Documentation Verification Report

Nontrivial

📁 Source: Mathlib/RingTheory/TensorProduct/Nontrivial.lean

Statistics

MetricCount
DefinitionsNontrivial, Nontrivial, Nontrivial, Nontrivial
4
Theoremsnontrivial_of_algebraMap_injective_of_isDomain
1
Total5

Algebra.TensorProduct

Theorems

NameKindAssumesProvesValidatesDepends On
nontrivial_of_algebraMap_injective_of_isDomain 📖mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.instFunLike
algebraMap
Nontrivial
TensorProduct
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Algebra.toModule
Function.Injective.isDomain
RingHomClass.toMonoidWithZeroHomClass
RingHom.instRingHomClass
Localization.isLocalization
OreLocalization.instIsScalarTower
IsScalarTower.right
IsFractionRing.injective
RingHom.domain_nontrivial
IsScalarTower.to_smulCommClass
IsScalarTower.left
IsScalarTower.to_smulCommClass'
IsScalarTower.of_algHom
TensorProduct.CompatibleSMul.isScalarTower
Module.FaithfullyFlat.lTensor_nontrivial
Module.FaithfullyFlat.instOfNontrivialOfFree
FractionRing.instNontrivial
IsDomain.toNontrivial
Module.Free.of_divisionRing
IsDomain.to_noZeroDivisors

Cycle

Definitions

NameCategoryTheorems
Nontrivial 📖MathDef
5 mathmath: nontrivial_reverse_iff, Equiv.Perm.IsCycle.existsUnique_cycle_nontrivial_subtype, Nodup.nontrivial_iff, Equiv.Perm.nontrivial_toCycle, nontrivial_coe_nodup_iff

Finset

Definitions

NameCategoryTheorems
Nontrivial 📖MathDef
18 mathmath: Nontrivial.of_coe, one_lt_card_iff_nontrivial, not_nontrivial_singleton, nontrivial_coe, map_nontrivial, eq_singleton_or_nontrivial, not_nontrivial_empty, nontrivial_iff_ne_singleton, erase_nonempty, Set.Finite.toFinset_nontrivial, univ_nontrivial_iff, nontrivial_prod_iff, Set.toFinset_nontrivial, nontrivial_def, range_nontrivial, Nat.not_isPrimePow_iff_nontrivial_of_two_le, Nonempty.exists_eq_singleton_or_nontrivial, univ_nontrivial

Set

Definitions

NameCategoryTheorems
Nontrivial 📖MathDef
55 mathmath: not_subsingleton_iff, Convex.nontrivial_iff_nonempty_interior, offDiag_nonempty, image_nontrivial, einfsep_lt_top_iff, Finset.infsep_pos_iff_nontrivial, Finset.nontrivial_coe, nontrivial_of_einfsep_ne_top, Finite.infsep, Infinite.nontrivial, nontrivial_pair, RelSeries.length_ne_zero, RelSeries.length_pos, one_lt_ncard_iff_nontrivial, nontrivial_iff_pair_subset, Finite.infsep_pos_iff_nontrivial, subsingleton_or_nontrivial, infsep_eq_iInf, nontrivial_of_lt, MonoidWithZeroHom.range_nontrivial, infsep_pos_iff_nontrivial_of_finite, nontrivial_of_einfsep_lt_top, nontrivial_iff_ne_singleton, infsep_of_fintype, InjOn.image_nontrivial_iff, Metric.ediam_pos_iff, nontrivial_of_exists_ne, nontrivial_of_mem_mem_ne, not_nontrivial_empty, Equiv.Perm.isCycle_iff_exists_isCycleOn, nontrivial_of_exists_lt, ConjClasses.mem_noncenter, EMetric.diam_pos_iff, not_nontrivial_singleton, Subsingleton.not_nontrivial, Finite.toFinset_nontrivial, Matroid.IsCircuit.isCocircuit_inter_nontrivial, Finset.Nontrivial.coe, nontrivial_of_pair_subset, not_nontrivial_iff, Nonempty.exists_eq_singleton_or_nontrivial, einfsep_ne_top_iff, toFinset_nontrivial, Matroid.loopless_iff_forall_isCircuit, Matroid.IsCircuit.isCocircuit_disjoint_or_nontrivial_inter, one_lt_ncard_iff_nontrivial_and_finite, eq_singleton_or_nontrivial, one_lt_encard_iff_nontrivial, nontrivial_iff_exists_lt, nontrivial_univ, nontrivial_coe_sort, nontrivial_univ_iff, nontrivial_iff_exists_ne, powersetCard.coe_nontrivial_iff, nontrivial_of_infsep_pos

(root)

Definitions

NameCategoryTheorems
Nontrivial 📖CompData
362 mathmath: IsPGroup.center_nontrivial, Ideal.Quotient.nontrivial_of_liesOver_of_isPrime, WithZero.instNontrivial, SkewMonoidAlgebra.instNontrivialOfNonempty, Submonoid.LocalizationMap.instNontrivialLocalizationOfIsCancelMul, Module.instNontrivialDual, Additive.instNontrivial, Set.Nontrivial.coe_sort, RootPairing.IsIrreducible.nontrivial, instNontrivialCarrierStalkCommRingCatPresheafSmoothSheafCommRing, RootPairing.instNontrivialSubtypeSubmoduleMemSublatticeInvtRootSubmodule, instNontrivialFreeRing, UniformSpace.Completion.instNontrivialOfT0Space, SubsemiringClass.nontrivial, QuadraticAlgebra.instNontrivial, Multiplicative.instNontrivial, isDomain_iff_cancelMulZero_and_nontrivial, FractionalIdeal.instNontrivialNonZeroDivisors, instNontrivialShrink, MonoidWithZeroHom.mrange_nontrivial, Hamming.instNontrivialOfNonemptyOfDefault, Nonneg.nontrivial, LinearOrderedCommGroupWithZero.toNontrivial, Function.nontrivial, MonoidAlgebra.nontrivial, Set.nontrivial_of_nontrivial_coe, instNontrivialQuaternion, ENat.one_lt_card_iff_nontrivial, Nontrivial.of_not_isCyclic, AddSubgroup.instNontrivialSubtypeMemTop, Polynomial.subalgebraNontrivial, HahnSeries.instNontrivialOfNonempty, Valuation.instNontrivialSubtypeUnitsMemSubmonoidValueMonoidOfIsNontrivial, Matrix.nonempty, Equiv.nontrivial, Subsemiring.nontrivial, NormOneClass.nontrivial, Subtype.nontrivial_iff_exists_ne, Function.Surjective.nontrivial, Sym2.instNontrivial, Associates.instNontrivial, LieModule.exists_nontrivial_weightSpace_of_lieIdeal, AddSubmonoid.nontrivial_iff_exists_ne_zero, OreLocalization.nontrivial, instNontrivialLinearMapId, Nontrivial.of_not_isAddCyclic, Representation.invtSubmodule.nontrivial_iff, MulOpposite.instNontrivial, IsSimpleModule.nontrivial, LieAlgebra.instNontrivialSubtypeMemLieSubmoduleLieSubalgebraGenWeightSpaceOfNatForall, TensorProduct.nontrivial_of_linearMap_injective_of_flat_right, Submonoid.LocalizationMap.nontrivial, Nontrivial.of_finiteRingKrullDim, NumberField.mixedEmbedding.instNontrivialMixedSpace, DihedralGroup.instNontrivial, Module.nontrivial_of_rankAtStalk_pos, CliffordAlgebra.instNontrivialOfInvertibleOfNat, Nat.instNontrivial, Equiv.nontrivial_congr, LucasLehmer.X.instNontrivialOfFactLtNatOfNat, SimpleGraph.Subgraph.nontrivial_verts_of_degree_ne_zero, CStarMatrix.instNontrivial, Subgroup.bot_or_nontrivial, ContinuousMap.instNontrivialOfNonempty, LieAlgebra.non_trivial_center_of_isNilpotent, Module.rank_pos_iff_of_free, Submonoid.nontrivial_iff, PerfectClosure.instNontrivialOfIsReduced, SimpleGraph.nontrivial_of_ediam_ne_zero, AlgebraicGeometry.LocallyRingedSpace.component_nontrivial, IsSimpleOrder.toNontrivial, CategoryTheory.Subobject.nontrivial_of_not_isZero, Ring.instNontrivialNormalClosure, Rat.nontrivial, Bialgebra.nontrivial, Function.Injective.nontrivial, Module.FaithfullyFlat.lTensor_nontrivial, Quaternion.instNontrivial, Pi.nontrivial, Cardinal.instNontrivial, domain_nontrivial, Module.FaithfullyFlat.nontrivial_tensorProduct_iff_right, Module.length_pos_iff, Filter.instNontrivialFilter, RatFunc.instNontrivial, selfAdjoint.instNontrivialSubtypeMemAddSubgroup, Subgroup.nontrivial_iff, SimpleGraph.nontrivial_of_diam_ne_zero, Subgroup.nontrivial_iff_exists_ne_one, Set.powersetCard.nontrivial_iff, Int.instNontrivial, Set.nontrivial_of_nontrivial, Polynomial.nontrivial_iff, TopologicalSpace.Compacts.nontrivial_iff, HahnSeries.instNontrivialSubalgebra, Ordinal.nontrivial, QuotientAddGroup.nontrivial_iff, QuotientGroup.nontrivial_iff, LieModule.nontrivial_max_triv_of_isNilpotent, Finsupp.nontrivial_of_nontrivial, SymmetricAlgebra.instNontrivial, not_nontrivial_iff_subsingleton, CommRingCat.nontrivial_of_isPushout_of_isField, AffineSubspace.instNontrivial, TwoSidedIdeal.instNontrivial, SeparatingDual.instNontrivialContinuousLinearMapIdOfContinuousSMul, ULift.nontrivial, Nonneg.instNontrivialSubtypeLeOfNatOfAddLeftMono, Valuation.IsNontrivial.nontrivial_codomain, WithZero.instNontrivialUnits, Field.DirectLimit.nontrivial, Ideal.Quotient.nontrivial_iff, isSimpleModule_self_iff_isUnit, instNontrivialUnitsOfDenselyOrdered, RingCon.instNontrivial, SymAlg.instNontrivial, Profinite.NobelingProof.instNontrivialLocallyConstantIntOfNonempty, rank_pos_iff_nontrivial, RingTheory.Sequence.IsRegular.nontrivial, subsingleton_or_nontrivial, AlgebraicGeometry.Scheme.component_nontrivial, NumberField.RingOfIntegers.instNontrivial, MulAction.BlockMem.instNontrivial, PowerSeries.instNontrivial, RootPairing.isIrreducible_iff, nontrivial_of_invariantBasisNumber, MvPolynomial.nontrivial_of_nontrivial, TopologicalSpace.NonemptyCompacts.nontrivial_iff, nontrivial_of_ne, Module.nontrivial_of_finrank_eq_succ, AddSubgroup.nontrivial_iff, IsFractionRing.nontrivial_iff_nontrivial, Submodule.nontrivial_iff, LieModule.exists_nontrivial_weightSpace_of_isNilpotent, RingCon.nontrivial_iff, nontrivial_of_lt, RootPairing.nontrivial', Tropical.instNontrivialWithTopOfZero, isSimpleRing_iff_isTwoSided_imp, Module.nontrivial_of_finrank_pos, Semifield.toNontrivial, Ideal.Quotient.nontrivial, IsPGroup.nontrivial_iff_card, nontrivial_prod_left, PowerSeries.instNontrivialSubalgebra, Module.nontrivial_dual_iff, Sym.instNontrivialHAddNatOfNat, Pi.nontrivial_at, HahnEmbedding.ArchimedeanStrata.nontrivial_stratum, Module.nontrivial, Algebra.TensorProduct.nontrivial_of_algebraMap_injective_of_isDomain, NumberField.mixedEmbedding.euclidean.instNontrivialMixedSpace, Filter.Germ.instNontrivial, Polynomial.nontrivial, TrivSqZeroExt.instNontrivial_of_left, nontrivial_iff_exists_ne, IsFractionRing.nontrivial, nontrivial_of_hasRankNullity, SeparationQuotient.instNontrivialOfNontrivialTopology, Bool.instNontrivial, ArchimedeanClass.instNontrivial, AddSubmonoid.LocalizationMap.instNontrivialLocalizationOfIsCancelAdd, Fintype.one_lt_card_iff_nontrivial, MulArchimedeanClass.instNontrivial, MvPowerSeries.instNontrivialSubalgebraOfNonempty, Subsemigroup.instNontrivialOfNonempty, TensorAlgebra.instNontrivial, ZMod.nontrivial', Zsqrtd.nontrivial, nontrivial_iff, not_isRightRegular_zero_iff, isSimpleAddGroup_iff, SimpleGraph.instNontrivialEdgeLabelingOfNonemptyElemSym2EdgeSet, SimpleGraph.ediam_ne_zero_iff_nontrivial, Algebra.IsAlgebraic.nontrivial, SeparationQuotient.nontrivial_iff, QuaternionGroup.instNontrivial, ValuationRing.instNontrivialValueGroup, instNontrivialLex, alternatingGroup.nontrivial_of_three_le_card, PrimeSpectrum.nonempty_iff_nontrivial, Submodule.Quotient.nontrivial_of_ne_top, CharP.nontrivial_of_char_ne_one, QuaternionAlgebra.instNontrivial, ExteriorAlgebra.instNontrivial, SimpleGraph.eccent_pos_iff, Submonoid.nontrivial_iff_exists_ne_one, instNontrivialColex, IsLocalization.AtPrime.nontrivial, OrderDual.instNontrivial, Submodule.nontrivial_span_singleton, Unitization.instNontrivialRight, AddAction.nontrivial_of_fixedPoints_ne_univ, AddSubgroup.nontrivial_iff_exists_ne_zero, ContinuousLinearMap.instNontrivialId, Unitization.instNontrivialLeft, Complex.instNontrivial, isSimpleOrder_iff, AddOpposite.instNontrivial, IsLocalRing.not_isLocalRing_tfae, WithOne.instNontrivial, RootPairing.nontrivial, MulActionWithZero.nontrivial, Polynomial.Nontrivial.of_polynomial_ne, Algebra.nontrivial_of_isAlgebraic, OreLocalization.nontrivial_of_nonZeroDivisorsRight, FreeAbelianGroup.instNontrivialOfNonempty, Set.nontrivial_of_univ_nontrivial, ZMod.nontrivial_iff, IsAlgebraic.nontrivial, Module.finrank_pos_iff, LieModule.exists_nontrivial_weightSpace_of_isSolvable, Set.nontrivial_of_nonempty, Submodule.nontrivial_iff_ne_bot, Module.FaithfullyFlat.rTensor_nontrivial, SimpleGraph.nontrivial_iff, IsStrictOrderedRing.toNontrivial, Finset.univ_nontrivial_iff, RootPairing.IsIrreducible.nontrivial', Fin.instNontrivial, Ideal.instNontrivial, EMetric.nontrivial_iff_nontrivialTopology, instNontrivialStalkPresheafSmoothSheaf, Valuation.instNontrivialSubtypeUnitsMemSubgroupValueGroupOfIsNontrivial, Filter.nontrivial_iff_nonempty, DivisionSemiring.toNontrivial, Finset.instNontrivial, ValuativeRel.isNontrivial_iff_nontrivial_units, nontrivial_prod_right, AddSubmonoid.instNontrivial, IsLocalRing.toNontrivial, IsSimpleGroup.toNontrivial, Cardinal.one_lt_iff_nontrivial, Module.mem_support_iff_nontrivial_residueField_tensorProduct, LieSubmodule.instNontrivial, ZNum.nontrivial, ZMod.nontrivial, AddSubgroup.bot_or_nontrivial, Set.powersetCard.nontrivial', RingCon.nontrivial_quotient, LieSubalgebra.instNontrivialSubtypeMemOfIsCartanSubalgebra, not_isLeftRegular_zero_iff, Set.powersetCard.nontrivial, TwoPointing.to_nontrivial, Valuation.integers_nontrivial, Valuation.Integers.nontrivial_iff, WithTop.nontrivial, LieSubmodule.nontrivial_iff_ne_bot, Ideal.Quotient.nontrivial_of_liesOver_of_ne_top, HomogeneousLocalization.instNontrivialAtPrime, AddSubgroup.nontrivial_iff_ne_bot, Order.krullDim_pos_iff_of_orderTop, RingTheory.Sequence.IsRegular.quot_ofList_smul_nontrivial, WithBot.nontrivial, LieSubmodule.nontrivial_iff, CategoryTheory.instNontrivialSubobjectOfSimple, Order.krullDim_pos_iff_of_orderBot, FreeGroup.instNontrivialOfNonempty, Interval.instNontrivialOfNonempty, FreeAlgebra.instNontrivial, alternatingGroup.instNontrivialSubtypePermFinHAddNatOfNatMemSubgroup, Module.mem_support_iff, LieModule.nontrivial_of_isIrreducible, AddSubgroup.instNontrivial, AddSubmonoid.bot_or_nontrivial, Module.nonempty_support_iff, TopologicalSpace.Opens.instNontrivialOfNonempty, SimpleGraph.instNontrivial, TopologicalSpace.Compacts.instNontrivialOfNonempty, Subring.instNontrivialSubtypeMem, SimpleGraph.instNontrivialColoringOfNonempty, WithAbs.instNontrivial, EuclideanDomain.toNontrivial, ModP.nontrivial, AdjoinRoot.nontrivial, Module.supportDim_ne_bot_iff_nontrivial, LieModule.nontrivial_lowerCentralSeriesLast, instNontrivialSubtypeMemSubmoduleValSetIsotypicComponents, IsSimpleRing.instNontrivial, Module.FaithfullyFlat.iff_flat_and_rTensor_faithful, WithCStarModule.instNontrivial, not_nontrivial, IsSimpleAddGroup.toNontrivial, isDomain_iff_noZeroDivisors_and_nontrivial, DivisionRing.toNontrivial, isSimpleRing_isArtinianRing_iff, GaussianInt.instNontrivial, CommGroupWithZero.toNontrivial, TwoPointing.nonempty_two_pointing_iff, IsSMulRegular.not_zero_iff, OrderType.instNontrivial, Submodule.instNontrivial, RingHom.domain_nontrivial, Module.finrank_pos_iff_of_free, isSimpleModule_iff_toSpanSingleton_surjective, WittVector.instNontrivial, AddMonoidAlgebra.nontrivial, unitInterval.instNontrivialElemReal, Algebra.TensorProduct.nontrivial_of_algebraMap_injective_of_flat_right, Submonoid.instNontrivial, NumberField.InfiniteAdeleRing.instNontrivial, TrivSqZeroExt.instNontrivial_of_right, Equiv.instNontrivialPerm, Surreal.instNontrivial, not_subsingleton_iff_nontrivial, Subgroup.instNontrivial, MulAction.nontrivial_of_fixedPoints_ne_univ, Representation.invtSubmodule.instNontrivialSubtypeSubmoduleMemSublattice, instNontrivialTensorProduct, FreeAddGroup.instNontrivialOfNonempty, AddSubmonoid.nontrivial_iff, nontrivial_iff_lt, Fin.nontrivial_iff_two_le, GroupWithZero.toNontrivial, IsDomain.toNontrivial, Module.FaithfullyFlat.nontrivial_tensorProduct_iff_left, stdSimplex.instNontrivialElemForall, TensorProduct.nontrivial_of_linearMap_injective_of_flat_left, OreLocalization.nontrivial_of_nonZeroDivisorsLeft, isSimpleGroup_iff, AddSubsemigroup.instNontrivialOfNonempty, Algebra.TensorProduct.nontrivial_of_algebraMap_injective_of_flat_left, CategoryTheory.instNontrivialEndOfSimple, Module.FaithfullyFlat.iff_flat_and_lTensor_faithful, OreLocalization.nontrivial_iff, WithLp.instNontrivial, SimpleGraph.nontrivial_of_degree_ne_zero, Subgroup.instNontrivialSubtypeMemTop, Submodule.Quotient.nontrivial_iff, denselyOrdered_iff_denselyOrdered_units_and_nontrivial_units, LinearOrderedAddCommGroupWithTop.toNontrivial, instNontrivialOfCharZero, Function.nontrivial_of_nontrivial, Submonoid.bot_or_nontrivial, Set.nontrivial_coe_sort, Subgroup.nontrivial_iff_ne_bot, IsField.nontrivial, OreLocalization.nontrivial_of_nonZeroDivisors, TopologicalSpace.NonemptyCompacts.instNontrivial, WithConv.instNontrivial, Real.instNontrivial, Infinite.instNontrivial, FractionRing.instNontrivial, Set.nontrivial_univ_iff, PrimeSpectrum.nontrivial, MvPowerSeries.instNontrivial, NNRat.instNontrivial, Finite.one_lt_card_iff_nontrivial, instNontrivialProp, NonemptyInterval.instNontrivial, PrimeSpectrum.nontrivial_iff_mem_rangeComap, Option.nontrivial, Submodule.Quotient.nontrivial_of_lt_top, UpperHalfPlane.instNontrivial, AddAction.BlockMem.instNontrivial, Digraph.instNontrivialOfNonempty, AlgebraicGeometry.Scheme.Pullback.Triplet.instNontrivialCarrierTensor, Finsupp.instNontrivial, Set.nontrivial_mono, CauSeq.Completion.Cauchy.instNonTrivial, Field.toNontrivial, Module.End.instNontrivial

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