Documentation Verification Report

TranscendenceBasis

πŸ“ Source: Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean

Statistics

MetricCount
Definitionsmatroid
1
TheoremsisDomain_of_adjoin_range, isTranscendenceBasis_of_le_trdeg, isTranscendenceBasis_of_le_trdeg_of_finite, isTranscendenceBasis_of_lift_le_trdeg, isTranscendenceBasis_of_lift_le_trdeg_of_finite, trdeg_le_cardinalMk, instFinitaryMatroid, isAlgebraic_adjoin_iff_of_matroid_isBasis, isTranscendenceBasis_iff, isTranscendenceBasis_iff_isAlgebraic, isTranscendenceBasis_of_lift_trdeg_le, isTranscendenceBasis_of_lift_trdeg_le_of_finite, isTranscendenceBasis_of_trdeg_le, isTranscendenceBasis_of_trdeg_le_of_finite, matroid_cRank_eq, matroid_closure_eq, matroid_closure_of_subsingleton, matroid_e, matroid_indep_iff, matroid_isBase_iff, matroid_isBasis_iff, matroid_isBasis_iff_of_subsingleton, matroid_isFlat_iff, matroid_isFlat_of_subsingleton, matroid_spanning_iff, matroid_spanning_iff_of_subsingleton, algebraMap_comp, cardinalMk_eq, cardinalMk_eq_trdeg, isAlgebraic, isAlgebraic_field, isAlgebraic_iff, isEmpty_iff_isAlgebraic, lift_cardinalMk_eq, lift_cardinalMk_eq_trdeg, mvPolynomial, mvPolynomial', nonempty_iff_transcendental, of_isAlgebraic_adjoin_image_compl, of_isAlgebraic_adjoin_insert_diff, polynomial, sumElim_comp, trdeg_of_isDomain, trdeg_of_isDomain, exists_isTranscendenceBasis, exists_isTranscendenceBasis', exists_isTranscendenceBasis_between, exists_isTranscendenceBasis_subset, exists_isTranscendenceBasis_superset, isAlgebraic_iff_exists_isTranscendenceBasis_subset, isTranscendenceBasis_iff_algebraicIndependent_isAlgebraic, lift_trdeg_add_eq, trdeg_add_eq, trdeg_eq_iSup_cardinalMk_isTranscendenceBasis, trdeg_lt_aleph0
55
Total56

Algebra.IsAlgebraic

Theorems

NameKindAssumesProvesValidatesDepends On
isDomain_of_adjoin_range πŸ“–mathematicalβ€”IsDomain
CommSemiring.toSemiring
CommRing.toCommSemiring
β€”nontrivial
isDomain_iff_noZeroDivisors_and_nontrivial
Function.Injective.nontrivial
Subtype.val_injective
isTranscendenceBasis_of_le_trdeg πŸ“–mathematicalCardinal
Preorder.toLT
PartialOrder.toPreorder
Cardinal.partialOrder
Algebra.trdeg
Cardinal.aleph0
Cardinal.instLE
IsTranscendenceBasisβ€”isTranscendenceBasis_of_lift_le_trdeg
Cardinal.lift_id
isTranscendenceBasis_of_le_trdeg_of_finite πŸ“–mathematicalCardinal
Cardinal.instLE
Algebra.trdeg
IsTranscendenceBasisβ€”isTranscendenceBasis_of_lift_le_trdeg_of_finite
Cardinal.lift_id
isTranscendenceBasis_of_lift_le_trdeg πŸ“–mathematicalCardinal
Preorder.toLT
PartialOrder.toPreorder
Cardinal.partialOrder
Algebra.trdeg
Cardinal.aleph0
Cardinal.instLE
Cardinal.lift
IsTranscendenceBasisβ€”Cardinal.mk_lt_aleph0_iff
Cardinal.lift_lt
LE.le.trans_lt
LT.lt.trans_eq
Cardinal.lift_aleph0
isTranscendenceBasis_of_lift_le_trdeg_of_finite
isTranscendenceBasis_of_lift_le_trdeg_of_finite πŸ“–mathematicalCardinal
Cardinal.instLE
Cardinal.lift
Algebra.trdeg
IsTranscendenceBasisβ€”Cardinal.lift_mk_le'
LE.le.trans
Cardinal.lift_le
trdeg_le_cardinalMk
Function.Surjective.bijective_of_nat_card_le
Set.rangeFactorization_surjective
Nat.card_le_card_of_injective
Finite.Set.finite_range
IsTranscendenceBasis.of_subtype_range
isDomain_of_adjoin_range
Matroid.Spanning.isBase_of_le_cRank_of_finite
IsDomain.to_noZeroDivisors
AlgebraicIndependent.matroid_spanning_iff
Cardinal.mk_range_le_lift
AlgebraicIndependent.matroid_cRank_eq
Set.finite_range
trdeg_le_cardinalMk πŸ“–mathematicalβ€”Cardinal
Cardinal.instLE
Algebra.trdeg
Set.Elem
β€”isDomain_of_adjoin_range
faithfulSMul_iff_algebraMap_injective
AlgebraicIndependent.matroid_cRank_eq
Matroid.Spanning.cRank_le_cardinalMk
IsDomain.to_noZeroDivisors
Matroid.invariantCardinalRank_of_finitary
AlgebraicIndependent.instFinitaryMatroid
AlgebraicIndependent.matroid_spanning_iff
trdeg_eq_zero_of_not_injective
Cardinal.canonicallyOrderedAdd

AlgebraicIndependent

Definitions

NameCategoryTheorems
matroid πŸ“–CompOp
13 mathmath: instFinitaryMatroid, matroid_indep_iff, matroid_isFlat_of_subsingleton, matroid_isBasis_iff, matroid_spanning_iff, matroid_cRank_eq, matroid_spanning_iff_of_subsingleton, matroid_isBase_iff, matroid_e, matroid_closure_eq, matroid_isFlat_iff, matroid_closure_of_subsingleton, matroid_isBasis_iff_of_subsingleton

Theorems

NameKindAssumesProvesValidatesDepends On
instFinitaryMatroid πŸ“–mathematicalβ€”Matroid.Finitary
matroid
β€”algebraicIndependent_of_finite
isAlgebraic_adjoin_iff_of_matroid_isBasis πŸ“–mathematicalMatroid.IsBasis
matroid
IsAlgebraic
Subalgebra
CommRing.toCommSemiring
CommSemiring.toSemiring
SetLike.instMembership
Subalgebra.instSetLike
Algebra.adjoin
Subalgebra.toCommRing
CommRing.toRing
Subalgebra.toAlgebra
Ring.toSemiring
Algebra.id
β€”subsingleton_or_nontrivial
is_transcendental_of_subsingleton
instSubsingletonSubtype_mathlib
isDomain_iff_noZeroDivisors_and_nontrivial
IsAlgebraic.adjoin_of_forall_isAlgebraic
IsDomain.to_noZeroDivisors
matroid_isBasis_iff
isTranscendenceBasis_iff πŸ“–mathematicalAlgebraicIndependentIsTranscendenceBasisβ€”coe_range
Set.range_comp_subset_range
Set.range_eq_univ
Set.image_injective
injective
Set.image_univ
Set.range_comp
Set.range_subset_iff
Function.Surjective.range_eq
Set.image_congr
Subtype.range_coe_subtype
Set.image_image
isTranscendenceBasis_iff_isAlgebraic πŸ“–mathematicalAlgebraicIndependentIsTranscendenceBasis
Algebra.IsAlgebraic
Subalgebra
CommRing.toCommSemiring
CommSemiring.toSemiring
SetLike.instMembership
Subalgebra.instSetLike
Algebra.adjoin
Set.range
Subalgebra.toCommRing
CommRing.toRing
Subalgebra.toAlgebra
Ring.toSemiring
Algebra.id
β€”IsTranscendenceBasis.isAlgebraic
of_not_not
HasSubset.Subset.antisymm
Set.instAntisymmSubset
Set.not_subset
transcendental_adjoin
Set.range_inclusion
Algebra.IsAlgebraic.isAlgebraic
Set.range_comp
Set.val_comp_inclusion
Subtype.range_val
isTranscendenceBasis_of_lift_trdeg_le πŸ“–mathematicalAlgebraicIndependent
Cardinal
Preorder.toLT
PartialOrder.toPreorder
Cardinal.partialOrder
Algebra.trdeg
Cardinal.aleph0
Cardinal.instLE
Cardinal.lift
IsTranscendenceBasisβ€”faithfulSMul_iff_algebraMap_injective
algebraMap_injective
IsTranscendenceBasis.of_subtype_range
injective
Matroid.Indep.isBase_of_cRank_le
Matroid.rankFinite_iff_cRank_lt_aleph0
matroid_cRank_eq
matroid_indep_iff
to_subtype_range
Cardinal.lift_le
LE.le.trans_eq
Cardinal.mk_range_eq_of_injective
isTranscendenceBasis_of_lift_trdeg_le_of_finite πŸ“–mathematicalAlgebraicIndependent
Cardinal
Cardinal.instLE
Cardinal.lift
Algebra.trdeg
IsTranscendenceBasisβ€”isTranscendenceBasis_of_lift_trdeg_le
Cardinal.lift_lt
LE.le.trans_lt
Cardinal.lift_aleph0
isTranscendenceBasis_of_trdeg_le πŸ“–mathematicalAlgebraicIndependent
Cardinal
Preorder.toLT
PartialOrder.toPreorder
Cardinal.partialOrder
Algebra.trdeg
Cardinal.aleph0
Cardinal.instLE
IsTranscendenceBasisβ€”isTranscendenceBasis_of_lift_trdeg_le
Cardinal.lift_id
isTranscendenceBasis_of_trdeg_le_of_finite πŸ“–mathematicalAlgebraicIndependent
Cardinal
Cardinal.instLE
Algebra.trdeg
IsTranscendenceBasisβ€”isTranscendenceBasis_of_lift_trdeg_le_of_finite
Cardinal.lift_id
matroid_cRank_eq πŸ“–mathematicalβ€”Matroid.cRank
matroid
Algebra.trdeg
β€”trdeg_eq_iSup_cardinalMk_isTranscendenceBasis
matroid_closure_eq πŸ“–mathematicalβ€”Matroid.closure
matroid
IsDomain.to_noZeroDivisors
CommSemiring.toSemiring
CommRing.toCommSemiring
SetLike.coe
Subalgebra
SetLike.instMembership
Subalgebra.instSetLike
Algebra.adjoin
Subalgebra.toCommRing
Subalgebra.toAlgebra
Algebra.id
Subalgebra.algebraicClosure
Subalgebra.isDomain
CommRing.toRing
β€”IsDomain.to_noZeroDivisors
Subalgebra.isDomain
Matroid.exists_isBasis
Matroid.IsBasis.closure_eq_closure
Matroid.Indep.closure_eq_setOf_isBasis_insert
isAlgebraic_adjoin_iff_of_matroid_isBasis
isAlgebraic_algebraMap
SubsemiringClass.nontrivial
Subalgebra.instSubsemiringClass
IsDomain.toNontrivial
Algebra.subset_adjoin
matroid_closure_of_subsingleton πŸ“–mathematicalβ€”Matroid.closure
matroid
Subsingleton.to_noZeroDivisors
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
β€”Subsingleton.to_noZeroDivisors
Set.inter_univ
subset_antisymm
Set.instAntisymmSubset
Set.sInter_subset_of_mem
subset_refl
Set.instReflSubset
Set.subset_sInter
matroid_e πŸ“–mathematicalβ€”Matroid.E
matroid
Set.univ
β€”β€”
matroid_indep_iff πŸ“–mathematicalβ€”Matroid.Indep
matroid
AlgebraicIndepOn
β€”β€”
matroid_isBase_iff πŸ“–mathematicalβ€”Matroid.IsBase
matroid
IsTranscendenceBasis
Set
Set.instMembership
β€”β€”
matroid_isBasis_iff πŸ“–mathematicalβ€”Matroid.IsBasis
matroid
IsDomain.to_noZeroDivisors
CommSemiring.toSemiring
CommRing.toCommSemiring
AlgebraicIndepOn
Set
Set.instHasSubset
IsAlgebraic
Subalgebra
SetLike.instMembership
Subalgebra.instSetLike
Algebra.adjoin
Subalgebra.toCommRing
CommRing.toRing
Subalgebra.toAlgebra
Ring.toSemiring
Algebra.id
β€”IsDomain.to_noZeroDivisors
Matroid.IsBasis.eq_1
Set.maximal_iff_forall_insert
Matroid.Indep.subset
HasSubset.Subset.trans
Set.instIsTransSubset
of_not_not
isAlgebraic_algebraMap
SubsemiringClass.nontrivial
Subalgebra.instSubsemiringClass
IsDomain.toNontrivial
Algebra.subset_adjoin
AlgebraicIndepOn.insert
Set.image_id
Set.insert_subset
AlgebraicIndepOn.insert_iff
Set.mem_insert
matroid_isBasis_iff_of_subsingleton πŸ“–mathematicalβ€”Matroid.IsBasis
matroid
Subsingleton.to_noZeroDivisors
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
β€”Function.Injective.subsingleton
FaithfulSMul.algebraMap_injective
Subsingleton.to_noZeroDivisors
matroid_isFlat_iff πŸ“–mathematicalβ€”Matroid.IsFlat
matroid
IsDomain.to_noZeroDivisors
CommSemiring.toSemiring
CommRing.toCommSemiring
SetLike.coe
Subalgebra
Subalgebra.instSetLike
Set
Set.instMembership
β€”IsDomain.to_noZeroDivisors
Matroid.isFlat_iff_closure_eq
Subalgebra.isDomain
matroid_closure_eq
IsScalarTower.subalgebra'
IsScalarTower.right
IsAlgebraic.restrictScalars
Subalgebra.noZeroDivisors
instIsAlgebraicSubtypeMemSubalgebraAlgebraicClosure
Set.ext
Algebra.adjoin_eq
isAlgebraic_algebraMap
SubsemiringClass.nontrivial
Subalgebra.instSubsemiringClass
IsDomain.toNontrivial
Algebra.subset_adjoin
matroid_isFlat_of_subsingleton πŸ“–mathematicalβ€”Matroid.IsFlat
matroid
Subsingleton.to_noZeroDivisors
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
β€”Subsingleton.to_noZeroDivisors
Eq.subset
Set.instReflSubset
matroid_spanning_iff πŸ“–mathematicalβ€”Matroid.Spanning
matroid
IsDomain.to_noZeroDivisors
CommSemiring.toSemiring
CommRing.toCommSemiring
Algebra.IsAlgebraic
Subalgebra
SetLike.instMembership
Subalgebra.instSetLike
Algebra.adjoin
Subalgebra.toCommRing
CommRing.toRing
Subalgebra.toAlgebra
Ring.toSemiring
Algebra.id
β€”IsDomain.to_noZeroDivisors
Subalgebra.isDomain
matroid_closure_eq
matroid_spanning_iff_of_subsingleton πŸ“–mathematicalβ€”Matroid.Spanning
matroid
Subsingleton.to_noZeroDivisors
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
Set.univ
β€”Subsingleton.to_noZeroDivisors
matroid_closure_of_subsingleton

IsTranscendenceBasis

Theorems

NameKindAssumesProvesValidatesDepends On
algebraMap_comp πŸ“–mathematicalIsTranscendenceBasisDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
algebraMap
β€”AlgebraicIndependent.isTranscendenceBasis_iff_isAlgebraic
AlgebraicIndependent.map
Function.Injective.injOn
FaithfulSMul.algebraMap_injective
Set.range_comp
AlgHom.map_adjoin
isAlgebraic
Algebra.IsAlgebraic.trans
IsScalarTower.subalgebra
Algebra.IsAlgebraic.extendScalars
IsScalarTower.of_algebraMap_eq
AlgEquiv.injective
cardinalMk_eq πŸ“–β€”IsTranscendenceBasisβ€”β€”Cardinal.lift_id
lift_cardinalMk_eq
cardinalMk_eq_trdeg πŸ“–mathematicalIsTranscendenceBasisAlgebra.trdegβ€”Cardinal.lift_id
lift_cardinalMk_eq_trdeg
isAlgebraic πŸ“–mathematicalIsTranscendenceBasisAlgebra.IsAlgebraic
Subalgebra
CommRing.toCommSemiring
CommSemiring.toSemiring
SetLike.instMembership
Subalgebra.instSetLike
Algebra.adjoin
Set.range
Subalgebra.toCommRing
CommRing.toRing
Subalgebra.toAlgebra
Ring.toSemiring
Algebra.id
β€”not_iff_comm
AlgebraicIndependent.option_iff_transcendental
AlgebraicIndependent.injective
algebraicIndependent_subtype_range
isAlgebraic_field πŸ“–mathematicalIsTranscendenceBasis
EuclideanDomain.toCommRing
Field.toEuclideanDomain
Algebra.IsAlgebraic
IntermediateField
SetLike.instMembership
IntermediateField.instSetLike
IntermediateField.adjoin
Set.range
IntermediateField.toField
DivisionRing.toRing
Field.toDivisionRing
IntermediateField.instAlgebraSubtypeMem
Ring.toSemiring
Algebra.id
Semifield.toCommSemiring
Field.toSemifield
β€”Algebra.IsAlgebraic.extendScalars
IntermediateField.algebra_adjoin_le_adjoin
IsScalarTower.of_algebraMap_eq
SubringClass.toSubsemiringClass
SubfieldClass.toSubringClass
IntermediateField.instSubfieldClass
Subalgebra.inclusion_injective
isAlgebraic
IsLocalRing.toNontrivial
Field.instIsLocalRing
isAlgebraic_iff πŸ“–mathematicalIsTranscendenceBasisAlgebra.IsAlgebraic
CommRing.toRing
IsAlgebraic
β€”Algebra.IsAlgebraic.isAlgebraic
Algebra.Subalgebra.restrictScalars_adjoin
le_sup_right
IsScalarTower.of_algebraMap_eq
RingHom.domain_nontrivial
IsDomain.toNontrivial
isAlgebraic
Algebra.IsAlgebraic.extendScalars
Subalgebra.inclusion_injective
Algebra.IsAlgebraic.trans
IsScalarTower.subalgebra'
IsScalarTower.right
Subalgebra.noZeroDivisors
isEmpty_iff_isAlgebraic πŸ“–mathematicalIsTranscendenceBasisIsEmpty
Algebra.IsAlgebraic
CommRing.toRing
β€”isAlgebraic
Subalgebra.algebra_isAlgebraic_of_algebra_isAlgebraic_bot_left
Algebra.adjoin_empty
Set.range_eq_empty
AlgebraicIndependent.isEmpty_of_isAlgebraic
lift_cardinalMk_eq πŸ“–mathematicalIsTranscendenceBasisCardinal.liftβ€”Cardinal.lift_lift
lift_cardinalMk_eq_trdeg
lift_cardinalMk_eq_trdeg πŸ“–mathematicalIsTranscendenceBasisCardinal.lift
Algebra.trdeg
β€”faithfulSMul_iff_algebraMap_injective
AlgebraicIndependent.algebraMap_injective
AlgebraicIndependent.matroid_cRank_eq
Matroid.IsBase.cardinalMk_eq_cRank
Matroid.invariantCardinalRank_of_finitary
AlgebraicIndependent.instFinitaryMatroid
AlgebraicIndependent.matroid_isBase_iff
to_subtype_range
Cardinal.lift_mk_eq'
AlgebraicIndependent.injective
mvPolynomial πŸ“–mathematicalβ€”IsTranscendenceBasis
MvPolynomial
CommRing.toCommSemiring
MvPolynomial.instCommRingMvPolynomial
MvPolynomial.algebra
Algebra.id
MvPolynomial.X
β€”isTranscendenceBasis_iff_algebraicIndependent_isAlgebraic
MvPolynomial.algebraicIndependent_X
MvPolynomial.adjoin_range_X
Algebra.isIntegral_of_surjective
Algebra.IsIntegral.isAlgebraic
SubsemiringClass.nontrivial
Subalgebra.instSubsemiringClass
MvPolynomial.nontrivial_of_nontrivial
mvPolynomial' πŸ“–mathematicalβ€”IsTranscendenceBasis
MvPolynomial
CommRing.toCommSemiring
MvPolynomial.instCommRingMvPolynomial
MvPolynomial.algebra
Algebra.id
MvPolynomial.X
β€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
mvPolynomial
nonempty_iff_transcendental πŸ“–mathematicalIsTranscendenceBasisAlgebra.Transcendental
CommRing.toRing
β€”not_isEmpty_iff
Algebra.transcendental_iff_not_isAlgebraic
isEmpty_iff_isAlgebraic
of_isAlgebraic_adjoin_image_compl πŸ“–β€”IsTranscendenceBasis
IsAlgebraic
Subalgebra
CommRing.toCommSemiring
CommSemiring.toSemiring
SetLike.instMembership
Subalgebra.instSetLike
Algebra.adjoin
Set.image
Compl.compl
Set
Set.instCompl
Set.instSingletonSet
Subalgebra.toCommRing
CommRing.toRing
Subalgebra.toAlgebra
Ring.toSemiring
Algebra.id
β€”β€”eq_or_ne
of_isAlgebraic_adjoin_insert_diff
Set.compl_eq_univ_diff
Set.insert_diff_self_of_mem
Set.mem_univ
of_isAlgebraic_adjoin_insert_diff πŸ“–β€”Set
Set.instMembership
Set.instInsert
IsTranscendenceBasis
Set.Elem
IsAlgebraic
Subalgebra
CommRing.toCommSemiring
CommSemiring.toSemiring
SetLike.instMembership
Subalgebra.instSetLike
Algebra.adjoin
Set.image
Set.instSDiff
Set.instSingletonSet
Subalgebra.toCommRing
CommRing.toRing
Subalgebra.toAlgebra
Ring.toSemiring
Algebra.id
β€”β€”IsAlgebraic.nontrivial
Function.Injective.nontrivial
Subsemiring.subtype_injective
isDomain_iff_noZeroDivisors_and_nontrivial
Module.nontrivial
Set.injOn_iff_injective
AlgebraicIndependent.injective
AlgebraicIndependent.matroid_isBase_iff
to_subtype_range
Set.image_eq_range
em
IsDomain.to_noZeroDivisors
Set.insert_diff_self_of_notMem
Subalgebra.isDomain
AlgebraicIndependent.matroid_closure_eq
SetLike.mem_coe
Subalgebra.mem_algebraicClosure
Matroid.Indep.notMem_closure_diff_of_mem
Matroid.IsBase.indep
Set.image_singleton
Set.InjOn.image_diff_subset
Set.singleton_subset_iff
Set.insert_eq_of_mem
eq_or_ne
Equiv.image_swap_of_mem_of_notMem
Equiv.swap_apply_right
Equiv.swap_apply_of_ne_of_ne
comp_equiv
Matroid.closure_subset_closure
Set.InjOn.ne
Set.injOn_insert
isTranscendenceBasis_subtype_range
Set.InjOn.mono
Set.diff_subset
Set.image_insert_eq
Matroid.IsBase.isBase_insert_diff_of_mem_closure
polynomial πŸ“–mathematicalβ€”IsTranscendenceBasis
Polynomial
CommSemiring.toSemiring
CommRing.toCommSemiring
Polynomial.commRing
Polynomial.algebraOfAlgebra
Algebra.id
Polynomial.X
β€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
nonempty_unique
isTranscendenceBasis_equiv
AlgEquiv.isTranscendenceBasis_iff
MvPolynomial.ext
Polynomial.evalβ‚‚_X
mvPolynomial
sumElim_comp πŸ“–mathematicalIsTranscendenceBasisDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
algebraMap
β€”subsingleton_or_nontrivial
isTranscendenceBasis_iff_of_subsingleton
AlgebraicIndependent.isTranscendenceBasis_iff_isAlgebraic
AlgebraicIndependent.sumElim_comp
IsScalarTower.subalgebra'
Algebra.adjoin_algebraMap_image_union_eq_adjoin_adjoin
Set.Sum.elim_range
Set.union_comm
Set.range_comp
Function.Injective.nontrivial
AlgebraicIndependent.algebraMap_injective
isAlgebraic
IsScalarTower.subalgebra
Algebra.adjoin_le
Algebra.subset_adjoin
IsScalarTower.of_algebraMap_eq
Algebra.adjoin_induction
isAlgebraic_algebraMap
SubsemiringClass.nontrivial
Subalgebra.instSubsemiringClass
IsAlgebraic.extendScalars
algebraMap_mem
AddSubmonoidWithOneClass.toOneMemClass
SubsemiringClass.addSubmonoidWithOneClass
Subalgebra.instSMulMemClass
IsAlgebraic.algHom
Algebra.IsAlgebraic.isAlgebraic
IsAlgebraic.add
Subalgebra.noZeroDivisors
IsAlgebraic.mul
Algebra.IsAlgebraic.trans

MvPolynomial

Theorems

NameKindAssumesProvesValidatesDepends On
trdeg_of_isDomain πŸ“–mathematicalβ€”Algebra.trdeg
MvPolynomial
CommRing.toCommSemiring
instCommRingMvPolynomial
algebra
Algebra.id
Cardinal.lift
β€”IsTranscendenceBasis.lift_cardinalMk_eq_trdeg
IsDomain.toNontrivial
instNoZeroDivisors
IsDomain.to_noZeroDivisors
IsTranscendenceBasis.mvPolynomial
Cardinal.lift_id
Cardinal.lift_lift
Cardinal.lift_id'

Polynomial

Theorems

NameKindAssumesProvesValidatesDepends On
trdeg_of_isDomain πŸ“–mathematicalβ€”Algebra.trdeg
Polynomial
CommSemiring.toSemiring
CommRing.toCommSemiring
commRing
algebraOfAlgebra
Algebra.id
Cardinal
Cardinal.instOne
β€”Cardinal.lift_uzero
Cardinal.mk_fintype
Fintype.card_unique
Nat.cast_one
Cardinal.lift_one
IsTranscendenceBasis.lift_cardinalMk_eq_trdeg
IsDomain.toNontrivial
instNoZeroDivisors
IsDomain.to_noZeroDivisors
IsTranscendenceBasis.polynomial

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
exists_isTranscendenceBasis πŸ“–mathematicalβ€”IsTranscendenceBasis
Set
Set.instMembership
β€”exists_isTranscendenceBasis_superset
algebraicIndependent_empty_iff
FaithfulSMul.algebraMap_injective
exists_isTranscendenceBasis' πŸ“–mathematicalβ€”IsTranscendenceBasisβ€”exists_isTranscendenceBasis
exists_isTranscendenceBasis_between πŸ“–mathematicalSet
Set.instHasSubset
AlgebraicIndepOn
IsTranscendenceBasis
Set.instMembership
β€”Algebra.IsAlgebraic.nontrivial
Function.Injective.nontrivial
Subtype.val_injective
isDomain_iff_noZeroDivisors_and_nontrivial
faithfulSMul_iff_algebraMap_injective
AlgebraicIndependent.algebraMap_injective
Matroid.Indep.exists_isBase_subset_spanning
AlgebraicIndependent.matroid_indep_iff
IsDomain.to_noZeroDivisors
AlgebraicIndependent.matroid_spanning_iff
exists_isTranscendenceBasis_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
IsTranscendenceBasis
Set.instMembership
β€”exists_isTranscendenceBasis_between
Set.empty_subset
algebraicIndependent_empty_iff
FaithfulSMul.algebraMap_injective
exists_isTranscendenceBasis_superset πŸ“–mathematicalAlgebraicIndepOnSet
Set.instHasSubset
IsTranscendenceBasis
Set.instMembership
β€”exists_maximal_algebraicIndependent
Set.subset_univ
isAlgebraic_iff_exists_isTranscendenceBasis_subset πŸ“–mathematicalβ€”Algebra.IsAlgebraic
Subalgebra
CommRing.toCommSemiring
CommSemiring.toSemiring
SetLike.instMembership
Subalgebra.instSetLike
Algebra.adjoin
Subalgebra.toCommRing
CommRing.toRing
Subalgebra.toAlgebra
Ring.toSemiring
Algebra.id
Set
Set.instHasSubset
IsTranscendenceBasis
Set.instMembership
β€”IsDomain.to_noZeroDivisors
Matroid.spanning_iff_exists_isBase_subset
Set.subset_univ
isTranscendenceBasis_iff_algebraicIndependent_isAlgebraic πŸ“–mathematicalβ€”IsTranscendenceBasis
AlgebraicIndependent
Algebra.IsAlgebraic
Subalgebra
CommRing.toCommSemiring
CommSemiring.toSemiring
SetLike.instMembership
Subalgebra.instSetLike
Algebra.adjoin
Set.range
Subalgebra.toCommRing
CommRing.toRing
Subalgebra.toAlgebra
Ring.toSemiring
Algebra.id
β€”AlgebraicIndependent.isTranscendenceBasis_iff_isAlgebraic
lift_trdeg_add_eq πŸ“–mathematicalβ€”Cardinal
Cardinal.instAdd
Cardinal.lift
Algebra.trdeg
β€”exists_isTranscendenceBasis
Function.Injective.noZeroDivisors
FaithfulSMul.algebraMap_injective
map_zero
MonoidWithZeroHomClass.toZeroHomClass
RingHomClass.toMonoidWithZeroHomClass
RingHom.instRingHomClass
map_mul
NonUnitalRingHomClass.toMulHomClass
RingHomClass.toNonUnitalRingHomClass
Function.Injective.nontrivial
IsTranscendenceBasis.cardinalMk_eq_trdeg
Cardinal.lift_umax
add_comm
IsTranscendenceBasis.lift_cardinalMk_eq_trdeg
IsTranscendenceBasis.sumElim_comp
Cardinal.mk_sum
Cardinal.lift_add
Cardinal.lift_lift
trdeg_add_eq πŸ“–mathematicalβ€”Cardinal
Cardinal.instAdd
Algebra.trdeg
β€”Cardinal.lift_id
lift_trdeg_add_eq
trdeg_eq_iSup_cardinalMk_isTranscendenceBasis πŸ“–mathematicalβ€”Algebra.trdeg
iSup
Cardinal
Set
IsTranscendenceBasis
Set.instMembership
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
Cardinal.instConditionallyCompleteLinearOrderBot
Set.Elem
β€”LE.le.antisymm
ciSup_le'
le_ciSup_of_le
Cardinal.bddAbove_range
small_subtype
small_set
UnivLE.small
UnivLE.self
Cardinal.mk_le_mk_of_subset
exists_isTranscendenceBasis_superset
le_rfl
trdeg_lt_aleph0 πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
Cardinal.partialOrder
Algebra.trdeg
Cardinal.aleph0
β€”Algebra.FiniteType.iff_quotient_mvPolynomial''
Cardinal.lift_lt
LE.le.trans_lt
lift_trdeg_le_of_surjective
MvPolynomial.trdeg_of_isDomain
Cardinal.mk_fintype
Fintype.card_fin
Cardinal.lift_natCast
Cardinal.lift_aleph0

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