Documentation Verification Report

Bijective

šŸ“ Source: Mathlib/RingTheory/RingHom/Bijective.lean

Statistics

MetricCount
DefinitionsBijective
1
TheoremscontainsIdentities, isStableUnderBaseChange, ofLocalizationSpan, respectsIso, stableUnderComposition
5
Total6

Function

Definitions

NameCategoryTheorems
Bijective šŸ“–MathDef
425 mathmath: LieEquiv.bijective, Algebra.IsUnramifiedAt.exists_primesOver_under_adjoin_eq_singleton_and_residueField_bijective, WittVector.ghostMap.bijective_of_invertible, SubAddAction.map_ofFixingAddSubgroupUnion_bijective, CategoryTheory.regularTopology.EqualizerCondition.bijective_mapToEqualizer_pullback', FirstOrder.Language.Equiv.bijective, AffineMap.restrict.bijective, PrimeSpectrum.sigmaToPi_bijective, IsNoetherian.bijective_of_surjective_endomorphism, ULift.up_bijective, AddUnits.addLeft_bijective, Polynomial.Gal.galActionHom_bijective_of_prime_degree', CategoryTheory.Limits.CofanTypes.isColimit_iff_bijective_fromSigma, MeasurableEquiv.symm_bijective, WithConv.toConv_bijective, LinearIsometryEquiv.symm_bijective, LinearMap.IsPerfPair.bijective_left, Path.symm_bijective, MeasurableEquiv.bijective, Path.Homotopy.symm_bijective, Path.bijective_cast, IsLocalization.bijective_lift_piRingHom_algebraMap_comp_piEvalRingHom, OrderRingIso.symm_bijective, AlgebraicGeometry.isIso_SpecMap_iff, Localization.AtPrime.mapPiEvalRingHom_bijective, toWeakSpaceCLM_bijective, ContinuousAlgEquiv.symm_bijective, CategoryTheory.Limits.CofanTypes.bijective_fromSigma_of_isColimit, TensorProduct.toIntegralClosure_bijective_of_isLocalization, RingHom.Bijective.respectsIso, ContinuousMap.HomotopyWith.symm_bijective, SimpleGraph.exists_bijective_of_forall_ncard_le, Composition.reverse_bijective, bijective_id, AddMonoid.Coprod.swap_bijective, CircleDeg1Lift.isUnit_iff_bijective, List.get_bijective_iff, AddSubgroup.isComplement_iff_bijective, PLift.down_bijective, SubAddAction.ofFixingAddSubgroup_of_singleton_bijective, Module.FaithfullyFlat.lTensor_bijective_iff_bijective, Polynomial.UniversalCoprimeFactorizationRing.exists_liesOver_residueFieldMap_bijective, Field.localization_map_bijective, eq_mpr_bijective, LinearEquiv.bijective, OrzechProperty.bijective_of_surjective_endomorphism, SymAlg.sym_bijective, AddCommGrpCat.isColimit_iff_bijective_desc, mul_left_bijective_of_finiteā‚€, Set.rangeFactorization_bijective, Prod.map_bijective, Polynomial.Gal.mapRoots_bijective, Multiset.bijective_iff_map_univ_eq_univ, CategoryTheory.Functor.CoconeTypes.IsColimit.bijective, Algebra.PreSubmersivePresentation.isUnit_jacobian_iff_aevalDifferential_bijective, Equiv.bijective, CochainComplex.IsKInjective.Qh_map_bijective, LinearMap.IsContPerfPair.bijective_left, CategoryTheory.Functor.FullyFaithful.map_bijective, OrderIso.bijective, gelfandTransform_bijective, MulAction.bijective, bijective_iterateFrobenius, TensorProduct.toIntegralClosure_bijective_of_smooth, Algebra.IsAlgebraic.algHom_bijective, CategoryTheory.isIso_iff_coyoneda_map_bijective, CategoryTheory.Functor.FullyFaithful.nonempty_iff_map_bijective, bijective_of_subsingleton, Injective.bijective_of_nat_card_le, conjneg_bijective, Finite.surjective_iff_bijective, FreeAddGroup.negRev_bijective, ContinuousLinearMap.IsInvertible.bijective, AddUnits.addRight_bijective, CochainComplex.HomComplex.CohomologyClass.bijective_toSmallShiftedHom_of_isKProjective, Algebra.QuasiFiniteAt.exists_fg_and_exists_notMem_and_awayMap_bijective, ContinuousAddEquiv.symm_bijective, IsLocalizedModule.fromLocalizedModule.bij, CochainComplex.HomComplex.CohomologyClass.bijective_toSmallShiftedHom_of_isKInjective, Equiv.bijective_comp, FirstOrder.Language.Equiv.symm_bijective, LocallyFiniteOrder.orderAddMonoidHom_bijective, CondensedSet.topCatAdjunctionCounit_bijective, RelIso.symm_bijective, PrimeSpectrum.comap_quotientMk_bijective_of_le_nilradical, IndexedPartition.piecewise_bij, Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRing, ContinuousMulEquiv.symm_bijective, SubAddAction.conjMap_ofFixingAddSubgroup_bijective, WeakDual.CharacterSpace.continuousMapEval_bijective, AddSubgroup.isComplement_addSubgroup_left_iff_bijective, Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed, RingHom.Bijective.isStableUnderBaseChange, WittVector.frobenius_bijective, ULift.map_bijective, RelIso.bijective, Subgroup.isComplement_iff_bijective, CategoryTheory.isIso_iff_bijective, CategoryTheory.Functor.bijective_sectionsPrecomp, IsPurelyInseparable.bijective_comp_algebraMap, ContinuousMap.Homotopy.symm_bijective, CategoryTheory.Functor.whiskerLeft_obj_map_bijective_of_isCoverDense, PerfectRing.bijective_frobenius, compl_bijective, Algebra.IsCentral.baseField_essentially_unique, DiscreteQuotient.proj_bot_bijective, DilationEquiv.symm_bijective, Prefunctor.IsCovering.star_bijective, DilationEquiv.bijective, Module.Invertible.bijective_curry, Prod.swap_bijective, HomotopicalAlgebra.LeftHomotopyClass.postcomp_bijective_of_weakEquivalence, HomotopicalAlgebra.LeftHomotopyClass.postcomp_bijective_of_fibration_of_weakEquivalence, IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed, TensorProduct.toIntegralClosure_mvPolynomial_bijective, mulLeft_bijectiveā‚€, ContinuousLinearEquiv.bijective, SubAddAction.ofFixingAddSubgroup_of_eq_bijective, RingHom.Bijective.ofLocalizationSpan, Representation.IsIrreducible.bijective_or_eq_zero, StarAlgebra.elemental.bijective_characterSpaceToSpectrum, IsField.localization_map_bijective, Units.mulRight_bijective, ContinuousMap.HomotopyRel.symm_bijective, FreeGroup.invRev_bijective, Algebra.Extension.CotangentSpace.map_toInfinitesimal_bijective, SimpleGraph.Walk.isHamiltonian_iff_support_get_bijective, NumberField.InfinitePlace.Completion.bijective_extensionEmbedding_of_isReal, LinearMap.flip_bijective_iff₁, LinearMap.flip_bijective_iffā‚‚, WithLp.toLp_bijective, Nat.Coprime.pow_left_bijective, Algebra.bijective_algebraMap_iff, Profinite.NobelingProof.iso_map_bijective, IsAzumaya.bij, Ideal.Quotient.mk_bijective_iff_eq_bot, ULift.down_bijective, TopologicalSpace.Opens.compl_bijective, AddGroup.addRight_bijective, Representation.apply_bijective, OrderIso.symm_bijective, CategoryTheory.Functor.isRepresentedBy_iff, Module.Invertible.bijective_of_surjective, Localization.mapPiEvalRingHom_bijective, CategoryTheory.PreGaloisCategory.evaluation_aut_bijective_of_isGalois, Module.IsReflexive.bijective_dual_eval', AdicCompletion.ofTensorProduct_bijective_of_finite_of_isNoetherian, Subspace.flip_quotDualCoannihilatorToDual_bijective, isHomeomorph_iff_continuous_bijective, AddSubgroup.isComplement_addSubgroup_right_iff_bijective, StarMulEquiv.symm_bijective, FirstOrder.Language.PartialEquiv.symm_bijective, LinearMap.linearProjOfIsCompl_comp_bijective_of_exact, IsLocalization.smul_bijective, AddEquiv.bijective, Module.End.isUnit_iff, SubAddAction.ofFixingAddSubgroup_insert_map_bijective, Algebra.Generators.cotangentRestrict_bijective_of_isCompl, RingHom.pi_bijective_of_isIdempotentElem, RingCon.lift_bijective_iff, AdicCompletion.of_bijective_iff, bijective_algebraMap_of_linearEquiv, AddEquiv.symm_bijective, PrimeSpectrum.toPiLocalization_bijective, Fintype.bijective_iff_surjective_and_card, Nat.bijective_iff_injective_and_card, AffineIsometryEquiv.symm_bijective, AddChar.doubleDualEmb_bijective, AdicCompletion.ofTensorProduct_bijective_of_pi_of_fintype, bijective_rootsOfUnityAddChar, LinearMap.dualMap_bijective_iff, CategoryTheory.Functor.CoconeTypes.IsColimit.iff_bijective, Fintype.bijective_iff_injective_and_card, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_map_bijective, StarAlgEquiv.symm_bijective, CategoryTheory.Pseudofunctor.bijective_toDescentData_map_iff, EquivLike.comp_bijective, Bijective.of_isEmpty, Embedding.schroeder_bernstein, AddOpposite.op_bijective, Set.sigmaToiUnion_bijective, IsometryEquiv.bijective, OrderAddMonoidIso.symm_bijective, Algebra.IsAlgebraic.algHom_bijectiveā‚‚, OpenPartialHomeomorph.MDifferentiable.mfderiv_bijective, List.getElem_bijective_iff, PartialEquiv.symm_bijective, Group.mulLeft_bijective, SimpleGraph.Walk.reverse_bijective, RingHom.SurjectiveOnStalks.residueFieldMap_bijective, IsArtinian.bijective_of_injective_endomorphism, RootPairing.Equiv.bijective_weightMap, PLift.map_bijective, Module.Invertible.rTensor_bijective_iff, CategoryTheory.ObjectProperty.leftOrthogonal.map_bijective_of_isTriangulated, Equiv.comp_bijective, AlgebraicGeometry.StructureSheaf.algebraMap_obj_top_bijective, ContinuousLinearEquiv.symm_bijective, Set.powersetCard.mulActionHom_compl_bijective, Module.Invertible.toModuleEnd_bijective, ENNReal.log_bijective, RingEquiv.symm_bijective, RingHom.prod_bijective_of_isIdempotentElem, CochainComplex.IsKProjective.Qh_map_bijective, Nat.bijective_iff_surjective_and_card, MulAction.bijectiveā‚€, Sum.map_bijective, Localization.exists_awayMap_bijective_of_residueField_surjective, bijective_iff_has_inverse, RootPairing.Equiv.bijective_coweightMap, Unique.bijective, LinearEquiv.symm_bijective, RingHom.Bijective.containsIdentities, Finite.injective_iff_bijective, Set.powersetCard.addActionHom_singleton_bijective, HomotopicalAlgebra.RightHomotopyClass.precomp_bijective_of_cofibration_of_weakEquivalence, CategoryTheory.Functor.bijective_colimitTypePrecomp, eq_mp_bijective, CategoryTheory.ConcreteCategory.hom_bijective, LightCondSet.topCatAdjunctionCounit_bijective, AlgebraicGeometry.isOpenImmersion_eq_inf, swap_bijective, Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq, OpenPartialHomeomorph.symm_bijective, Group.mulRight_bijective, AddSubmonoid.isLocalizationMap_iff_bijective, AlgebraicGeometry.instIsZariskiLocalAtTargetStalkwiseBijectiveCoeRingHom, Algebra.FormallyEtale.iff_comp_bijective, IsContDiffImplicitAt.bijective, CategoryTheory.MorphismProperty.isLocal_iff, Language.reverse_bijective, Surjective.bijectiveā‚‚_of_injective, Configuration.HasLines.exists_bijective_of_card_eq, SubAddAction.ofStabilizer.conjMap_bijective, Algebra.Extension.Cotangent.map_toInfinitesimal_bijective, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.sheafCondition_iff_bijective_toPullbackObj, CharacterModule.dual_bijective_iff_bijective, CochainComplex.HomComplex.CohomologyClass.toHom_bijective, LinearMap.bijective_or_eq_zero, IsAddUnit.isAddUnit_iff_addLeft_bijective, CategoryTheory.ConcreteCategory.isIso_iff_bijective, Subtype.coind_bijective, Injective.bijective_of_finite, StandardEtalePresentation.lift_bijective, SemimoduleCat.hom_bijective, LinearMap.lTensor_bij_iff_rTensor_bij, NumberField.InfinitePlace.Completion.bijective_extensionEmbeddingOfIsReal, LinearMap.IsPerfPair.bijective_right, IsLocalization.bijective_linearMap_mul', Set.bijective_iff_bijective_of_iUnion_eq_univ, IsPurelyInseparable.bijective_restrictDomain, IsLocalizedModule.map_bijective_iff_localizedModuleMap_bijective, eq_rec_on_bijective, SymAlg.unsym_bijective, SSet.IsStrictSegal.segal, Algebra.IsAlgebraic.bijective_of_isScalarTower', IsAlgClosed.algebraMap_bijective_of_isIntegral, Prefunctor.IsCovering.costar_bijective, MulEquiv.toMonoidWithZeroHom_bijective, Module.bijective_dual_eval, PEquiv.symm_bijective, NumberField.InfinitePlace.Completion.bijective_extensionEmbedding_of_isComplex, cast_bijective, LieEquiv.symm_bijective, Module.Relations.Solution.IsPresentation.bijective, Surjective.bijective_of_finite, LinearMap.IsContPerfPair.bijective_right, CategoryTheory.Functor.IsRepresentedBy.map_bijective, bijective_frobenius, RingHom.rangeRestrictField_bijective, AffineMap.linear_bijective_iff, Subgroup.isComplement_subgroup_left_iff_bijective, mulRight_bijectiveā‚€, IsCoveringMap.monodromy_bijective, Real.arsinh_bijective, Algebra.exists_etale_bijective_residueFieldMap_and_map_eq_mul_and_isCoprime, FixedPoints.toAlgHom_bijective, Equiv.symm_bijective, CategoryTheory.ObjectProperty.rightOrthogonal.map_bijective_of_isTriangulated, Submonoid.isLocalizationMap_iff_bijective, IsHomeomorph.bijective, Polynomial.Gal.galActionHom_bijective_of_prime_degree, ZMod.castHom_bijective, ENNReal.rpow_left_bijective, CategoryTheory.FreeGroupoid.of_obj_bijective, StarRingEquiv.symm_bijective, EquivLike.bijective, LieModuleEquiv.symm_bijective, WithLp.ofLp_bijective, CategoryTheory.isIso_iff_yoneda_map_bijective, AlgebraicGeometry.StructureSheaf.toOpenā‚—_top_bijective, AlgHom.bijective, HomotopicalAlgebra.RightHomotopyClass.precomp_bijective_of_weakEquivalence, SubAddAction.ofFixingAddSubgroupEmpty_equivariantMap_bijective, FiniteField.bijective_frobeniusAlgHom_pow, List.reverse_bijective, SubMulAction.ofFixingSubgroup_of_singleton_bijective, SubMulAction.ofFixingSubgroupEmpty_equivariantMap_bijective, Involutive.bijective, Algebra.Generators.cotangentRestrict_bijective_of_basis_kaehlerDifferential, UniformEquiv.bijective, Units.mulLeft_bijective, AdicCompletion.of_bijective, IsUnit.smul_bijective, TopologicalSpace.Closeds.compl_bijective, AffineIsometryEquiv.bijective, ConjClasses.mk_bijective, SubMulAction.map_ofFixingSubgroupUnion_bijective, Set.powersetCard.mulActionHom_singleton_bijective, Submodule.linearProjOfIsCompl_comp_bijective_of_exact, Monoid.Coprod.swap_bijective, SubMulAction.ofFixingSubgroup_of_eq_bijective, Set.BijOn.bijective, CompleteOrthogonalIdempotents.bijective_pi', AlgHom.normal_bijective, MulOpposite.unop_bijective, MulOpposite.op_bijective, Equiv.instCanLiftForallCoeBijective, Module.Invertible.bijective, Module.Invertible.lTensor_bijective_iff, Localization.localRingHom_bijective_of_saturated_inf_eq_top, List.Nodup.getBijectionOfForallMemList_coe, FiniteField.bijective_frobeniusAlgEquivOfAlgebraic_pow, Embedding.schroeder_bernstein_of_rel, SheafOfModules.bijective_pushforwardSections, ContextFreeRule.reverse_bijective, IsUnit.isUnit_iff_mulRight_bijective, Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq_aux, ContinuousAffineEquiv.bijective, IsometryEquiv.symm_bijective, SSet.Truncated.IsStrictSegal.spine_bijective, CategoryTheory.Abelian.Ext.mkā‚€_bijective, MulActionHom.oneEmbeddingMap_bijective, MulEquiv.bijective, ComplexShape.symm_bijective, OrderMonoidIso.symm_bijective, isHomeomorph_iff_continuous_isClosedMap_bijective, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.SheafCondition.bijective_toPullbackObj, bijective_rangeRestrict_comp_of_valuationRing, Prefunctor.IsCovering.pathStar_bijective, AddGroup.addLeft_bijective, FixedPoints.toAlgAut_bijective, PrimeSpectrum.piLocalizationToMaximal_bijective, MvPolynomial.esymmAlgHom_fin_bijective, Set.bijOn_univ, WithConv.ofConv_bijective, MulEquiv.symm_bijective, IsPurelyInseparable.bijective_algebraMap_of_isSeparable, AlgebraicGeometry.isomorphisms_eq_stalkwise, SubMulAction.ofStabilizer.conjMap_bijective, LinearMap.bijective_of_ne_zero, Localization.localRingHom_bijective_of_not_conductor_le, LinearMap.bijective_of_linearIndependent_of_span_eq_top, ModuleCat.hom_bijective, Nat.Coprime.nsmul_right_bijective, Profinite.NobelingProof.GoodProducts.range_equiv_smaller_toFun_bijective, List.map_bijective_iff, ContinuousAddEquiv.bijective, ContextFreeGrammar.reverse_bijective, mul_right_bijective_of_finiteā‚€, Set.bijective_iff_bijOn_univ, AddAction.bijective, CategoryTheory.PreGaloisCategory.toAut_bijective, AddOpposite.unop_bijective, PLift.up_bijective, RingEquiv.bijective, RingHom.Bijective.stableUnderComposition, CategoryTheory.regularTopology.EqualizerCondition.bijective_mapToEqualizer_pullback, Ideal.bijective_algebraMap_quotient_residueField, AlgEquiv.symm_bijective, AlgEquiv.bijective, Algebra.IsAlgebraic.bijective_of_isScalarTower, CategoryTheory.PreGaloisCategory.exists_galois_representative, IsAddUnit.vadd_bijective, Bool.not_bijective, IsLocalization.bijective, Injective.bijectiveā‚‚_of_surjective, Equidecomp.symm_bijective, Homeomorph.symm_bijective, CategoryTheory.Iso.symm_bijective, ComplexShape.Embedding.AreComplementary.fromSum_bijective, ContinuousMulEquiv.bijective, AffineEquiv.bijective, ContinuousLinearMap.bijective_iff_dense_range_and_antilipschitz, ContinuousAffineEquiv.symm_bijective, Subspace.quotDualCoannihilatorToDual_bijective, EquivLike.bijective_comp, unitInterval.symm_bijective, IsUnit.isUnit_iff_mulLeft_bijective, EuclideanGeometry.inversion_bijective, Subgroup.isComplement_subgroup_right_iff_bijective, CategoryTheory.Limits.Cofan.nonempty_isColimit_iff_bijective_fromSigma, Surjective.bijective_of_nat_card_le, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_bijective, SSet.Truncated.IsStrictSegal.segal, SubMulAction.ofFixingSubgroup_insert_map_bijective, CategoryTheory.Presieve.isSheafFor_ofArrows_iff_bijective_toCompabible, Prefunctor.bijective_costar_iff_bijective_star, Homeomorph.bijective, CategoryTheory.Limits.Types.isLimit_iff_bijective_sectionOfCone, not_bijective, Valuation.Integers.bijective_algebraMap_of_subsingleton_units_mrange, Fin.rev_bijective, CategoryTheory.MorphismProperty.isColocal_iff, SubMulAction.conjMap_ofFixingSubgroup_bijective, Matrix.vec_bijective, Algebra.FormallyEtale.comp_bijective, CategoryTheory.ConcreteCategory.bijective_of_isIso, bijective_iff_existsUnique, CategoryTheory.Limits.Multifork.isLimit_types_iff, Set.setOf_bijective, PrimeSpectrum.discreteTopology_iff_toPiLocalization_bijective, ContinuousLinearMap.isUnit_iff_bijective, Real.sinh_bijective, RingHom.FaithfullyFlat.codescendsAlong_bijective, CompleteOrthogonalIdempotents.bijective_pi, IsFractionRing.self_iff_bijective, IsAlgClosed.ringHom_bijective_of_isIntegral, IsAzumaya.AlgHom.mulLeftRight_bij, LinearIsometryEquiv.bijective, NNReal.rpow_left_bijective, IsAddUnit.isAddUnit_iff_addRight_bijective, Algebra.Extension.H1Cotangent.map_toInfinitesimal_bijective, AddActionHom.zeroEmbeddingMap_bijective, Algebra.ZariskisMainProperty.exists_fg_and_exists_notMem_and_awayMap_bijective

RingHom.Bijective

Theorems

NameKindAssumesProvesValidatesDepends On
containsIdentities šŸ“–mathematical—RingHom.ContainsIdentities
Function.Bijective
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
—Function.bijective_id
isStableUnderBaseChange šŸ“–mathematical—RingHom.IsStableUnderBaseChange
Function.Bijective
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
—respectsIso
Algebra.TensorProduct.includeLeft_bijective
IsScalarTower.left
ofLocalizationSpan šŸ“–mathematical—RingHom.OfLocalizationSpan
Function.Bijective
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
—bijective_of_isLocalization_of_span_eq_top
Localization.isLocalization
respectsIso šŸ“–mathematical—RingHom.RespectsIso
Function.Bijective
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
—RingHom.StableUnderComposition.respectsIso
stableUnderComposition
RingEquiv.bijective
stableUnderComposition šŸ“–mathematical—RingHom.StableUnderComposition
Function.Bijective
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
—Function.Bijective.comp

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