”inv_iff
invOf πŸ“–mathematicalIsAlgebraic
CommRing.toRing
Invertible.invOf
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
β€”Polynomial.aeval_def
Polynomial.evalβ‚‚_reverse_eq_zero_iff
invOf_iff πŸ“–mathematicalβ€”IsAlgebraic
CommRing.toRing
Invertible.invOf
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
β€”invOf
inv_iff πŸ“–mathematicalβ€”IsAlgebraic
DivisionRing.toRing
Field.toDivisionRing
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
Field.toSemifield
β€”inv_zero
invOf_iff
nontrivial πŸ“–mathematicalIsAlgebraicNontrivialβ€”Mathlib.Tactic.Contrapose.contrapose₁
is_transcendental_of_subsingleton
of_aeval πŸ“–β€”Submonoid
MulZeroOneClass.toMulOneClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
SetLike.instMembership
Submonoid.instSetLike
nonZeroDivisors
Polynomial.leadingCoeff
IsAlgebraic
DFunLike.coe
AlgHom
Polynomial
Polynomial.semiring
Ring.toSemiring
Polynomial.algebraOfAlgebra
Algebra.id
AlgHom.funLike
Polynomial.aeval
β€”β€”Polynomial.leadingCoeff_eq_zero
mul_right_mem_nonZeroDivisors_eq_zero_iff
pow_mem
Submonoid.instSubmonoidClass
Polynomial.coeff_comp_degree_mul_degree
Polynomial.aeval_comp
of_aeval_of_transcendental πŸ“–β€”IsAlgebraic
DFunLike.coe
AlgHom
Polynomial
CommSemiring.toSemiring
CommRing.toCommSemiring
Polynomial.semiring
Ring.toSemiring
Polynomial.algebraOfAlgebra
Algebra.id
AlgHom.funLike
Polynomial.aeval
Transcendental
Polynomial.ring
CommRing.toRing
β€”β€”Mathlib.Tactic.Contrapose.contrapose₁
Transcendental.aeval_of_transcendental
of_pow πŸ“–β€”IsAlgebraic
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
β€”β€”Polynomial.expand_ne_zero
Polynomial.expand_aeval
of_ringHom_of_comp_eq πŸ“–β€”IsAlgebraic
DFunLike.coe
RingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
β€”β€”Polynomial.map_surjective
Polynomial.map_zero
map_zero
MonoidWithZeroHomClass.toZeroHomClass
RingHomClass.toMonoidWithZeroHomClass
Polynomial.map_aeval_eq_aeval_map
ringHom_of_comp_eq πŸ“–β€”IsAlgebraic
DFunLike.coe
RingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
β€”β€”Polynomial.map_ne_zero_iff
Polynomial.map_aeval_eq_aeval_map
map_zero
MonoidWithZeroHomClass.toZeroHomClass
RingHomClass.toMonoidWithZeroHomClass
RingHom.instRingHomClass
tower_top πŸ“–β€”IsAlgebraic
Field.toCommRing
β€”β€”extendScalars
RingHom.injective
DivisionRing.isSimpleRing
IsSimpleRing.instNontrivial
tower_top_of_subalgebra_le πŸ“–β€”Subalgebra
CommRing.toCommSemiring
CommSemiring.toSemiring
Preorder.toLE
PartialOrder.toPreorder
Subalgebra.instPartialOrder
IsAlgebraic
SetLike.instMembership
Subalgebra.instSetLike
Subalgebra.toCommRing
CommRing.toRing
Subalgebra.toAlgebra
Ring.toSemiring
Algebra.id
β€”β€”extendScalars
IsScalarTower.of_algebraMap_eq
Subalgebra.inclusion_injective

IsLocalization

Theorems

NameKindAssumesProvesValidatesDepends On
isAlgebraic πŸ“–mathematicalβ€”Algebra.IsAlgebraic
CommRing.toRing
β€”eq_or_ne
isAlgebraic_zero
surj
mk'_zero
Polynomial.coeff_sub
Polynomial.mul_coeff_zero
Polynomial.coeff_C_zero
Polynomial.coeff_X_zero
MulZeroClass.mul_zero
zero_sub
eq_mk'_iff_mul_eq
Polynomial.aeval_sub
map_mul
NonUnitalAlgSemiHomClass.toMulHomClass
AlgHom.instNonUnitalAlgHomClassOfAlgHomClass
AlgHom.algHomClass
Polynomial.aeval_C
Polynomial.aeval_X
mul_comm

Polynomial

Theorems

NameKindAssumesProvesValidatesDepends On
transcendental πŸ“–mathematicalSubmonoid
MulZeroOneClass.toMulOneClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
SetLike.instMembership
Submonoid.instSetLike
nonZeroDivisors
leadingCoeff
Transcendental
Polynomial
ring
CommRing.toRing
algebraOfAlgebra
Algebra.id
β€”aeval_X_left
Transcendental.aeval
transcendental_X
transcendental_X πŸ“–mathematicalβ€”Transcendental
Polynomial
CommSemiring.toSemiring
CommRing.toCommSemiring
ring
CommRing.toRing
algebraOfAlgebra
Algebra.id
X
β€”aeval_X_left

Subalgebra

Theorems

NameKindAssumesProvesValidatesDepends On
algebra_isAlgebraic_bot_left_iff πŸ“–mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
algebraMap
Algebra.IsAlgebraic
Subalgebra
SetLike.instMembership
instSetLike
Bot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
Algebra.instCompleteLatticeSubalgebra
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
toCommRing
CommRing.toRing
toAlgebra
Ring.toSemiring
Algebra.id
β€”isAlgebraic_bot_iff
algebra_isAlgebraic_bot_right πŸ“–mathematicalβ€”Algebra.IsAlgebraic
Subalgebra
CommRing.toCommSemiring
CommSemiring.toSemiring
SetLike.instMembership
instSetLike
Bot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
Algebra.instCompleteLatticeSubalgebra
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
toRing
CommRing.toRing
algebra
β€”isAlgebraic_algebraMap
algebra_isAlgebraic_of_algebra_isAlgebraic_bot_left πŸ“–mathematicalβ€”Algebra.IsAlgebraic
CommRing.toRing
β€”Algebra.IsAlgebraic.of_ringHom_of_comp_eq
RingHom.instRingHomClass
RingHom.ext
inv_mem_of_algebraic πŸ“–mathematicalIsAlgebraic
Field.toCommRing
DivisionRing.toRing
Field.toDivisionRing
Subalgebra
Semifield.toCommSemiring
Field.toSemifield
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
SetLike.instMembership
instSetLike
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
β€”AddSubmonoidClass.toZeroMemClass
SubsemiringClass.toAddSubmonoidClass
instSubsemiringClass
inv_mem_of_root_of_coeff_zero_ne_zero
Polynomial.coeff_add
zero_add
Polynomial.coeff_C
mul_eq_zero
noZeroDivisors
IsDomain.to_noZeroDivisors
instIsDomain
Polynomial.aeval_X
map_mul
NonUnitalAlgSemiHomClass.toMulHomClass
AlgHom.instNonUnitalAlgHomClassOfAlgHomClass
AlgHom.algHomClass
coe_zero
inv_zero
zero_mem
coe_eq_zero
aeval_coe
inv_mem_of_root_of_coeff_zero_ne_zero πŸ“–mathematicalDFunLike.coe
AlgHom
Polynomial
CommSemiring.toSemiring
Semifield.toCommSemiring
Field.toSemifield
Subalgebra
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
SetLike.instMembership
instSetLike
Polynomial.semiring
toSemiring
Polynomial.algebraOfAlgebra
Algebra.id
algebra
AlgHom.funLike
Polynomial.aeval
ZeroMemClass.zero
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
AddSubmonoidClass.toZeroMemClass
AddMonoid.toAddZeroClass
AddMonoidWithOne.toAddMonoid
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
DivisionRing.toRing
Field.toDivisionRing
SubsemiringClass.toAddSubmonoidClass
Semiring.toNonAssocSemiring
instSubsemiringClass
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
β€”AddSubmonoidClass.toZeroMemClass
SubsemiringClass.toAddSubmonoidClass
instSubsemiringClass
aeval_coe
coe_zero
inv_eq_of_root_of_coeff_zero_ne_zero
div_eq_inv_mul
Algebra.smul_def
map_invβ‚€
RingHomClass.toMonoidWithZeroHomClass
RingHom.instRingHomClass
map_neg
RingHomClass.toAddMonoidHomClass
inv_neg
neg_mul
smul_mem
isAlgebraic_bot_iff πŸ“–mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
algebraMap
IsAlgebraic
Subalgebra
SetLike.instMembership
instSetLike
Bot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
Algebra.instCompleteLatticeSubalgebra
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
toCommRing
CommRing.toRing
toAlgebra
Ring.toSemiring
Algebra.id
β€”isAlgebraic_ringHom_iff_of_comp_eq
AlgEquivClass.toRingEquivClass
AlgEquiv.instAlgEquivClass
RingHom.instRingHomClass
RingEquivClass.toRingHomClass
isAlgebraic_iff_isAlgebraic_val πŸ“–mathematicalβ€”IsAlgebraic
Subalgebra
CommRing.toCommSemiring
Ring.toSemiring
SetLike.instMembership
instSetLike
toRing
algebra
β€”isAlgebraic_algHom_iff
Subtype.val_injective
isAlgebraic_of_isAlgebraic_bot πŸ“–β€”IsAlgebraic
Subalgebra
CommRing.toCommSemiring
CommSemiring.toSemiring
SetLike.instMembership
instSetLike
Bot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
Algebra.instCompleteLatticeSubalgebra
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
toCommRing
CommRing.toRing
toAlgebra
Ring.toSemiring
Algebra.id
β€”β€”IsAlgebraic.of_ringHom_of_comp_eq
RingHom.instRingHomClass
isField_of_algebraic πŸ“–mathematicalβ€”IsField
Subalgebra
Semifield.toCommSemiring
Field.toSemifield
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
SetLike.instMembership
instSetLike
toSemiring
β€”SubsemiringClass.nontrivial
instSubsemiringClass
IsSimpleRing.instNontrivial
DivisionRing.isSimpleRing
Nontrivial.exists_pair_ne
CommRing.mul_comm
inv_mem_of_algebraic
Algebra.IsAlgebraic.isAlgebraic
mul_inv_cancelβ‚€
AddSubmonoidClass.toZeroMemClass
SubsemiringClass.toAddSubmonoidClass
coe_eq_zero

Transcendental

Theorems

NameKindAssumesProvesValidatesDepends On
aeval πŸ“–mathematicalTranscendental
Submonoid
MulZeroOneClass.toMulOneClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
CommSemiring.toSemiring
CommRing.toCommSemiring
SetLike.instMembership
Submonoid.instSetLike
nonZeroDivisors
Polynomial.leadingCoeff
DFunLike.coe
AlgHom
Polynomial
Polynomial.semiring
Ring.toSemiring
Polynomial.algebraOfAlgebra
Algebra.id
AlgHom.funLike
Polynomial.aeval
β€”IsAlgebraic.of_aeval
aeval_of_transcendental πŸ“–mathematicalTranscendental
Polynomial
CommSemiring.toSemiring
CommRing.toCommSemiring
Polynomial.ring
CommRing.toRing
Polynomial.algebraOfAlgebra
Algebra.id
DFunLike.coe
AlgHom
Polynomial.semiring
Ring.toSemiring
AlgHom.funLike
Polynomial.aeval
β€”transcendental_iff
Polynomial.comp_eq_aeval
Polynomial.aeval_comp
infinite πŸ“–mathematicalTranscendentalInfiniteβ€”Infinite.of_injective
Polynomial.infinite
transcendental_iff_injective
of_aeval πŸ“–mathematicalTranscendental
DFunLike.coe
AlgHom
Polynomial
CommSemiring.toSemiring
CommRing.toCommSemiring
Polynomial.semiring
Ring.toSemiring
Polynomial.algebraOfAlgebra
Algebra.id
AlgHom.funLike
Polynomial.aeval
Polynomial.ring
CommRing.toRing
β€”transcendental_iff
Polynomial.aeval_comp
Polynomial.comp_eq_aeval
map_zero
MonoidWithZeroHomClass.toZeroHomClass
RingHomClass.toMonoidWithZeroHomClass
AlgHomClass.toRingHomClass
AlgHom.algHomClass
of_ringHom_of_comp_eq πŸ“–β€”Transcendental
DFunLike.coe
RingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
β€”β€”IsAlgebraic.ringHom_of_comp_eq
of_tower_top πŸ“–β€”Transcendental
Field.toCommRing
β€”β€”IsAlgebraic.tower_top
of_tower_top_of_subalgebra_le πŸ“–β€”Subalgebra
CommRing.toCommSemiring
CommSemiring.toSemiring
Preorder.toLE
PartialOrder.toPreorder
Subalgebra.instPartialOrder
Transcendental
SetLike.instMembership
Subalgebra.instSetLike
Subalgebra.toCommRing
CommRing.toRing
Subalgebra.toAlgebra
Ring.toSemiring
Algebra.id
β€”β€”IsAlgebraic.tower_top_of_subalgebra_le
pow πŸ“–mathematicalTranscendentalMonoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
β€”IsAlgebraic.of_pow
restrictScalars πŸ“–β€”DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
algebraMap
Transcendental
β€”β€”IsAlgebraic.extendScalars
ringHom_of_comp_eq πŸ“–β€”Transcendental
DFunLike.coe
RingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
β€”β€”IsAlgebraic.of_ringHom_of_comp_eq

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
inv_eq_of_aeval_divX_ne_zero πŸ“–mathematicalβ€”InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
Field.toSemifield
DivInvMonoid.toDiv
DivisionRing.toDivInvMonoid
Field.toDivisionRing
DFunLike.coe
AlgHom
Polynomial
CommSemiring.toSemiring
Semifield.toCommSemiring
Polynomial.semiring
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Polynomial.algebraOfAlgebra
Algebra.id
AlgHom.funLike
Polynomial.aeval
Polynomial.divX
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
DivisionRing.toRing
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
algebraMap
Polynomial.coeff
β€”inv_eq_iff_eq_inv
inv_div
div_eq_iff
sub_eq_iff_eq_add
mul_comm
Polynomial.divX_mul_X_add
map_add
SemilinearMapClass.toAddHomClass
NonUnitalAlgHomClass.instLinearMapClass
AlgHom.instNonUnitalAlgHomClassOfAlgHomClass
AlgHom.algHomClass
map_mul
NonUnitalAlgSemiHomClass.toMulHomClass
Polynomial.aeval_X
Polynomial.aeval_C
inv_eq_of_root_of_coeff_zero_ne_zero πŸ“–mathematicalDFunLike.coe
AlgHom
Polynomial
CommSemiring.toSemiring
Semifield.toCommSemiring
Field.toSemifield
Polynomial.semiring
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Polynomial.algebraOfAlgebra
Algebra.id
AlgHom.funLike
Polynomial.aeval
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Ring.toAddCommGroup
DivisionRing.toRing
Field.toDivisionRing
DivInvMonoid.toDiv
DivisionRing.toDivInvMonoid
Polynomial.divX
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
algebraMap
Polynomial.coeff
β€”zero_sub
div_neg
inv_eq_of_aeval_divX_ne_zero
RingHom.injective
DivisionRing.isSimpleRing
IsSimpleRing.instNontrivial
RingHom.map_zero
Polynomial.divX_mul_X_add
map_add
SemilinearMapClass.toAddHomClass
NonUnitalAlgHomClass.instLinearMapClass
AlgHom.instNonUnitalAlgHomClassOfAlgHomClass
AlgHom.algHomClass
map_mul
NonUnitalAlgSemiHomClass.toMulHomClass
MulZeroClass.zero_mul
zero_add
Polynomial.aeval_C
isAlgebraic_algHom_iff πŸ“–mathematicalDFunLike.coe
AlgHom
CommRing.toCommSemiring
Ring.toSemiring
AlgHom.funLike
IsAlgebraicβ€”map_zero
MonoidWithZeroHomClass.toZeroHomClass
RingHomClass.toMonoidWithZeroHomClass
AlgHomClass.toRingHomClass
AlgHom.algHomClass
AlgHom.comp_apply
Polynomial.aeval_algHom
IsAlgebraic.algHom
isAlgebraic_algebraMap πŸ“–mathematicalβ€”IsAlgebraic
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
β€”Polynomial.X_sub_C_ne_zero
map_sub
DistribMulActionSemiHomClass.toAddMonoidHomClass
NonUnitalAlgSemiHomClass.toDistribMulActionSemiHomClass
AlgHom.instNonUnitalAlgHomClassOfAlgHomClass
AlgHom.algHomClass
Polynomial.aeval_X
Polynomial.aeval_C
sub_self
isAlgebraic_algebraMap_iff πŸ“–mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
IsAlgebraic
CommRing.toRing
β€”isAlgebraic_algHom_iff
isAlgebraic_iff_not_injective πŸ“–mathematicalβ€”IsAlgebraic
Polynomial
CommSemiring.toSemiring
CommRing.toCommSemiring
DFunLike.coe
AlgHom
Polynomial.semiring
Ring.toSemiring
Polynomial.algebraOfAlgebra
Algebra.id
AlgHom.funLike
Polynomial.aeval
β€”DistribMulActionSemiHomClass.toAddMonoidHomClass
NonUnitalAlgSemiHomClass.toDistribMulActionSemiHomClass
AlgHom.instNonUnitalAlgHomClassOfAlgHomClass
AlgHom.algHomClass
isAlgebraic_int πŸ“–mathematicalβ€”IsAlgebraic
AddGroupWithOne.toIntCast
Ring.toAddGroupWithOne
β€”map_intCast
RingHom.instRingHomClass
isAlgebraic_algebraMap
isAlgebraic_nat πŸ“–mathematicalβ€”IsAlgebraic
AddMonoidWithOne.toNatCast
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
β€”map_natCast
RingHom.instRingHomClass
isAlgebraic_algebraMap
isAlgebraic_of_mem_rootSet πŸ“–mathematicalSet
Set.instMembership
Polynomial.rootSet
Field.toCommRing
instIsDomain
Field.toSemifield
IsAlgebraic
DivisionRing.toRing
Field.toDivisionRing
β€”instIsDomain
Polynomial.ne_zero_of_mem_rootSet
Polynomial.aeval_eq_zero_of_mem_rootSet
isAlgebraic_one πŸ“–mathematicalβ€”IsAlgebraic
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
β€”map_one
MonoidHomClass.toOneHomClass
MonoidWithZeroHomClass.toMonoidHomClass
RingHomClass.toMonoidWithZeroHomClass
RingHom.instRingHomClass
isAlgebraic_algebraMap
isAlgebraic_rat πŸ“–mathematicalβ€”IsAlgebraic
Field.toCommRing
DivisionRing.toRing
DivisionRing.toRatCast
β€”map_ratCast
RingHom.instRingHomClass
isAlgebraic_algebraMap
IsSimpleRing.instNontrivial
DivisionRing.isSimpleRing
isAlgebraic_ringHom_iff_of_comp_eq πŸ“–mathematicalDFunLike.coe
RingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
EquivLike.toFunLike
RingEquivClass.toRingHomClass
IsAlgebraicβ€”RingEquivClass.toRingHomClass
IsAlgebraic.of_ringHom_of_comp_eq
EquivLike.surjective
IsAlgebraic.ringHom_of_comp_eq
EquivLike.injective
isAlgebraic_zero πŸ“–mathematicalβ€”IsAlgebraic
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”Polynomial.X_ne_zero
Polynomial.aeval_X
is_transcendental_of_subsingleton πŸ“–mathematicalβ€”Transcendentalβ€”Unique.instSubsingleton
transcendental_algebraMap_iff πŸ“–mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
Transcendental
CommRing.toRing
β€”isAlgebraic_algebraMap_iff
transcendental_iff_injective πŸ“–mathematicalβ€”Transcendental
Polynomial
CommSemiring.toSemiring
CommRing.toCommSemiring
DFunLike.coe
AlgHom
Polynomial.semiring
Ring.toSemiring
Polynomial.algebraOfAlgebra
Algebra.id
AlgHom.funLike
Polynomial.aeval
β€”Iff.not_left
isAlgebraic_iff_not_injective
transcendental_iff_ker_eq_bot πŸ“–mathematicalβ€”Transcendental
RingHom.ker
Polynomial
CommSemiring.toSemiring
CommRing.toCommSemiring
AlgHom
Polynomial.semiring
Ring.toSemiring
Polynomial.algebraOfAlgebra
Algebra.id
AlgHom.funLike
AlgHomClass.toRingHomClass
AlgHom.algHomClass
Polynomial.aeval
Bot.bot
Ideal
Submodule.instBot
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Semiring.toModule
β€”AlgHomClass.toRingHomClass
AlgHom.algHomClass
transcendental_iff_injective
RingHom.injective_iff_ker_eq_bot
transcendental_ringHom_iff_of_comp_eq πŸ“–mathematicalDFunLike.coe
RingHom.comp
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
Ring.toSemiring
algebraMap
RingHomClass.toRingHom
EquivLike.toFunLike
RingEquivClass.toRingHomClass
Transcendentalβ€”RingEquivClass.toRingHomClass
isAlgebraic_ringHom_iff_of_comp_eq

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