symm

📁 Source: MathlibTest/symm.lean

Statistics

Metric	Count
Definitions symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm	85
Theorems symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm, symm	196
Total	281

AbsoluteValue.IsEquiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AbsoluteValue.IsEquiv	—	—	—

AbsolutelyContinuousOnInterval

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AbsolutelyContinuousOnInterval	—	—	disjWithin_comm

AddCommGroup.ModEq

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AddCommGroup.ModEq	—	—	AddCommGroup.modEq_iff_nsmul

AddCommute

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AddCommute	—	—	—

AddCon

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	DFunLike.coe AddCon instFunLikeForallProp	—	—	Setoid.symm'

AddConstEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	7 math math: symm_symm, coe_symm_toEquiv, toEquiv_symm, symm_trans_self, symm_refl, self_trans_symm, equivUnits_symm_apply_symm_apply

AddEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	327 math math: CochainComplex.HomComplex.Cochain.rightShiftAddEquiv_symm_apply, AddMonoidAlgebra.mapDomainRingEquiv_apply, AddSubmonoid.LocalizationMap.symm_comp_ofAddEquivOfLocalizations_apply, WithConv.addEquiv_symm_apply_ofConv, AddAction.stabilizerEquivStabilizer_symm, CochainComplex.HomComplex.Cocycle.leftShiftAddEquiv_symm_apply, addMonoidHomCongrLeftEquiv_apply, AddAut.inv_symm, toMultiplicative_apply_symm_apply, AddMonCat.uliftFunctor_map, CochainComplex.HomComplex.Cocycle.equivHom_symm_apply, addSubgroupMap_symm_apply, Multiset.equivDFinsupp_symm_apply, AddAutAdditive_apply_symm_apply, Complex.equivRealProdAddHom_symm_apply, AddMonoidHom.ker_comp_addEquiv, invFun_eq_symm, AddMonoidHom.toAddEquiv_symm_apply, Matrix.entryAddHom_eq_comp, Finsupp.sumFinsuppAddEquivProdFinsupp_symm_inl, coprodPUnit_symm_apply, HahnSeries.addOppositeEquiv_symm_support, AddAut.mulRight_symm_apply, FreeAddGroup.freeAddGroupCongr_symm, AddSubmonoid.fromLeftNeg_leftNegEquiv_symm, AddSubgroup.comap_equiv_eq_map_symm, CochainComplex.HomComplex.Cocycle.equivHomShift_symm_postcomp, AddSubmonoid.val_neg_addUnitsTypeEquivIsAddUnitAddSubmonoid_symm_apply, Matrix.transposeAddEquiv_symm, QuotientAddGroup.equivQuotientZSMulOfEquiv_symm, MultilinearMap.domDomCongrEquiv_symm_apply, AddSubmonoid.LocalizationMap.addEquivOfLocalizations_symm_apply, AddEquivClass.apply_coe_symm_apply, AddSubmonoid.add_leftNegEquiv_symm, uliftMultiplesHom_symm_apply, funUnique_symm_apply, coe_prodComm_symm, symm_comp_eq, AddHom.toAddEquiv_symm_apply, RingEquiv.op_apply_symm_apply, MulOpposite.coe_opLinearEquiv_symm_addEquiv, AddEquivClass.coe_symm_apply_apply, AddAutAdditive_symm_apply_symm_apply, AddSubmonoid.LocalizationMap.ofAddEquivOfLocalizations_eq_iff_eq, AddSubsemigroup.mem_map_equiv, mk_coe', MonoidAlgebra.opRingEquiv_symm_apply, comp_symm_eq, neg_symm, AddMonoidHom.apply_ofInjective_symm, MonoidAlgebra.symm_mapDomainAddEquiv, Finsupp.sumFinsuppAddEquivProdFinsupp_symm_apply, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.biprodAddEquiv_symm_biprodIsoProd_hom_toBiprod_apply, sigmaFinsuppAddEquivDFinsupp_symm_apply, zmultiplesAddHom_symm_apply, finsuppUnique_symm, CochainComplex.HomComplex.Cocycle.equivHomShift'_symm_apply, AddMonoidAlgebra.opRingEquiv_symm_apply, AddGrpCat.uliftFunctor_map, AddCommGrpCat.Colimits.colimitCocone_ι_app, multiplesAddHom_symm_apply, AddSubmonoid.centerCongr_symm_apply_coe, symmEquiv_symm_apply_symm_apply, CategoryTheory.ProjectiveResolution.extEquivCohomologyClass_symm_mk_hom, Matrix.coe_ofAddEquiv_symm, symm_ofLexAddEquiv, toMultiplicativeLeft_apply_symm_apply, coe_addMonoidHom_comp_coe_addMonoidHom_symm, self_comp_symm, symm_addMonoidHomCongrRight, CochainComplex.HomComplex.CohomologyClass.toSmallShiftedHom_mk, Matrix.uniqueAddEquiv_symm_apply, AddSubmonoid.centerToAddOpposite_symm_apply_coe, AddMonoidHom.ofLeftInverse_symm_apply, Subalgebra.mopAlgEquivOp_symm_apply, AddSubsemigroup.centerToAddOpposite_symm_apply_coe, CategoryTheory.ProjectiveResolution.extAddEquivCohomologyClass_symm_apply, AddAut.conj_symm_apply, toMultiplicativeRight_apply_symm_apply, Matrix.entryAddMonoidHom_eq_comp, uniqueProd_symm_apply, AddSubsemigroup.comap_equiv_eq_map_symm, FreeAbelianGroup.liftAddEquiv_symm_apply, addMonoidHomCongrLeft_apply, LinearEquiv.funUnique_symm_apply, AddSubmonoid.LocalizationMap.symm_comp_ofAddEquivOfLocalizations_apply', symm_apply_apply, AddAut.coe_inv, toAddMagmaCatIso_inv, toAdditive_toMultiplicative_symm_apply, AlgEquiv.op_apply_symm_apply, withZeroCongr_symm, DirectSum.addEquivProdDirectSum_symm_apply_toFun, CategoryTheory.Iso.addCommGroupIsoToAddEquiv_symm_apply, coe_addEquiv_lpPiLp_symm, toAddMonCatIso_inv, MulAutMultiplicative_symm_apply_symm_apply, FreeAbelianGroup.equivFinsupp_symm_apply, ModuleCat.homAddEquiv_symm_apply_hom, piUnique_symm_apply, toAddCommMonCatIso_inv, AddCommGrpCat.homAddEquiv_symm_apply_hom, CategoryTheory.ProjectiveResolution.extMk_hom, AddSubsemigroup.topEquiv_symm_apply_coe, op_symm_apply_symm_apply, Rep.toAdditive_symm_apply, symm_mapAddSubgroup, AddSubmonoid.comap_equiv_eq_map_symm, LinearEquiv.coe_toAddEquiv_symm, AddAction.zmultiplesQuotientStabilizerEquiv_symm_apply, AddSubgroup.map_equiv_eq_comap_symm, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.mk₀_f_comp_biprodAddEquiv_symm_biprodIsoProd_hom, AddAction.stabilizerEquivStabilizer_symm_apply, IsPrimitiveRoot.zmodEquivZPowers_symm_apply_pow', toEquiv_symm, QuadraticForm.dualProdIsometry_invFun, AddCon.comapQuotientEquivOfSurj_symm_mk, symmEquiv_apply_symm_apply, Subring.mopRingEquivOp_symm_apply, AddMonoid.IsTorsion.torsionAddEquiv_symm_apply_coe, add_submonoid_map_symm_apply, symm_toLexAddEquiv, AddSubmonoid.val_addUnitsTypeEquivIsAddUnitAddSubmonoid_symm_apply, mapAddSubgroup_symm_apply, Complex.equivRealProdAddHom_symm_apply_re, Complex.equivRealProdAddHom_symm_apply_im, Subsemiring.addEquivOp_symm_apply_coe, mulOp_symm_apply, Finsupp.mapRange.addEquiv_symm, toMultiplicativeLeft_symm_apply_symm_apply, coprodAssoc_symm_apply_inr_inr, groupCohomology.norm_ofAlgebraAutOnUnits_eq, symm_bijective, AddSubmonoid.leftNegEquiv_symm_add, CategoryTheory.InducedCategory.homAddEquiv_symm_apply_hom, piCongrRight_symm, AddGroupExtension.Equiv.coe_symm, eq_comp_symm, CochainComplex.HomComplex.CohomologyClass.equiv_toSmallShiftedHom_mk, AlgEquiv.symm_toAddEquiv, mulOp_apply, AddSubgroup.addSubgroupOfEquivOfLe_symm_apply_coe_coe, piAdditive_symm_apply, symm_toContinuousAddEquiv, AddCommGrpCat.Colimits.Quot.desc_colimitCocone, Finsupp.domCongr_symm, AddSubmonoid.leftNegEquiv_symm_fromLeftNeg, DirectSum.addEquivProdDirectSum_symm_apply_support', AddSubsemigroup.map_equiv_eq_comap_symm, starL_symm_apply, AddSubgroup.topEquiv_symm_apply_coe, SkewMonoidAlgebra.toFinsuppAddEquiv_symm_apply, apply_symm_apply, symm_addMonoidHomCongrLeftEquiv, AddAction.stabilizerEquivStabilizer_neg, addMonoidHomCongrRightEquiv_symm_apply, AddSubgroup.map_equiv_eq_comap_symm', equivLike_neg_eq_symm, DFinsupp.mapRange.addEquiv_symm, toIntLinearEquiv_symm, CochainComplex.HomComplex.Cocycle.equivHomShift_symm_apply, AlgEquiv.op_symm_apply_symm_apply, IsPrimitiveRoot.zmodEquivZPowers_symm_apply_pow, Multiset.toFinsupp_symm_apply, AddSubmonoid.topEquiv_symm_apply_coe, MonoidAlgebra.symm_mapRangeAddEquiv, Representation.ofModule_asModule_act, Matrix.piAddEquiv_symm_apply, QuotientAddGroup.quotientKerEquivOfRightInverse_symm_apply, coprodAssoc_symm_apply_inl, punitCoprod_symm_apply, symmEquiv_symm_apply_apply, starL'_symm_apply, AddSubgroup.centerCongr_symm_apply_coe, prodAdditive_symm_apply, CochainComplex.HomComplex.Cocycle.rightShiftAddEquiv_symm_apply, coe_toEquiv_symm, coe_toLinearEquiv_symm, coe_prodAssoc_symm, AddLocalization.addEquivOfQuotient_symm_mk', RestrictScalars.lsmul_apply_apply, groupHomology.H1AddEquivOfIsTrivial_symm_apply, toMultiplicativeRight_symm_apply_symm_apply, AddSubmonoid.LocalizationMap.addEquivOfLocalizations_symm_eq_addEquivOfLocalizations, coe_addEquiv_lpBCF_symm, addEquivOfAddOrderOfEq_symm_apply_gen, addSubmonoidMap_symm_apply, AddOpposite.coe_symm_opAddEquiv, AddSubsemigroup.centerCongr_symm_apply_coe, symm_trans_apply, symm_comp_self, opOp_symm_apply, RestrictScalars.addEquiv_symm_map_smul_smul, RingEquiv.op_symm_apply_symm_apply, symm_symm, RingEquiv.nonUnitalSubsemiringMap_symm_apply_coe, AddSubgroup.mem_map_equiv, CategoryTheory.HomOrthogonal.matrixDecompositionAddEquiv_symm_apply, HahnSeries.addOppositeEquiv_symm_orderTop, symm_apply_eq, AddCommGroup.DirectLimit.congr_symm_apply_of, toAddGrpIso_inv, AddCommGrpCat.uliftFunctor_map, IsPrimitiveRoot.zmodEquivZPowers_symm_apply_zpow', QuaternionAlgebra.coe_symm_addEquivProd, CategoryTheory.InjectiveResolution.extMk_hom, AddAut.inv_apply, ofLeftInverse'_symm_apply, AddMonoidAlgebra.domCongr_symm, AddSubgroup.centerToAddOpposite_symm_apply_coe, AddOpposite.opAddEquiv_symm_apply, toMultiplicative_symm_apply_symm_apply, CategoryTheory.Adjunction.homAddEquiv_symm_apply, val_piAddUnits_symm_apply, cast_symm_apply, AddMonoidAlgebra.symm_mapDomainRingEquiv, additiveMultiplicative_symm_apply, AddLocalization.addEquivOfQuotient_symm_mk, DistribMulAction.toAddEquiv_symm_apply, AddAut.inv_def, coe_symm_toNatLinearEquiv, finsuppUnique_symm_apply_support_val, HahnSeries.addOppositeEquiv_symm_leadingCoeff, CategoryTheory.Abelian.Ext.addEquiv₀_symm_apply, Representation.ofModule_asAlgebraHom_apply_apply, QuotientAddGroup.quotientBot_symm_apply, MulEquiv.toAdditive_symm_apply_symm_apply, refl_symm, Finsupp.lift_symm_apply, AddSubmonoid.LocalizationMap.ofAddEquivOfDom_comp_symm, addSubgroupCongr_symm_apply, AddAut.symm_inv, MulEquiv.toAdditive_apply_symm_apply, coe_addMonoidHom_symm_comp_coe_addMonoidHom, AddLocalization.addEquivOfQuotient_symm_addMonoidOf, AlgEquiv.opComm_symm_apply_symm_apply, symmEquiv_apply_apply, DirectSum.decomposeAddEquiv_symm_apply, MulAutMultiplicative_apply_symm_apply, AddChar.doubleDualEquiv_symm_doubleDualEmb_apply, symm_comapAddSubgroup, symm_addMonoidHomCongrRightEquiv, AddCommGrpCat.Colimits.Quot.desc_quotQuotUliftAddEquiv, subsemigroupMap_symm_apply_coe, ContinuousMap.addEquivBoundedOfCompact_symm_apply, AddMonoidAlgebra.domCongr_apply, AddSubgroup.map_symm_eq_iff_map_eq, mapMatrix_symm, finsuppAddEquivDFinsupp_symm_apply, Finsupp.liftAddHom_symm_apply_apply, MulOpposite.opLinearEquiv_symm_toAddEquiv, CategoryTheory.Abelian.Ext.addEquivBiprod_symm_apply, neg'_symm_apply, AddCon.comapQuotientEquivOfSurj_symm_mk', Finsupp.sumFinsuppAddEquivProdFinsupp_symm_inr, HahnSeries.addOppositeEquiv_symm_apply_coeff, val_neg_piAddUnits_symm_apply, Finsupp.curryAddEquiv_symm_apply, Finsupp.liftAddHom_symm_apply, CochainComplex.HomComplex.Cochain.leftShiftAddEquiv_symm_apply, AddSubmonoid.map_equiv_eq_comap_symm, AddSubgroup.comap_equiv_eq_map_symm', AddAut.mulLeft_apply_symm_apply, eq_symm_apply, WeierstrassCurve.Jacobian.Point.toAffineAddEquiv_symm_apply, RingEquiv.coe_toAddEquiv_symm, CochainComplex.HomComplex.Cocycle.equivHomShift_symm_precomp, addMonoidAlgebraAddEquivDirectSum_symm_apply, coprodCongr_symm_apply, finsuppUnique_symm_apply_apply, prodUnique_symm_apply, coprodComm_symm_apply, toNatLinearEquiv_symm, Equiv.addEquiv_symm_apply, WithLp.addEquiv_symm_apply, AddChar.doubleDualEmb_doubleDualEquiv_symm_apply, toAddUnits_symm_apply, LinearEquiv.arrowCongrAddEquiv_symm_apply, AddUnits.coe_opEquiv_symm, Subsemiring.mopRingEquivOp_symm_apply, WeierstrassCurve.Projective.Point.toAffineAddEquiv_symm_apply, AlgEquiv.subalgebraMap_symm_apply_coe, ModuleCat.semilinearMapAddEquiv_symm_apply_apply, coprodAssoc_symm_apply_inr_inl, symm_mk, symm_trans_self, RestrictScalars.addEquiv_symm_map_algebraMap_smul, AddCircle.equivAddCircle_symm_apply_mk, CategoryTheory.Abelian.Ext.biprodAddEquiv_symm_apply, DFinsupp.liftAddHom_symm_apply, AddMonoidAlgebra.symm_mapDomainAddEquiv, toAddSemigrpIso_inv, groupHomology.H1AddEquivOfIsTrivial_symm_tmul, AddCon.quotientKerEquivOfRightInverse_symm_apply, toContinuousAddEquiv_symm_apply, CochainComplex.HomComplex.CohomologyClass.toHom_mk, MulOpposite.opAddEquiv_symm_apply, apply_eq_iff_symm_apply, toAddCommGrpIso_inv, AddSubmonoid.mem_map_equiv, CategoryTheory.InjectiveResolution.extEquivCohomologyClass_symm_mk_hom, ofBijective_apply_symm_apply, AddAut.congr_apply, ofLeftInverse_symm_apply, Matrix.conjTransposeAddEquiv_symm, AddCommMonCat.uliftFunctor_map, AlternatingMap.domDomCongrEquiv_symm_apply, WithLp.coe_symm_addEquiv, self_trans_symm, Matrix.compAddEquiv_symm_apply, AddSubmonoid.leftNegEquiv_symm_eq_neg, CategoryTheory.InjectiveResolution.extAddEquivCohomologyClass_symm_apply, AddMonoidAlgebra.symm_mapRangeAddEquiv, strictMono_symm, IsPrimitiveRoot.zmodEquivZPowers_symm_apply_zpow, op_apply_symm_apply, RestrictScalars.smul_def, Subring.addEquivOp_symm_apply_coe, QuaternionAlgebra.coe_symm_addEquivTuple, comapAddSubgroup_symm_apply, addEquivOfAddOrderOfEq_symm, AddAut.congr_symm_apply, symm_addMonoidHomCongrLeft, SemimoduleCat.homAddEquiv_symm_apply_hom, coe_symm_toIntLinearEquiv, eq_symm_comp, uliftZMultiplesHom_symm_apply

AddGroupExtension.Equiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	4 math math: refl_symm_apply, symm_symm_apply, coe_symm, trans_symm_apply

AddGroupExtension.IsConj

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AddGroupExtension.IsConj	—	—	map_neg AddMonoidHom.instAddMonoidHomClass neg_one_zsmul add_assoc add_one_zsmul neg_add_cancel zero_zsmul zero_add mul_zsmul' mul_one neg_neg one_zsmul Mathlib.Tactic.Group.zsmul_trick_zero' add_zero

AddSubgroup.Commensurable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AddSubgroup.Commensurable	—	—	—

AddSubgroup.IsComplement'

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AddSubgroup.IsComplement'	—	—	neg_neg neg_add_rev AddSubgroup.isComplement'_def AddSubgroup.isComplement_iff_bijective Equiv.bijective_comp Equiv.comp_bijective

AddValuation.IsEquiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AddValuation.IsEquiv	—	—	Valuation.IsEquiv.symm

AffineEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	35 math math: coe_symm_toEquiv, symm_trans_self, AffineIsometryEquiv.coe_symm_toAffineEquiv, self_trans_symm, AffineSubspace.comap_symm, prodAssoc_symm_apply, ContinuousAffineEquiv.prodComm_symm_apply, AffineSubspace.map_symm, toEquiv_symm, coe_homothetyUnitsMulHom_apply_symm, linear_symm, inv_def, apply_symm_apply, AffineSubspace.topEquiv_symm_apply_coe, equivUnitsAffineMap_symm_apply_symm_apply, symm_apply_apply, coe_constVSub_symm, apply_eq_iff_eq_symm_apply, ofEq_symm, constVAdd_symm, image_symm, prodComm_symm, prodCongr_symm, AffineBasis.coord_vadd, vaddConst_symm_apply, ContinuousAffineEquiv.toAffineEquiv_symm, coe_toHomeomorphOfFiniteDimensional_symm, ContinuousAffineEquiv.coe_symm_toAffineEquiv, val_inv_equivUnitsAffineMap_apply, preimage_symm, AffineIsometryEquiv.toAffineEquiv_symm, pointReflection_symm, symm_refl, ContinuousAffineEquiv.prodAssoc_symm_apply, ofBijective.symm_eq

AffineIsometryEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	23 math math: coe_symm_toAffineEquiv, symm_trans_self, EuclideanGeometry.reflection_symm, coe_symm_trans, toContinuousAffineEquiv_symm, ofTop_symm_apply_coe, coe_inv, ofEq_symm, apply_symm_apply, symm_bijective, symm_constVSub, symm_symm, symm_apply_apply, self_trans_symm, coe_vaddConst_symm, pointReflection_symm, coe_symm_toContinuousAffineEquiv, AffineSubspace.isometryEquivMap.apply_symm_apply, coe_symm_toIsometryEquiv, toAffineEquiv_symm, toHomeomorph_symm, coe_symm_toHomeomorph, toIsometryEquiv_symm

AffineSubspace.Parallel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AffineSubspace.Parallel	—	—	AffineSubspace.map_map AffineEquiv.coe_trans_to_affineMap AffineEquiv.constVAdd_add neg_add_cancel AffineEquiv.constVAdd_zero AffineEquiv.coe_refl_to_affineMap AffineSubspace.map_id

AffineSubspace.SOppSide

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AffineSubspace.SOppSide	—	—	AffineSubspace.sOppSide_comm

AffineSubspace.SSameSide

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AffineSubspace.SSameSide	—	—	AffineSubspace.sSameSide_comm

AffineSubspace.WOppSide

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AffineSubspace.WOppSide	—	—	AffineSubspace.wOppSide_comm

AffineSubspace.WSameSide

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AffineSubspace.WSameSide	—	—	AffineSubspace.wSameSide_comm

AlgEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	315 math math: MvPolynomial.pUnitAlgEquiv_symm_monomial, PowerSeries.IsWeierstrassFactorizationAt.algEquivQuotient_symm_apply, coe_restrictScalars_symm, AddMonoidAlgebra.symm_mapRangeAlgEquiv, leftInverse_symm, coe_adjoinSingletonEquivAdjoinRootMinpoly_symm, IsLocalization.algEquiv_symm_mk', Quaternion.snd_imJ_dualNumberEquiv_symm, AlgCat.hom_inv_associator, MvPolynomial.comapEquiv_symm_coe, self_trans_symm, Submodule.coe_mapAlgEquiv_symm_apply, DoubleQuot.coe_quotQuotEquivQuotOfLEₐ_symm, IntermediateField.adjoinRootEquivAdjoin_symm_apply_gen, Algebra.IsPushout.equiv_symm_algebraMap_left, DirectSum.decomposeAlgEquiv_symm_apply, AdicCompletion.of_ofAlgEquiv_symm, Algebra.TensorProduct.congr_symm_apply, Polynomial.algEquivOfCompEqX_symm, MvPolynomial.pUnitAlgEquiv_symm_apply, Polynomial.quotientSpanXSubCAlgEquiv_symm_apply, LinearEquiv.conjAlgEquiv_symm_apply_apply, toOpposite_symm_apply, CliffordAlgebraDualNumber.equiv_symm_eps, IsAdjoinRoot.algEquiv_symm, StandardEtalePresentation.toPresentation_algebra_smul, IsAdjoinRoot.adjoinRootAlgEquiv_symm_apply_eq_mk, Algebra.TensorProduct.prodRight_symm_tmul, Algebra.TensorProduct.tensorTensorTensorComm_symm, IsGaloisGroup.mulEquivAlgEquiv_apply_symm_apply, IntermediateField.botEquiv_symm, AlgCat.hom_inv_rightUnitor, Polynomial.pUnitAlgEquiv_symm_toPowerSeries, adjoinRootXPowSubCEquiv_symm_eq_root, Quaternion.snd_re_dualNumberEquiv_symm, coe_inv, symm_apply_eq, CliffordAlgebra.equivBaseChange_symm_apply, MvPolynomial.supportedEquivMvPolynomial_symm_X, DoubleQuot.quotQuotEquivQuotOfLEₐ_symm_toRingEquiv, DoubleQuot.quotQuotEquivComm_symmₐ, PowerBasis.equivAdjoinSimple_symm_aeval, NumberField.InfinitePlace.smul_apply, symm_toEquiv_eq_symm, Algebra.TensorProduct.algEquivIncludeRange_symm_tmul, adjoinSingletonEquivAdjoinRootMinpoly_symm_toAlgHom, AdjoinRoot.quotEquivQuotMap_symm_apply_mk, aut_inv, ofBijective_symm_apply_apply, Polynomial.algEquivCMulXAddC_symm_eq, FractionalIdeal.mapEquiv_symm, arrowCongr_apply, Algebra.Presentation.quotientEquiv_symm, MvPolynomial.eval₂_const_pUnitAlgEquiv_symm, IsPrimitiveRoot.adjoinEquivRingOfIntegers_symm_apply, NumberField.InfinitePlace.smul_eq_comap, Algebra.TensorProduct.tensorTensorTensorComm_symm_tmul, equivCongr_symm, NumberField.InfinitePlace.smul_mk, LocallyConstant.congrRightₐ_symm_apply_apply, Subalgebra.mopAlgEquivOp_symm_apply, StarAlgEquiv.toAlgEquiv_symm, TensorAlgebra.equivFreeAlgebra_symm_ι, matPolyEquiv_symm_apply_coeff, IntermediateField.equivOfEq_symm, PiTensorProduct.constantBaseRingEquiv_symm, equivCongr_apply, Polynomial.toMvPolynomial_eq_rename_comp, Subalgebra.LinearDisjoint.mulMapLeftOfSupEqTop_symm_apply, Quaternion.fst_imK_dualNumberEquiv_symm, QuaternionAlgebra.imK_swapEquiv_symm_apply, StandardEtalePair.equivMvPolynomialQuotient_symm_apply, CategoryTheory.Iso.toAlgEquiv_symm_apply, PowerBasis.quotientEquivQuotientMinpolyMap_symm_apply_mk, sumArrowEquivProdArrow_symm_apply_inr, CommAlgCat.algEquivOfIso_symm_apply, MvPolynomial.optionEquivRight_symm_apply, op_apply_symm_apply, AdjoinRoot.equiv'_symm_toAlgHom, LinearEquiv.symm_conjAlgEquiv, MulSemiringAction.toAlgEquiv_symm_apply, NumberField.ComplexEmbedding.IsConj.symm, eq_symm_apply, piCongrRight_symm, symm_toMulEquiv, toLinearEquiv_symm, IsSymmetricAlgebra.equiv_symm_comp, MonoidAlgebra.symm_mapRangeAlgEquiv, ofAlgHom_symm_apply, FiniteField.frobeniusAlgEquiv_symm_apply, polyEquivTensor_symm_apply_tmul, Polynomial.Bivariate.equivMvPolynomial_symm_X_0, MvPolynomial.algebraTensorAlgEquiv_symm_comp_aeval, cast_symm_apply, NumberField.InfinitePlace.smul_coe_apply, Ideal.quotientKerAlgEquivOfRightInverse_symm_apply, CliffordAlgebraQuaternion.equiv_symm_apply, MonoidAlgebra.tensorEquiv_symm_single, Matrix.charpoly.optionEquivLeft_symm_univ_isHomogeneous, addMonoidAlgebraAlgEquivDirectSum_symm_apply, restrictScalars_symm_apply, AdicCompletion.mk_ofAlgEquiv_symm, apply_symm_apply, coe_symm_toLinearEquiv, FiniteField.frobeniusAlgEquivOfAlgebraic_symm_apply, coe_apply_coe_coe_symm_apply, Polynomial.Bivariate.equivMvPolynomial_symm_C, coe_polyEquivTensor'_symm, PowerBasis.equivAdjoinSimple_symm_gen, MvPolynomial.algebraTensorAlgEquiv_symm_X, Matrix.kroneckerTMulAlgEquiv_symm_apply, toUnder_inv_right_apply, mapMatrix_symm, NumberField.InfinitePlace.ComplexEmbedding.exists_comp_symm_eq_of_comp_eq, symm_toAddEquiv, symm_trans_apply, QuaternionAlgebra.re_swapEquiv_symm_apply, Quaternion.fst_imJ_dualNumberEquiv_symm, PowerBasis.equivOfRoot_symm, AlgCat.hom_inv_leftUnitor, IsAdjoinRoot.algEquiv_def, toLieEquiv_symm_apply, CliffordAlgebra.reverseOpEquiv_opComm, Polynomial.algEquivAevalNegX_symm_apply, symm_comp, Localization.algEquiv_symm_mk', autCongr_symm, Matrix.dualNumberEquiv_symm_apply, Polynomial.Bivariate.equivMvPolynomial_symm_X_1, galLiftEquiv_symm_apply, Algebra.pushoutDesc_apply, Quaternion.fst_imI_dualNumberEquiv_symm, DoubleQuot.coe_quotQuotEquivQuotSupₐ_symm, Polynomial.algEquivOfCompEqX_symm_apply, IsSymmetricAlgebra.equiv_symm_apply, HahnSeries.toPowerSeriesAlg_symm_apply_coeff, Localization.algEquiv_symm_mk, Polynomial.algEquivAevalXAddC_symm_apply, mk_coe', op_symm_apply_symm_apply, LocallyConstant.congrLeftₐ_symm_apply_apply, DoubleQuot.quotQuotEquivQuotOfLE_symm_comp_mkₐ, Matrix.reindexAlgEquiv_symm, PowerBasis.quotientEquivQuotientMinpolyMap_symm_apply, QuaternionAlgebra.imI_swapEquiv_symm_apply, SymmetricAlgebra.equivMvPolynomial_symm_X, LinearMap.toMatrixAlgEquiv_symm, CliffordAlgebraComplex.equiv_symm_apply, Algebra.TensorProduct.cancelBaseChange_symm_tmul, NormedRing.algEquivComplexOfComplete_symm_apply, algHomUnitsEquiv_apply_symm_apply, coe_coe_symm_apply_coe_apply, extendScalarsOfSurjective_symm, symm_mk, RingCon.coe_comapQuotientEquivRangeₐ_symm_mk, Algebra.TensorProduct.assoc_symm_tmul, Algebra.TensorProduct.opAlgEquiv_symm_apply, matrixEquivTensor_apply_symm, symm_apply_apply, StarAlgEquiv.coe_symm_toAlgEquiv, LinearEquiv.algEquivOfRing_symm_apply, toAlgebraIso_inv, IsLocalization.algEquivOfAlgEquiv_symm_apply, WithLp.unitizationAlgEquiv_symm_apply_ofLp, IsPrimitiveRoot.adjoinEquivRingOfIntegersOfPrimePow_symm_apply, Subalgebra.lTensorBot_symm_apply, MvPolynomial.finSuccEquiv_comp_C_eq_C, Quaternion.snd_imK_dualNumberEquiv_symm, AdjoinRoot.algEquivOfAssociated_symm, matPolyEquiv_symm_C, opComm_apply_symm_apply, LaurentPolynomial.invert_symm, CommAlgCat.associator_inv_hom, MonoidAlgebra.domCongr_symm, Matrix.compAlgEquiv_symm_apply, IsAdjoinRoot.adjoinRootAlgEquiv_symm_apply_root, AddMonoidAlgebra.symm_commAlgEquiv, CliffordAlgebra.prodEquiv_symm_apply, MvPolynomial.renameSymmetricSubalgebra_symm_apply_coe, HahnSeries.ofPowerSeriesAlg_apply_coeff, MvPolynomial.mapAlgEquiv_symm, RingCon.quotientQuotientEquivQuotientₐ_symm_mk, Polynomial.IsDistinguishedAt.algEquivQuotient_symm_apply, AddMonoidAlgebra.tensorEquiv_symm_single, AdjoinRoot.equiv'_symm_apply, AdicCompletion.mk_smul_top_ofAlgEquiv_symm, ContinuousAlgEquiv.symm_toAlgEquiv, Algebra.TensorProduct.comm_symm_tmul, NumberField.ComplexEmbedding.isConj_symm, Ideal.quotientEquivAlg_symm, MonoidAlgebra.scalarTensorEquiv_symm_single, Algebra.TensorProduct.quotIdealMapEquivTensorQuot_symm_tmul, autCongr_apply, Polynomial.toFinsuppIsoAlg_symm_apply_toFinsupp, StandardEtalePresentation.toPresentation_σ', Matrix.piAlgEquiv_symm_apply, Polynomial.taylorEquiv_symm, NumberField.InfinitePlace.comap_smul, IntermediateField.topEquiv_symm_apply_coe, PowerBasis.equivOfMinpoly_symm, AddMonoidAlgebra.curryAlgEquiv_symm_single, AddMonoidAlgebra.domCongr_symm, Subalgebra.topEquiv_symm_apply_coe, AdicCompletion.mk_ofAlgEquiv_symm_eq_evalOneₐ, MvPolynomial.algebraTensorAlgEquiv_symm_map, rightInverse_symm, QuaternionAlgebra.imJ_swapEquiv_symm_apply, ofLinearEquiv_symm, Matrix.kroneckerAlgEquiv_symm_apply, Ideal.quotientKerAlgEquivOfSurjective_symm_apply, FreeLieAlgebra.universalEnvelopingEquivFreeAlgebra_symm_apply, Algebra.Presentation.algebraTensorAlgEquiv_symm_relation, ofLeftInverse_symm_apply, Algebra.TensorProduct.Algebra.TensorProduct.commRight_symm_tmul, quotientBot_symm_mk, val_inv_algHomUnitsEquiv_symm_apply, moduleEndSelfOp_symm_apply, Subalgebra.rTensorBot_symm_apply, LinearMap.toMatrixAlgEquiv'_symm, Matrix.uniqueAlgEquiv_symm_apply, MvPolynomial.sumAlgEquiv_symm_apply, TensorAlgebra.equivDirectSum_symm_apply, MvPolynomial.eval₂_pUnitAlgEquiv_symm, Algebra.Presentation.tensorModelOfHasCoeffsEquiv_symm_tmul, MonoidAlgebra.symm_commAlgEquiv, AddMonoidAlgebra.scalarTensorEquiv_symm_single, ofAlgHom_symm, MonoidAlgebra.curryAlgEquiv_symm_single, comp_symm, moduleEndSelf_symm_apply, refl_symm, StandardEtalePresentation.toPresentation_algebra_algebraMap_apply, MvPolynomial.optionEquivLeft_symm_apply, opComm_symm_apply_symm_apply, IsAdjoinRoot.ofAlgEquiv_algEquiv, matPolyEquiv_symm_map_eval, CommAlgCat.isoMk_inv, endVecAlgEquivMatrixEnd_symm_apply_apply, IsLocalization.algEquiv_symm_apply, AdjoinRoot.quotEquivQuotMap_symm_apply, piCongrLeft_symm_apply, StandardEtalePresentation.toPresentation_val, mopMatrix_symm_apply, Algebra.TensorProduct.algEquivIncludeRange_symm_toAlgHom, Matrix.transposeAlgEquiv_symm_apply, Polynomial.algEquivCMulXAddC_symm_apply, CliffordAlgebra.evenEquivEvenNeg_symm_apply, Matrix.toLinAlgEquiv'_symm, Quaternion.snd_imI_dualNumberEquiv_symm, arrowCongr_symm, Algebra.IsPushout.equiv_symm_algebraMap_right, matPolyEquiv_symm_X, FractionalIdeal.map_map_symm, IsFractionRing.algEquivOfAlgEquiv_symm, MvPolynomial.optionEquivLeft_symm_C_X, CliffordAlgebra.equivOfIsometry_symm, CliffordAlgebra.involuteEquiv_symm_apply, FractionalIdeal.map_symm_map, StandardEtalePresentation.toPresentation_relation, prodUnique_symm_apply, invFun_eq_symm, symm_trans_self, InfiniteGalois.limitToAlgEquiv_symm_apply, symm_symm, uniqueProd_symm_apply, symm_toRingEquiv, IsLocalization.algEquivOfAlgEquiv_symm, Polynomial.residueFieldMapCAlgEquiv_symm_X, coe_algEquiv_lpBCF_symm, MvPolynomial.algebraTensorAlgEquiv_symm_monomial, Subalgebra.equivOfEq_symm, AdjoinRoot.symm_mapAlgEquiv, MvPolynomial.optionEquivLeft_symm_X, MvPolynomial.renameEquiv_symm, subalgebraMap_symm_apply_coe, funUnique_symm_apply, symm_bijective, MvPolynomial.supportedEquivMvPolynomial_symm_C, AdjoinRoot.algEquivOfEq_symm, Algebra.TensorProduct.opAlgEquiv_symm_tmul, prodCongr_symm_apply, MvPolynomial.optionEquivLeft_symm_C_C, Ideal.quotientEquivAlgOfEq_symm, Subalgebra.algEquivOpMop_symm_apply_coe, Algebra.TensorProduct.lid_symm_apply, DoubleQuot.quotQuotEquivQuotSupₐ_symm_toRingEquiv, ofBijective_apply_symm_apply, opOp_symm_apply, Algebra.TensorProduct.leftComm_symm_tmul, IntermediateField.intermediateFieldMap_symm_apply_coe, Matrix.toLpLinAlgEquiv_symm_apply, toRingEquiv_symm, AlgebraicGeometry.localRingHom_comp_stalkIso, StarAlgEquiv.ofAlgEquiv_symm, Algebra.TensorProduct.congr_symm, piCongrLeft'_symm_apply, ofRingEquiv_symm_apply, Algebra.TensorProduct.rid_symm_apply, Quaternion.fst_re_dualNumberEquiv_symm, coe_restrictScalars_symm', Equiv.algEquiv_symm_apply, SkewMonoidAlgebra.domCongr_symm, polyEquivTensor_symm_apply_tmul_eq_smul, Algebra.TensorProduct.comm_symm, Matrix.toLinAlgEquiv_symm, Algebra.botEquiv_symm_apply, Polynomial.residueFieldMapCAlgEquiv_symm_C, StandardEtalePresentation.equivRing_symm_X, NumberField.ComplexEmbedding.exists_comp_symm_eq_of_comp_eq, Polynomial.algEquivAevalXAddC_symm, MvPolynomial.esymmAlgEquiv_symm_apply, Shrink.algEquiv_symm_apply, Localization.coe_algEquiv_symm, Localization.algEquiv_symm_apply, AdicCompletion.ofAlgEquiv_symm_of

Algebra.IsPushout

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	Algebra.IsPushout	—	IsBaseChange.of_equiv Algebra.to_smulCommClass RingHomInvPair.ids AddEquiv.map_add' TensorProduct.induction_on smul_zero map_zero MonoidWithZeroHomClass.toZeroHomClass IsScalarTower.right RingHomClass.toMonoidWithZeroHomClass AlgHomClass.toRingHomClass AlgEquivClass.toAlgHomClass AlgEquiv.instAlgEquivClass Algebra.smul_def one_mul equiv_tmul map_mul NonUnitalRingHomClass.toMulHomClass RingHomClass.toNonUnitalRingHomClass RingHom.instRingHomClass mul_left_comm smul_add map_add SemilinearMapClass.toAddHomClass NonUnitalAlgHomClass.instLinearMapClass AlgHom.instNonUnitalAlgHomClassOfAlgHomClass Equiv.left_inv Equiv.right_inv map_one MonoidHomClass.toOneHomClass MonoidWithZeroHomClass.toMonoidHomClass mul_one

AntisymmRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AntisymmRel	—	—	—

Antivary

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Antivary PartialOrder.toPreorder SemilatticeInf.toPartialOrder Lattice.toSemilatticeInf DistribLattice.toLattice instDistribLatticeOfLinearOrder	—	—	le_of_not_gt LT.lt.not_ge

AntivaryOn

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	AntivaryOn PartialOrder.toPreorder SemilatticeInf.toPartialOrder Lattice.toSemilatticeInf DistribLattice.toLattice instDistribLatticeOfLinearOrder	—	—	le_of_not_gt LT.lt.not_ge

Associated

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Associated	—	—	mul_assoc Units.mul_inv mul_one

Asymptotics.IsBigO

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Asymptotics.IsBigO SeminormedAddCommGroup.toNorm SubNegMonoid.toSub AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup SeminormedAddCommGroup.toSeminormedAddGroup	—	—	congr_left neg_left neg_sub

Asymptotics.IsBigOTVS

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Asymptotics.IsBigOTVS ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing NontriviallyNormedField.toNormedField SeminormedAddGroup.toContinuousENorm NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField Module.toDistribMulAction AddCommGroup.toAddCommMonoid Pi.instSub SubNegMonoid.toSub	—	—	neg_sub neg_left

Asymptotics.IsBigOWith

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Asymptotics.IsBigOWith SeminormedAddCommGroup.toNorm SubNegMonoid.toSub AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup SeminormedAddCommGroup.toSeminormedAddGroup	—	—	congr_left neg_left neg_sub

Asymptotics.IsEquivalent

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Asymptotics.IsEquivalent NormedAddCommGroup.toSeminormedAddCommGroup	—	—	Asymptotics.IsLittleO.symm Asymptotics.IsLittleO.trans_isBigO isLittleO isBigO_symm

Asymptotics.IsLittleO

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Asymptotics.IsLittleO SeminormedAddCommGroup.toNorm SubNegMonoid.toSub AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup SeminormedAddCommGroup.toSeminormedAddGroup	—	—	neg_sub neg_left

Asymptotics.IsLittleOTVS

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Asymptotics.IsLittleOTVS ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing NontriviallyNormedField.toNormedField SeminormedAddGroup.toContinuousENorm NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField Module.toDistribMulAction AddCommGroup.toAddCommMonoid Pi.instSub SubNegMonoid.toSub	—	—	neg_sub neg_left

Asymptotics.IsTheta

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Asymptotics.IsTheta	—	—	—

Asymptotics.IsThetaTVS

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Asymptotics.IsThetaTVS ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing NontriviallyNormedField.toNormedField SeminormedAddGroup.toContinuousENorm NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField Module.toDistribMulAction AddCommGroup.toAddCommMonoid	—	—	—

BialgEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	20 math math: symm_comp, apply_symm_apply, invFun_eq_symm, toEquiv_symm, toBialgIso_symm, CategoryTheory.Iso.toBialgEquiv_symm, toHopfAlgIso_symm, CategoryTheory.Iso.toHopfAlgEquiv_symm, ofBialgHom_symm, symm_toCoalgEquiv, coe_toEquiv_symm, coe_symm_toEquiv, Bialgebra.TensorProduct.rid_symm_apply, CommBialgCat.isoMk_inv, symm_apply_apply, Bialgebra.TensorProduct.lid_symm_apply, comp_symm, toHopfAlgIso_inv, Bialgebra.TensorProduct.assoc_symm_tmul, toBialgIso_inv

Bundle.ContMDiffRiemannianMetric

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	DFunLike.coe ContinuousLinearMap Real Real.semiring RingHom.id Semiring.toNonAssocSemiring AddCommGroup.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Real.instAddCommMonoid NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup instIsTopologicalAddGroupReal ContinuousLinearMap.addCommMonoid IsTopologicalSemiring.toContinuousAdd NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal ContinuousLinearMap.module Algebra.to_smulCommClass Real.instCommSemiring CommSemiring.toSemiring Algebra.id UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddCommMonoid.toAddMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Module.toDistribMulAction Ring.uniformContinuousConstSMul Real.instRing instIsUniformAddGroupReal IsTopologicalSemiring.toContinuousMul NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing inner	—	—

Bundle.ContinuousRiemannianMetric

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	DFunLike.coe ContinuousLinearMap Real Real.semiring RingHom.id Semiring.toNonAssocSemiring AddCommGroup.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Real.instAddCommMonoid NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup instIsTopologicalAddGroupReal ContinuousLinearMap.addCommMonoid IsTopologicalSemiring.toContinuousAdd NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal ContinuousLinearMap.module Algebra.to_smulCommClass Real.instCommSemiring CommSemiring.toSemiring Algebra.id UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddCommMonoid.toAddMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Module.toDistribMulAction Ring.uniformContinuousConstSMul Real.instRing instIsUniformAddGroupReal IsTopologicalSemiring.toContinuousMul NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing inner	—	—

Bundle.RiemannianMetric

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	DFunLike.coe ContinuousLinearMap Real Real.semiring RingHom.id Semiring.toNonAssocSemiring AddCommGroup.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Real.instAddCommMonoid NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup instIsTopologicalAddGroupReal ContinuousLinearMap.addCommMonoid IsTopologicalSemiring.toContinuousAdd NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal ContinuousLinearMap.module Algebra.to_smulCommClass Real.instCommSemiring CommSemiring.toSemiring Algebra.id UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddCommMonoid.toAddMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Module.toDistribMulAction Ring.uniformContinuousConstSMul Real.instRing instIsUniformAddGroupReal IsTopologicalSemiring.toContinuousMul NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing inner	—	—

CategoryTheory.Equivalence

Definitions

Name	Category	Theorems
symm 📖	CompOp	79 math math: CategoryTheory.ComposableArrows.opEquivalence_counitIso_inv_app_app, CategoryTheory.Localization.uniq_symm, symm_unit, symmEquivFunctor_obj, CategoryTheory.WithInitial.isColimitEquiv_apply_desc_right, rightOp_counitIso_inv_app, CategoryTheory.MonoidalClosed.ofEquiv_uncurry_def, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_left, Action.leftUnitor_inv_hom, CategoryTheory.Comon.monoidal_rightUnitor_inv_hom, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_right, Action.whiskerRight_hom, TwoP.swapEquiv_symm, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_single, mkHom_id_inverse, Rep.homEquiv_apply_hom, symm_unitIso, CategoryTheory.CatCommSq.hInv_hInv, Rep.MonoidalClosed.linearHomEquivComm_hom, CategoryTheory.ComposableArrows.opEquivalence_unitIso_inv_app, CategoryTheory.Comon.monoidal_whiskerLeft_hom, symmEquivFunctor_map, CategoryTheory.Comon.monoidal_leftUnitor_hom_hom, rightOp_unitIso_hom_app, TopCat.Presheaf.presheafEquivOfIso_unitIso_hom_app_app, Action.diagonalSuccIsoTensorTrivial_inv_hom_apply, CategoryTheory.ComposableArrows.opEquivalence_functor_map_app, CategoryTheory.Monoidal.instIsMonoidalTransportedSymmEquivalenceTransported, CategoryTheory.Comon.monoidal_whiskerRight_hom, CategoryTheory.WithTerminal.isLimitEquiv_symm_apply_lift, CategoryTheory.ComposableArrows.opEquivalence_unitIso_hom_app, CategoryTheory.Comon.monoidal_associator_hom_hom, CategoryTheory.Limits.IsColimit.coconePointsIsoOfEquivalence_inv, CategoryTheory.toOverIteratedSliceForwardIsoPullback_hom_app_left, CategoryTheory.Adjunction.rightOp_eq, CategoryTheory.Iso.isoFunctorOfIsoInverse_inv_app, IsTriangulated.instIsTriangulatedFunctorSymmOfInverse, TopCat.Presheaf.presheafEquivOfIso_unitIso_inv_app_app, isMonoidal_symm, symmEquiv_counitIso, symm_counit, symm_functor, CategoryTheory.Adjunction.leftOp_eq, CategoryTheory.CatCommSq.vInv_vInv, rightOp_counitIso_hom_app, Action.tensorHom_hom, Action.associator_hom_hom, symm_counitIso, TopCat.Presheaf.presheafEquivOfIso_counitIso_inv_app_app, IsTriangulated.instIsTriangulatedInverseSymmOfFunctor, Action.whiskerLeft_hom, rightOp_unitIso_inv_app, Rep.MonoidalClosed.linearHomEquiv_hom, CategoryTheory.Limits.IsLimit.conePointsIsoOfEquivalence_hom, CategoryTheory.Comon.monoidal_leftUnitor_inv_hom, CategoryTheory.Comon.monoidal_rightUnitor_hom_hom, Rep.homEquiv_symm_apply_hom, pow_neg_one, TopCat.Presheaf.presheafEquivOfIso_counitIso_hom_app_app, CategoryTheory.Iso.isoFunctorOfIsoInverse_hom_app, Rep.ihom_coev_app_hom, CategoryTheory.toOverIteratedSliceForwardIsoPullback_inv_app_left, Action.rightUnitor_hom_hom, IsTriangulated.symm, CategoryTheory.Comon.monoidal_associator_inv_hom, Action.associator_inv_hom, TopCat.Presheaf.presheafEquivOfIso_functor_map_app, CategoryTheory.MonoidalClosed.ofEquiv_curry_def, Action.rightUnitor_inv_hom, CategoryTheory.ComposableArrows.opEquivalence_counitIso_hom_app_app, Bipointed.swapEquiv_symm, CategoryTheory.Comon.monoidal_tensorHom_hom, TopCat.Presheaf.presheafEquivOfIso_inverse_map_app, CategoryTheory.Comon.monoidal_tensorObj_comon_counit, CommShift.instSymm, symm_inverse, Action.leftUnitor_hom_hom, symmEquivInverse_map_app, CategoryTheory.Comon.monoidal_tensorObj_comon_comul

CategoryTheory.Equivalence.IsTriangulated

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	CategoryTheory.Functor.Additive CategoryTheory.shiftFunctor Int.instAddMonoid	CategoryTheory.Equivalence.IsTriangulated CategoryTheory.Equivalence.symm CategoryTheory.Equivalence.CommShift.instCommShiftFunctorSymm CategoryTheory.Equivalence.CommShift.instCommShiftInverseSymm	—	CategoryTheory.Equivalence.CommShift.instSymm instIsTriangulatedFunctorSymmOfInverse instIsTriangulatedInverse instIsTriangulatedInverseSymmOfFunctor instIsTriangulatedFunctor

CategoryTheory.Iso

Definitions

Name	Category	Theorems
symm 📖	CompOp	279 math math: CategoryTheory.SingleFunctors.shiftIso_add, SheafOfModules.pushforward_assoc, Bicategory.Opposite.op2_associator, CategoryTheory.Monoidal.InducingFunctorData.rightUnitor_eq, CategoryTheory.equivOfTensorIsoUnit_unitIso, CategoryTheory.Equivalence.mapGrp_counitIso, CategoryTheory.Functor.isoWhiskerRight_twice_assoc, toCoalgEquiv_symm, CategoryTheory.Bicategory.eqToHomTransIso_refl_left, ModuleCat.restrictScalarsCongr_symm, Bicategory.Opposite.op2_associator_inv, CategoryTheory.equivEssImageOfReflective_unitIso, conjAut_apply, CategoryTheory.MonoOver.congr_unitIso, isoCongr_symm_apply, CategoryTheory.Bicategory.Prod.sectL_mapId_inv, imageToKernel_unop, CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_obj_map_snd_app, CategoryTheory.Monoidal.transportStruct_associator, CategoryTheory.NatIso.op_rightUnitor, CategoryTheory.Join.mapPairEquiv_unitIso, CategoryTheory.Join.mapPairEquiv_counitIso, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionAssocIso, CategoryTheory.Functor.CorepresentableBy.equivOfIsoObj_symm_apply, CategoryTheory.shiftFunctorAdd'_zero_add, CategoryTheory.Bicategory.eqToHomTransIso_refl_refl, CategoryTheory.BicategoricalCoherence.left'_iso, CategoryTheory.NatIso.unop_rightUnitor, CategoryTheory.SingleFunctors.shiftIso_add', CategoryTheory.Functor.commShiftOp_iso_eq, symm_symm_eq, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionAssocIso, CategoryTheory.Center.tensorUnit_β, self_symm_id, CategoryTheory.MonoidalCoherence.assoc'_iso, CategoryTheory.MonoOver.mapIso_unitIso, CategoryTheory.Bicategory.leftZigzagIso_symm, CategoryTheory.Pi.equivalenceOfEquiv_unitIso, CategoryTheory.Monad.algebraEquivOfIsoMonads_unitIso, CategoryTheory.Pi.equivalenceOfEquiv_counitIso, CategoryTheory.MonoidalCategory.endofunctorMonoidalCategory.evaluationRightAction_actionUnitIso, CategoryTheory.IsSifted.factorization_prodComparison_colim, imageToKernel_op, SimplexCategory.revEquivalence_unitIso, CategoryTheory.MonoOver.mapIso_counitIso, CategoryTheory.NatIso.op_isoWhiskerLeft, CategoryTheory.Monoidal.InducingFunctorData.associator_eq, CategoryTheory.Functor.isoWhiskerLeft_twice, CategoryTheory.Pseudofunctor.mapComp_id_left, CategoryTheory.MonoidalCoherence.tensor_right_iso, op_symm, CategoryTheory.Equivalence.symm_unitIso, CategoryTheory.Equivalence.mapHomologicalComplex_unitIso, symm_eq_iff, CategoryTheory.BicategoricalCoherence.tensorRight_iso, CategoryTheory.Bicategory.Prod.sectR_mapComp_inv, CategoryTheory.mop_associator, CategoryTheory.Limits.fiberwiseColimitLimitIso_hom_app, eHomCongr_symm, CategoryTheory.Localization.Lifting.ofIsos_iso, CategoryTheory.Over.postAdjunctionLeft_unit_app_left, CategoryTheory.Center.tensor_β, AlgebraicGeometry.Scheme.homeoOfIso_symm, CategoryTheory.Pseudofunctor.mapComp_id_right, CategoryTheory.skeletonEquivalence_unitIso, CategoryTheory.Pseudofunctor.mapComp'_comp_id, CategoryTheory.Pseudofunctor.isoMapOfCommSq_horiz_id, CategoryTheory.Bicategory.Prod.sectL_mapComp_hom, CategoryTheory.Pi.isoApp_symm, CategoryTheory.ChosenPullbacksAlong.isoInv_pullback_map_left, CategoryTheory.ShortComplex.RightHomologyData.mapOpcyclesIso_eq, symm_hom, CategoryTheory.MonoidalCoherence.right'_iso, CategoryTheory.Equivalence.mapMon_unitIso, CategoryTheory.Bicategory.Equivalence.right_triangle, CategoryTheory.Bicategory.Equivalence.left_triangle, AlgebraicGeometry.Scheme.coe_homeoOfIso_symm, CategoryTheory.op_leftUnitor, CategoryTheory.Limits.CategoricalPullback.functorEquiv_functor_map_fst_app, CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_map_app_fst_app, CategoryTheory.Bicategory.Prod.sectR_mapId_inv, CategoryTheory.ChosenPullbacksAlong.isoInv_mapPullbackAdj_unit_app_left, CategoryTheory.Aut.Aut_inv_def, CategoryTheory.Functor.isoWhiskerLeft_right, CategoryTheory.NatIso.op_isoWhiskerRight, toLinearEquiv_symm, CategoryTheory.Center.tensorUnit_snd_β, CategoryTheory.Pseudofunctor.whiskerLeftIso_mapId, CategoryTheory.Functor.RepresentableBy.equivOfIsoObj_symm_apply, CategoryTheory.Limits.Cones.equivalenceOfReindexing_counitIso, CategoryTheory.Endofunctor.Coalgebra.equivOfNatIso_counitIso, CategoryTheory.GradedObject.comapEquiv_unitIso, CategoryTheory.BicategoricalCoherence.right'_iso, CategoryTheory.Limits.IsLimit.equivIsoLimit_symm_apply, PresheafOfModules.pullback_assoc, CategoryTheory.Sum.natIsoOfWhiskerLeftInlInr_eq, CategoryTheory.Functor.isoWhiskerRight_left_assoc, CategoryTheory.Limits.CategoricalPullback.functorEquiv_functor_map_snd_app, CategoryTheory.Functor.isoWhiskerRight_twice, CategoryTheory.Bicategory.eqToHomTransIso_refl_right, CategoryTheory.NatIso.op_symm, coreAssociator, symm_self_id, CategoryTheory.BraidedCategory.yang_baxter_iso, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionUnitIso, CategoryTheory.Bicategory.adjointifyCounit_left_triangle, CategoryTheory.Equivalence.unop_counitIso, symm_inv, CategoryTheory.Equivalence.mapCommGrp_unitIso, BialgEquiv.toBialgIso_symm, CategoryTheory.Functor.commShiftPullback_iso_eq, CategoryTheory.Limits.Cones.functorialityEquivalence_unitIso, CategoryTheory.NatIso.unop_symm, AlgebraicGeometry.PresheafedSpace.sheafIsoOfIso_inv, CategoryTheory.Equivalence.mapCommMon_unitIso, Mathlib.Tactic.Bicategory.naturality_inv, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionAssocIso_unop, CategoryTheory.Bicategory.Prod.sectL_mapId_hom, CategoryTheory.Pseudofunctor.isoMapOfCommSq_vert_id, CategoryTheory.Functor.CoreMonoidal.ofOplaxMonoidal_εIso, unop_symm, CategoryTheory.Functor.isoWhiskerLeft_right_assoc, toBialgEquiv_symm, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionUnitIso, CategoryTheory.Limits.fiberwiseColimitLimitIso_inv_app, ModuleCat.matrixEquivalence_unitIso, CategoryTheory.Limits.IsColimit.coconePointsIsoOfEquivalence_inv, symm_mk, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionAssocIso_op, CategoryTheory.shiftFunctorAdd'_add_zero, BialgEquiv.toHopfAlgIso_symm, CategoryTheory.op_rightUnitor, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_hom_τ₃, CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_map_app_snd_app, CategoryTheory.Oplax.StrongTrans.categoryStruct_comp_naturality, CategoryTheory.Equivalence.mapGrp_unitIso, CategoryTheory.NatIso.unop_whiskerRight, toHopfAlgEquiv_symm, CategoryTheory.BraidedCategory.hexagon_reverse_iso, CategoryTheory.Limits.PullbackCone.unop_ι_app, CategoryTheory.ShortComplex.LeftHomologyData.mapLeftHomologyIso_eq, CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose_obj_map_fst_app, CategoryTheory.Lax.StrongTrans.categoryStruct_id_naturality, CategoryTheory.NatIso.op_associator, CategoryTheory.Monoidal.transportStruct_leftUnitor, CategoryTheory.Pseudofunctor.StrongTrans.naturality_id_iso, CategoryTheory.Monoidal.transportStruct_rightUnitor, CategoryTheory.Limits.colimitPointwiseProductToProductColimit_app, CategoryTheory.ShortComplex.HomologyData.left_homologyIso_eq_right_homologyIso_trans_iso_symm, CategoryTheory.Equivalence.inverseFunctorMapIso_symm_eq_isoInverseOfIsoFunctor, CategoryTheory.unop_leftUnitor, CategoryTheory.rightDistributor_assoc, CategoryTheory.NatIso.unop_leftUnitor, CategoryTheory.Endofunctor.Algebra.equivOfNatIso_counitIso, CategoryTheory.NatIso.unop_associator, CategoryTheory.Bicategory.Prod.sectL_mapComp_inv, CategoryTheory.Limits.Cones.functorialityEquivalence_inverse, CategoryTheory.Monad.algebraEquivOfIsoMonads_counitIso, CategoryTheory.NatIso.op_leftUnitor, CategoryTheory.Equivalence.mapCommMon_counitIso, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_inv_τ₁, CategoryTheory.MonoOver.congr_inverse, CategoryTheory.Limits.CategoricalPullback.toCatCommSqOver_map_fst_app, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionAssocIso_unop, CategoryTheory.Endofunctor.Coalgebra.equivOfNatIso_unitIso, CategoryTheory.Limits.Cocones.functorialityEquivalence_counitIso, CategoryTheory.Equivalence.mapMon_counitIso, CategoryTheory.leftDistributor_rightDistributor_assoc, SimplicialObject.opEquivalence_unitIso, SheafOfModules.pushforwardCongr_symm, CategoryTheory.Equivalence.trans_counitIso, isoCongr_apply, HomologicalComplex.mapBifunctorFlipIso_flip, CategoryTheory.ShortComplex.LeftHomologyData.homologyIso_leftHomologyData, CategoryTheory.Pseudofunctor.StrongTrans.naturality_naturality_iso, CategoryTheory.Bicategory.rightZigzagIso_symm, SSet.opEquivalence_unitIso, CategoryTheory.ShortComplex.RightHomologyData.mapRightHomologyIso_eq, Mathlib.Tactic.Bicategory.structuralIso_inv, CategoryTheory.Limits.Cones.functorialityEquivalence_counitIso, CategoryTheory.unop_rightUnitor, CategoryTheory.ChosenPullbacksAlong.isoInv_mapPullbackAdj_counit_app_left, CategoryTheory.Limits.Cone.equivCostructuredArrow_counitIso, symm_self_conj, SheafOfModules.pullback_assoc, CategoryTheory.shiftEquiv'_unitIso, SheafOfModules.Presentation.of_isIso_relations, CategoryTheory.Functor.mapIso_symm, HomologicalComplex₂.flip_totalFlipIso, CategoryTheory.Endofunctor.Algebra.equivOfNatIso_unitIso, CategoryTheory.ShortComplex.LeftHomologyData.mapCyclesIso_eq, Mathlib.Tactic.Monoidal.naturality_inv, refl_symm, CategoryTheory.Functor.isoWhiskerRight_symm, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionAssocIso, CategoryTheory.Core.isoMk_inv_iso, CategoryTheory.Center.tensorObj_snd_β, CategoryTheory.shiftComm_symm, CategoryTheory.DifferentialObject.isoApp_symm, CategoryTheory.Equivalence.symm_counitIso, CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose_map_app_fst_app, CategoryTheory.Bicategory.Prod.sectR_mapComp_hom, CategoryTheory.MonoidalCoherence.left'_iso, self_symm_conj, CategoryTheory.Limits.IsLimit.conePointsIsoOfEquivalence_hom, CategoryTheory.NatTrans.CommShift.of_iso_symm, CategoryTheory.Limits.IsColimit.equivIsoColimit_symm_apply, trans_symm, CategoryTheory.Idempotents.karoubiUniversal₁_unitIso, CategoryTheory.Pseudofunctor.mapComp'_id_comp, CategoryTheory.Monoidal.InducingFunctorData.leftUnitor_eq, PresheafOfModules.pushforward_assoc, CategoryTheory.ShortComplex.LeftHomologyData.mapHomologyIso_eq, CategoryTheory.Functor.isoWhiskerRight_left, CategoryTheory.MonoidalCategory.endofunctorMonoidalCategory.evaluationRightAction_actionAssocIso, CategoryTheory.Limits.Cones.equivalenceOfReindexing_unitIso, homCongr_symm, CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose_obj_map_snd_app, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionAssocIso_op, CategoryTheory.shiftFunctorComm_symm, CategoryTheory.shiftFunctorComm_eq, CategoryTheory.Equivalence.mapCommGrp_counitIso, CategoryTheory.Limits.Cocones.functorialityEquivalence_inverse, CategoryTheory.unmop_associator, CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose_map_app_snd_app, CategoryTheory.Equivalence.unop_unitIso, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_inv_τ₂, CategoryTheory.MonoidalCategory.whiskerRightIso_symm, CategoryTheory.Oplax.StrongTrans.categoryStruct_id_naturality, CoalgEquiv.toCoalgIso_symm, CategoryTheory.Functor.commShiftIso_eq_ofInduced, CategoryTheory.MonoOver.congr_counitIso, CategoryTheory.Equivalence.op_unitIso, CategoryTheory.Lax.StrongTrans.categoryStruct_comp_naturality, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionAssocIso, CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_obj_map_fst_app, coreWhiskerRight, CategoryTheory.FreeMonoidalCategory.normalizeIsoApp_tensor, CategoryTheory.Pseudofunctor.StrongTrans.naturality_comp_iso, self_symm_id_assoc, symm_self_id_assoc, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_hom_τ₁, CategoryTheory.equivEssImageOfReflective_counitIso, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_hom_τ₂, CategoryTheory.MonoidalCategory.whiskerLeftIso_symm, CategoryTheory.Equivalence.trans_unitIso, CategoryTheory.Bicategory.Prod.sectR_mapId_hom, CategoryTheory.op_associator, CategoryTheory.NatIso.unop_whiskerLeft, symm_bijective, CategoryTheory.Pseudofunctor.DescentData.pullFunctorEquivalence_unitIso, CategoryTheory.BicategoricalCoherence.assoc'_iso, CategoryTheory.Equivalence.op_counitIso, CategoryTheory.Pseudofunctor.isoMapOfCommSq_eq, CategoryTheory.Limits.CategoricalPullback.toCatCommSqOver_map_snd_app, CategoryTheory.leftDistributor_assoc, CategoryTheory.Limits.Cocone.equivStructuredArrow_counitIso, CategoryTheory.unop_associator, AlgebraicTopology.DoldKan.N₁Γ₀_app, CategoryTheory.Functor.commShiftUnop_commShiftIso, AlgebraicGeometry.Scheme.Pullback.Triplet.tensorCongr_symm, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionUnitIso, Bicategory.Opposite.op2_associator_hom, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_inv_τ₃, CategoryTheory.Limits.Cocones.functorialityEquivalence_unitIso, coreWhiskerLeft, CategoryTheory.Subfunctor.equivalenceMonoOver_counitIso, AlgebraicGeometry.Scheme.residueFieldCongr_symm, QuadraticModuleCat.ofIso_symm, CategoryTheory.Limits.widePushoutShapeOpEquiv_unitIso, CategoryTheory.GradedObject.comapEq_symm, CategoryTheory.Pseudofunctor.whiskerRightIso_mapId, CategoryTheory.ShortComplex.RightHomologyData.mapHomologyIso'_eq, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionUnitIso, toIsometryEquiv_symm, CategoryTheory.ShortComplex.RightHomologyData.homologyIso_rightHomologyData, CategoryTheory.Functor.isoWhiskerLeft_symm, CategoryTheory.Functor.CoreMonoidal.ofOplaxMonoidal_μIso, CategoryTheory.Limits.widePullbackShapeOpEquiv_unitIso

CategoryTheory.Limits.Types.Pushout.Rel'

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	CategoryTheory.Limits.Types.Pushout.Rel'	—	—	—

CategoryTheory.MorphismProperty.LeftFractionRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	CategoryTheory.MorphismProperty.LeftFractionRel	—	—	—

CategoryTheory.MorphismProperty.LeftFraction₂

Definitions

Name	Category	Theorems
symm 📖	CompOp	1 math math: symm_add

CategoryTheory.MorphismProperty.RightFractionRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	CategoryTheory.MorphismProperty.RightFractionRel	—	—	CategoryTheory.MorphismProperty.LeftFractionRel.unop CategoryTheory.MorphismProperty.LeftFractionRel.symm op

CategoryTheory.PresheafOfGroups.OneCohomologyRelation

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	CategoryTheory.PresheafOfGroups.OneCohomologyRelation	CategoryTheory.PresheafOfGroups.ZeroCochain InvOneClass.toInv DivInvOneMonoid.toInvOneClass DivisionMonoid.toDivInvOneMonoid Group.toDivisionMonoid CategoryTheory.PresheafOfGroups.instGroupZeroCochain	—	RightCancelSemigroup.toIsRightCancelMul mul_assoc CategoryTheory.PresheafOfGroups.Cochain₀.inv_apply map_inv MonoidHom.instMonoidHomClass inv_mul_cancel_left inv_mul_cancel mul_one

CategoryTheory.ShortComplex.Homotopy

Definitions

Name	Category	Theorems
symm 📖	CompOp	4 math math: symm_h₀, symm_h₃, symm_h₂, symm_h₁

CategoryTheory.ShortComplex.HomotopyEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	4 math math: symm_homotopyHomInvId, symm_homotopyInvHomId, symm_hom, symm_inv

CategoryTheory.Zag

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	CategoryTheory.Zag	—	—	CategoryTheory.zag_symmetric

CategoryTheory.Zigzag

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	CategoryTheory.Zigzag	—	—	CategoryTheory.zigzag_symmetric

CoalgEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	22 math math: ofCoalgHom_symm, CategoryTheory.Iso.toCoalgEquiv_symm, BialgEquiv.refl_symm_apply, toCoalgIso_inv, invFun_eq_symm, refl_symm_apply, coe_symm_toEquiv, coe_toEquiv_symm, BialgEquiv.symm_toCoalgEquiv, symm_toLinearEquiv, symm_toCoalgHom, coe_symm_toLinearEquiv, BialgEquiv.trans_symm_apply, CommBialgCat.bialgEquivOfIso_symm_apply, Coalgebra.TensorProduct.assoc_symm_tmul, trans_symm_apply, Coalgebra.TensorProduct.rid_symm_apply, symm_apply_apply, apply_symm_apply, toCoalgIso_symm, toEquiv_symm, Coalgebra.TensorProduct.lid_symm_apply

Codisjoint

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Codisjoint	—	—	codisjoint_comm

Commute

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Commute	—	—	—

CompRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	CompRel	—	—	Relation.SymmGen.symm

ComplexShape

Definitions

Name	Category	Theorems
symm 📖	CompOp	79 math math: HomologicalComplex.op_d, HomologicalComplex.opSymm_d, HomologicalComplex.unopFunctor_obj, HomologicalComplex.opcyclesOpIso_inv_naturality_assoc, Embedding.op_f, HomologicalComplex.instHasHomologyOppositeOp, symm_symm, HomologicalComplex.extend_op_d_assoc, HomologicalComplex.isStrictlySupportedOutside_op_iff, HomologicalComplex.unopInverse_map, HomologicalComplex.unopEquivalence_functor, Embedding.instIsRelIffOp, HomologicalComplex.opcyclesOpIso_inv_naturality, HomologicalComplex.extend.XOpIso_hom_d_op, HomologicalComplex.opEquivalence_unitIso, HomologicalComplex.extend.XOpIso_hom_d_op_assoc, HomologicalComplex.opFunctor_obj, HomologicalComplex.instQuasiIsoAtMapOppositeSymmUnopFunctorOp, HomologicalComplex.cyclesOpIso_inv_naturality_assoc, HomologicalComplex.Acyclic.op, HomologicalComplex.cyclesOpNatIso_inv_app, HomologicalComplex.instQuasiIsoMapOppositeSymmUnopFunctorOp, HomologicalComplex.opcyclesOpIso_hom_naturality_assoc, HomologicalComplex.quasiIsoAt_unopFunctor_map_iff, instHasNoLoopSymm, Embedding.instIsTruncLEOpOfIsTruncGE, HomologicalComplex.opcyclesOpIso_hom_toCycles_op, HomologicalComplex.unop_X, HomologicalComplex.opFunctor_map_f, HomologicalComplex.instQuasiIsoAtOppositeMapSymmOpFunctorOp, HomologicalComplex.isSupportedOutside_op_iff, HomologicalComplex.unopEquivalence_unitIso, HomologicalComplex.exactAt_op_iff, HomologicalComplex.opEquivalence_counitIso, HomologicalComplex.fromOpcycles_op_cyclesOpIso_inv_assoc, symm_Rel, HomologicalComplex.homologyOp_hom_naturality_assoc, HomologicalComplex.unopSymm_X, HomologicalComplex.cyclesOpIso_inv_naturality, HomologicalComplex.opcyclesOpIso_hom_naturality, HomologicalComplex.instHasHomologyOppositeObjSymmOpFunctorOp, HomologicalComplex.extend_op_d, HomologicalComplex.opInverse_obj, HomologicalComplex.unopSymm_d, HomologicalComplex.isStrictlySupported_op_iff, Embedding.op_boundaryLE_iff, HomologicalComplex.cyclesOpIso_hom_naturality_assoc, Embedding.op_boundaryGE_iff, HomologicalComplex.cyclesOpIso_hom_naturality, Embedding.instIsTruncGEOpOfIsTruncLE, HomologicalComplex.instHasHomologyUnopOfOpposite, HomologicalComplex.instQuasiIsoOppositeMapSymmOpFunctorOp, HomologicalComplex.isSupported_op_iff, HomologicalComplex.acyclic_op_iff, HomologicalComplex.ExactAt.op, HomologicalComplex.opEquivalence_functor, HomologicalComplex.op_X, HomologicalComplex.quasiIso_opFunctor_map_iff, HomologicalComplex.unopInverse_obj, HomologicalComplex.opEquivalence_inverse, HomologicalComplex.cyclesOpNatIso_hom_app, HomologicalComplex.fromOpcycles_op_cyclesOpIso_inv, symm_bijective, HomologicalComplex.quasiIso_unopFunctor_map_iff, HomologicalComplex.instIsStrictlySupportedOppositeOpOp, HomologicalComplex.unopEquivalence_inverse, HomologicalComplex.unop_d, HomologicalComplex.opFunctor_additive, HomologicalComplex.Acyclic.unop, HomologicalComplex.unopFunctor_map_f, HomologicalComplex.homologyOp_hom_naturality, HomologicalComplex.opcyclesOpIso_hom_toCycles_op_assoc, HomologicalComplex.quasiIsoAt_opFunctor_map_iff, HomologicalComplex.unopFunctor_additive, HomologicalComplex.unopEquivalence_counitIso, HomologicalComplex.opInverse_map, HomologicalComplex.ExactAt.unop, HomologicalComplex.opSymm_X, HomologicalComplex.instHasHomologyObjOppositeSymmUnopFunctorOp

ComplexShape.Embedding.AreComplementary

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	ComplexShape.Embedding.AreComplementary	—	—	disjoint union

CompositionSeries.Equivalent

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	CompositionSeries.Equivalent	—	—	JordanHolderLattice.iso_symm Equiv.apply_symm_apply

Computation.Equiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Computation.Equiv	—	—	—

Computation.LiftRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	Symmetric	Computation Computation.LiftRel	—	—

Con

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	DFunLike.coe Con instFunLikeForallProp	—	—	Setoid.symm'

Congruent

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Congruent	—	—	—

ContinuousAddEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	23 math math: eq_comp_symm, symm_comp_eq, symm_bijective, symm_apply_eq, eq_symm_comp, symm_comp_self, self_trans_symm, AddEquiv.symm_toContinuousAddEquiv, self_comp_symm, symm_symm, invFun_eq_symm, eq_symm_apply, apply_eq_iff_symm_apply, symm_trans_self, MeasureTheory.addEquivAddHaarChar_symm, AddSubgroup.addSubgroupOfContinuousAddEquivOfLe_symm_apply, symm_apply_apply, comp_symm_eq, equivLike_neg_eq_symm, AddEquiv.toContinuousAddEquiv_symm_apply, apply_symm_apply, coe_toHomeomorph_symm, symm_trans_apply

ContinuousAffineEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	26 math math: image_symm_eq_preimage, symm_symm_apply, symm_apply_apply, symm_trans_self, toEquiv_symm, self_trans_symm, AffineIsometryEquiv.toContinuousAffineEquiv_symm, symm_apply_eq, preimage_symm, apply_symm_apply, apply_eq_iff_eq_symm_apply, refl_symm, prodCongr_symm, coe_symm_toEquiv, image_eq_preimage_symm, symm_symm, prodComm_symm, toAffineEquiv_symm, AffineIsometryEquiv.coe_symm_toContinuousAffineEquiv, coe_symm_toAffineEquiv, image_symm, symm_refl, symm_bijective, symm_image_image, image_symm_image, eq_symm_apply

ContinuousAlgEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	28 math math: symm_apply_eq, symm_bijective, Rat.HeightOneSpectrum.adicCompletionIntegers.coe_padicIntEquiv_symm_apply, toContinuousLinearEquiv_symm, symm_trans_apply, symm_map_nhds_eq, PadicInt.coe_adicCompletionIntegersEquiv_symm_apply, refl_symm, cast_symm_apply, symm_comp_self, image_eq_preimage_symm, symm_symm_apply, symm_apply_apply, coe_comp_coe_symm, symm_image_image, eq_symm_apply, symm_toAlgEquiv, image_symm_eq_preimage, self_comp_symm, preimage_symm_preimage, apply_symm_apply, image_symm_image, ContinuousLinearEquiv.symm_conjContinuousAlgEquiv, symm_symm, symm_toHomeomorph, symm_preimage_preimage, coe_symm_comp_coe, ContinuousLinearEquiv.symm_conjContinuousAlgEquiv_apply_apply

ContinuousLinearEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	181 math math: Pi.comul_eq_adjoint, preimage_symm_preimage, antilipschitz, PiLp.continuousLinearEquiv_symm_apply, StarModule.decomposeProdAdjointL_symm_apply, symm_preimage_preimage, symm_neg, toLinearEquiv_symm, toCompactConvergenceCLM_symm_apply, Matrix.l2_opNorm_mulVec, HasFDerivWithinAt.of_local_left_inverse, OpenPartialHomeomorph.hasStrictFDerivAt_symm, symm_toDiffeomorph, subsingleton_or_norm_symm_pos, coe_toSpanNonzeroSingleton_symm, NumberField.mixedEmbedding.fundamentalCone.abs_det_completeBasis_equivFunL_symm, coe_comp_coe_symm, mdifferentiableOn_symm_coordChangeL, coe_toDiffeomorph_symm, prodProdProdComm_symm, WithLp.prodContinuousLinearEquiv_symm_apply, LinearEquiv.toLinearEquiv_toContinuousLinearEquiv_symm, apply_symm_apply, contDiffAt_comp_iff, ContinuousAlgEquiv.toContinuousLinearEquiv_symm, symm_trans_apply, Euclidean.closedBall_eq_image, MeasureTheory.charFunDual_pi', continuousAlternatingMapCongrLeftEquiv_apply, Pi.counit_eq_adjoint, conjContinuousAlgEquiv_apply, SchwartzMap.smulLeftCLM_compCLMOfContinuousLinearEquiv, ContinuousLinearMap.inverse_equiv, piUnique_symm_apply, uniqueProd_symm_apply, VectorField.pullbackWithin_eq_of_fderivWithin_eq, finTwoArrow_symm_apply, one_le_norm_mul_norm_symm, Trivialization.comp_continuousLinearEquivAt_eq_coord_change, arrowCongr_apply, Complex.equivRealProdCLM_symm_apply, prodCongr_symm, Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm, HasStrictFDerivAt.of_local_left_inverse, PiLp.hasFDerivAt_toLp, LinearEquiv.coe_toContinuousLinearEquiv_symm, Matrix.l2_opNNNorm_mulVec, arrowCongr_symm, symm_smul_apply, ContinuousLinearMap.equivRange_symm_apply, coe_symm_comp_coe, eq_toContinuousLinearMap_symm_comp, Module.Basis.equivFunL_symm_apply_repr, MulOpposite.opContinuousLinearEquiv_symm_apply, Submodule.coe_orthogonalDecomposition, PointwiseConvergenceCLM.equivWeakDual_symm_apply, WithLp.prodContinuousLinearEquiv_symm_apply_ofLp, HasFDerivAt.of_local_left_inverse, continuousAlternatingMapCongrRight_symm, LinearEquiv.coe_toContinuousLinearEquiv_symm', LinearIsometryEquiv.toContinuousLinearEquiv_symm, symm_smul, eq_comp_toContinuousLinearMap_symm, hasSum, comp_hasFDerivWithinAt_iff', ProbabilityTheory.indepFun_iff_charFunDual_prod', starL_symm_apply, toSpanNonzeroSingleton_symm_apply, contMDiffOn_symm_coordChangeL, Trivialization.symm_coordChangeL, LinearEquiv.extend_symm_apply, symm_bijective, norm_symm_pos, continuousMultilinearMapCongrRight_symm, subsingleton_or_nnnorm_symm_pos, LinearEquiv.extend_symm_eq, prodUnique_symm_apply, refl_symm, SchwartzMap.fourierTransformCLE_symm_apply, starL'_symm_apply, coe_symm_toHomeomorph, comp_right_hasFDerivWithinAt_iff', unitsEquivAut_apply_symm, symm_comp_self, NumberField.mixedEmbedding.negAt_symm, FormalMultilinearSeries.leftInv_coeff_one, image_eq_preimage_symm, ModelWithCorners.coe_extChartAt_transContinuousLinearEquiv_symm, contDiffWithinAt_comp_iff, self_comp_symm, image_symm_eq_preimage, Module.Basis.parallelepiped_eq_map, det_coe_symm, Trivialization.coordChangeL_symm_apply, rightInverse_hasRightInverse, prodComm_symm, comp_hasFDerivAt_iff', HasStrictFDerivAt.approximates_deriv_on_open_nhds, continuousAlternatingMapCongrLeft_apply, symm_symm_apply, Trivialization.continuousLinearEquivAt_symm_apply, LinearIsometryEquiv.coe_symm_toContinuousLinearEquiv, LinearEquiv.norm_extend_symm_le, ContinuousLinearMap.inverse_comp_equiv, ContinuousLinearMap.inCoordinates_eq, ProbabilityTheory.iIndepFun_iff_charFunDual_pi', HasStrictFDerivAt.to_localInverse, symm_image_image, MeasureTheory.Measure.addHaar_preimage_continuousLinearEquiv, arrowCongrEquiv_symm_apply, leftInverse_hasLeftInverse, eq_symm_apply, arrowCongrSL_apply, submoduleMap_symm_apply, MeasureTheory.charFunDual_prod', skewProd_symm_apply, Shrink.continuousLinearEquiv_symm_apply, Trivialization.symm_continuousLinearEquivAt_eq, tsum_eq_iff, arrowCongrEquiv_apply, Complex.equivRealProdCLM_symm_apply_re, MeasureTheory.charFunDual_eq_pi_iff', toHomeomorph_symm, ContinuousLinearMap.inverse_equiv_comp, FourierTransform.fourierCLE_symm_apply, FormalMultilinearSeries.radius_compContinuousLinearMap_le, symm_trans_self, ContinuousLinearMap.inverse_eq_ringInverse, symm_equivOfInverse, ofBijective_symm_apply_apply, Submodule.ClosedComplemented.exists_submodule_equiv_prod, Module.Basis.ofZLatticeBasis_comap, ofSubmodule'_symm_apply, symm_symm, HasStrictFDerivAt.to_local_left_inverse, VectorField.pullback_eq_of_fderiv_eq, ContinuousLinearMap.coe_symm_flipMultilinearEquiv, PiLp.coe_symm_continuousLinearEquiv, continuousAlternatingMapCongr_apply, symm_equivOfInverse', Complex.equivRealProdCLM_symm_apply_im, FormalMultilinearSeries.radius_rightInv_pos_of_radius_pos_aux2, nnnorm_symm_pos, symm_apply_apply, WithCStarModule.equivL_symm_apply, NumberField.mixedEmbedding.euclidean.volumePreserving_toMixed_symm, arrowCongrSL_symm_apply, continuousMultilinearMapCongrLeft_symm, symm_conjContinuousAlgEquiv, ModelWithCorners.coe_transContinuousLinearEquiv_symm, self_trans_symm, symm_apply_eq, LinearEquiv.coeFn_toContinuousLinearEquivOfContinuous_symm, comp_right_hasFDerivAt_iff', exists_continuousLinearEquiv_fderiv_symm_eq, FormalMultilinearSeries.rightInv_coeff, image_symm_image, arrowCongrSL_toLinearEquiv_apply, exists_continuousLinearEquiv_fderivWithin_symm_eq, FormalMultilinearSeries.rightInv_coeff_one, prodAssoc_symm_apply, ContinuousLinearMap.toSpanSingletonCLE_symm_apply, symm_map_nhds_eq, OpenPartialHomeomorph.hasFDerivAt_symm, MeasureTheory.charFunDual_eq_prod_iff', equivOfRightInverse_symm_apply, Fin.consEquivL_symm_apply, arrowCongrSL_toLinearEquiv_symm_apply, piCongrRight_symm_apply, conjContinuousAlgEquiv_apply_apply, ContinuousLinearMap.coprodEquivL_symm_apply, coe_funUnique_symm, mem_contMDiffFiberwiseLinear_iff, piFinTwo_symm_apply, coe_symm_toLinearEquiv, symm_conjContinuousAlgEquiv_apply_apply, Submodule.coe_prodEquivOfClosedCompl_symm, ModelWithCorners.extChartAt_transContinuousLinearEquiv_target, ofBijective_apply_symm_apply, ofSubmodules_symm_apply, PiLp.hasStrictFDerivAt_toLp

ContinuousMap.Homotopic

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	ContinuousMap.Homotopic	—	Nonempty.map

ContinuousMap.HomotopicRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	ContinuousMap.HomotopicRel	—	Nonempty.map

ContinuousMap.HomotopicWith

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	ContinuousMap.HomotopicWith	—	—

ContinuousMap.Homotopy

Definitions

Name	Category	Theorems
symm 📖	CompOp	4 math math: symm_trans, symm_symm, symm_bijective, symm_apply

ContinuousMap.HomotopyEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	5 math math: refl_symm_apply, Homeomorph.symm_toHomotopyEquiv, coe_invFun, symm_trans, trans_symm_apply

ContinuousMap.HomotopyRel

Definitions

Name	Category	Theorems
symm 📖	CompOp	4 math math: symm_bijective, symm_apply, symm_trans, symm_symm

ContinuousMap.HomotopyWith

Definitions

Name	Category	Theorems
symm 📖	CompOp	4 math math: symm_bijective, symm_trans, symm_symm, symm_apply

ContinuousMulEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	25 math math: symm_trans_self, MulEquiv.toContinuousMulEquiv_symm_apply, MeasureTheory.mulEquivHaarChar_symm, eq_comp_symm, symm_apply_eq, symm_comp_self, eq_symm_apply, symm_bijective, ContAction.resEquiv_inverse, eq_symm_comp, symm_trans_apply, invFun_eq_symm, Units.symm_mapContinuousMulEquiv, symm_apply_apply, apply_symm_apply, MulEquiv.symm_toContinuousMulEquiv, self_trans_symm, self_comp_symm, Subgroup.subgroupOfContinuousMulEquivOfLe_symm_apply, equivLike_inv_eq_symm, symm_comp_eq, coe_toHomeomorph_symm, apply_eq_iff_symm_apply, symm_symm, comp_symm_eq

CurveIntegrable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	CurveIntegrable	Path.symm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup	—	curveIntegralFun_symm sub_self sub_zero IntervalIntegrable.symm IntervalIntegrable.neg IntervalIntegrable.comp_sub_left enorm_ne_top

Diffeomorph

Definitions

Name	Category	Theorems
symm 📖	CompOp	23 math math: symm_refl, contMDiffOn_comp_diffeomorph_iff, symm_trans', ContinuousLinearEquiv.symm_toDiffeomorph, ContinuousLinearEquiv.coe_toDiffeomorph_symm, apply_symm_apply, sumCongr_symm_symm, image_eq_preimage_symm, range_comp, prodCongr_symm, symm_apply_apply, symm_image_image, symm_toEquiv, contMDiffWithinAt_comp_diffeomorph_iff, self_trans_symm, symm_image_eq_preimage, symm_toHomeomorph, prodComm_symm, symm_trans_self, toEquiv_coe_symm, image_symm_image, sumComm_symm, coe_toHomeomorph_symm

DilationEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	17 math math: smulTorsor_symm_apply, IsometryEquiv.toDilationEquiv_symm, coe_symm_toHomeomorph, symm_bijective, refl_symm, mulRight_symm_apply, symm_apply_apply, mulLeft_symm_apply, toHomeomorph_symm, apply_symm_apply, self_trans_symm, symm_trans_self, IsometryEquiv.coe_symm_toDilationEquiv, inv_def, coe_inv, ratio_symm, symm_symm

DiscreteQuotient

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	toSetoid	—	—	Setoid.symm'

Disjoint

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Disjoint	—	—	disjoint_comm

ENNReal.HolderConjugate

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	ENNReal.HolderConjugate	—	ENNReal.HolderTriple.symm

ENNReal.HolderTriple

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	ENNReal.HolderTriple	—	inv_add_inv_eq_inv add_comm

Equidecomp

Definitions

Name	Category	Theorems
symm 📖	CompOp	7 math math: map_target, restr_refl_symm, symm_involutive, symm_symm, symm_bijective, symm_toPartialEquiv, refl_symm

Equiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	2108 math math: Perm.decomposeFin_symm_of_refl, sumEmpty_symm_apply, MultilinearMap.domDomCongrLinearEquiv'_symm_apply, toColex_symm_eq, SSet.op_δ, NumberField.Ideal.ramificationIdx_primesOverSpanEquivMonicFactorsMod_symm_apply', FreeAddMonoid.lift_symm_apply, AffineEquiv.coe_symm_toEquiv, CategoryTheory.Adjunction.adjunctionOfEquivLeft_counit_app, DomMulAct.smul_monoidHom_apply, NumberField.mixedEmbedding.fundamentalCone.expMapBasis_apply', CategoryTheory.Quiv.equivOfIso_symm_apply, CategoryTheory.nerve.edgeMk_edge, sumAssoc_symm_apply_inl, CategoryTheory.TwoSquare.equivNatTrans_symm_apply, unitsEquivNeZero_symm_apply, Rep.invariantsAdjunction_homEquiv_symm_apply_hom, FreeMonoid.map_apply_map_symm_eq, invFun_as_coe, CategoryTheory.mateEquiv_counit_symm, HahnModule.support_smul_subset_vadd_support', PEquiv.vecMul_toMatrix_toPEquiv, CategoryTheory.Limits.IsLimit.ofConeEquiv_symm_apply_desc, CategoryTheory.coyonedaEquiv_symm_app_apply, CategoryTheory.tensorLeftHomEquiv_symm_coevaluation_comp_whiskerLeft, CategoryTheory.Functor.partialRightAdjointHomEquiv_comp_symm, CategoryTheory.Sheaf.ΓHomEquiv_naturality_left_symm, FirstOrder.Language.LEquiv.onTerm_symm_apply, PiTensorProduct.lift_reindex, AddSubgroup.quotientEquivProdOfLE_symm_apply, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_three_assoc, PerfectionMap.lift_symm_apply, QuaternionAlgebra.coe_linearEquivTuple_symm, Metric.Snowflaking.symm_ofSnowflaking, sigmaAssocProd_apply_fst, Matrix.transposeInvertibleEquivInvertible_symm_apply, CategoryTheory.Arrow.equivSigma_symm_apply_left, CategoryTheory.InjectiveResolution.extEquivCohomologyClass_symm_neg, DFA.reindex_apply_accept, InfTopHom.symm_dual_id, CategoryTheory.Limits.Types.Small.productIso_hom_comp_eval, CategoryTheory.InjectiveResolution.extEquivCohomologyClass_symm_add, IndexedPartition.proj_fiber, List.equivSigmaTuple_symm_apply, IntermediateField.algHomEquivAlgHomOfSplits_symm_apply, Nat.Primes.prodNatEquiv_symm_apply, RootPairing.reflectionPerm_symm, uniqueProd_symm_apply, WithZero.logEquiv_symm, LinearEquiv.extendScalarsOfIsLocalizationEquiv_symm_apply, Submonoid.equivOp_symm_apply_coe, CategoryTheory.Limits.Types.Small.productIso_hom_comp_eval_apply, inv_def, CategoryTheory.Localization.Preadditive.homEquiv_symm_apply, NumberField.Ideal.inertiaDeg_primesOverSpanEquivMonicFactorsMod_symm_apply, FundamentalGroupoid.equiv_symm_apply_as, CategoryTheory.Classifier.SubobjectRepresentableBy.pullback_homEquiv_symm_obj_Ω₀, val_inv_unitsEquivProdSubtype_symm_apply, MeasureTheory.ComplexMeasure.equivSignedMeasure_symm_apply, Set.union_symm_apply_left, PrimeSpectrum.preimageEquivFiber_symm_apply_coe, Topology.IsGeneratedBy.homeomorph_symm_coe, LightCondensed.ihomPoints_apply, sumCompl_symm_apply_pos, optionIsSomeEquiv_symm_apply_coe, coe_constVSub_symm, nsmul_finEquivZMultiples_symm_apply, Topology.WithLower.ofLower_symm, PartENat.withTopEquiv_symm_apply, CategoryTheory.Functor.curryingEquiv_symm_apply_obj_obj, AlgebraicGeometry.coprodSpec_apply, Polynomial.Gal.smul_def, MvPolynomial.comapEquiv_symm_coe, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_unitIso, MulArchimedeanOrder.val_symm_eq, Quiver.starEquivCostar_symm_apply_fst, SSet.ι₀_snd_assoc, symm_preimage_preimage, Function.Involutive.toPerm_symm, Combinatorics.Subspace.IsMono.reindex, CategoryTheory.ReflQuiv.adj.homEquiv_symm_apply, CategoryTheory.Iso.isoCongr_symm_apply, CategoryTheory.ExponentiableMorphism.homEquiv_symm_apply_eq, CategoryTheory.FunctorToTypes.functorHomEquiv_symm_apply_app_app, CategoryTheory.iterated_mateEquiv_conjugateEquiv_symm, AddEquiv.addSubgroupMap_symm_apply, CategoryTheory.CountableCategory.instObjAsType, CategoryTheory.MonoidalClosed.enrichedOrdinaryCategorySelf_homEquiv_symm, NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace_completeFamily_of_ne, contDiff_prodAssoc_symm, MeasureTheory.integral_domSMul, SupBotHom.symm_dual_comp, finEquivZPowers_symm_apply, CategoryTheory.shrinkYonedaEquiv_symm_map_assoc, TopHom.symm_dual_id, neg_def, CategoryTheory.GrothendieckTopology.yonedaEquiv_symm_app_apply, PiTensorProduct.lift_reindex_symm, CategoryTheory.Functor.CorepresentableBy.uniqueUpToIso_inv, Multiset.cast_symm_apply_fst, sSupHom.symm_dual_id, ExteriorAlgebra.lift_symm_apply, PiTensorProduct.map_reindex_symm, DFinsupp.equivFunOnFintype_symm_coe, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_symm_apply, CategoryTheory.FunctorToTypes.shrinkMap_app, CategoryTheory.Quiv.homEquivOfIso_symm_apply, HahnModule.coeff_single_smul_vadd, InfHom.symm_dual_comp, boolArrowEquivProd_symm_apply, NumberField.Ideal.inertiaDeg_primesOverSpanEquivMonicFactorsMod_symm_apply', DirectSum.decomposeAlgEquiv_symm_apply, CompleteLatticeHom.dual_symm_apply_toFun, piCongrLeft_apply, Finset.map_filter, PrimeSpectrum.primeSpectrumProd_symm_inl, AddMagma.FreeAddSemigroup.lift_symm_apply, NumberField.IsCMField.equivInfinitePlace_symm_apply, Subgroup.equivOp_symm_apply_coe, CategoryTheory.Bicategory.Adjunction.homEquiv₂_symm_apply, Quaternion.imK_equivProd_symm_apply, OrderAddMonoidIso.toEquiv_symm, CategoryTheory.Functor.CorepresentableBy.equivOfIsoObj_symm_apply, PiTensorProduct.reindex_reindex, CategoryTheory.Limits.FormalCoproduct.homPullbackEquiv_symm_apply_φ, piOptionEquivProd_symm_apply, LightCondensed.ihomPoints_symm_comp, WithZero.lift'_symm_apply_apply, Function.fromTypes_cons_equiv_symm_apply, WithTop.ofDual_symm, Polynomial.sylvester_comm, apply_symm_apply, CategoryTheory.MonoidalClosed.ofEquiv_uncurry_def, piSplitAt_symm_apply, SSet.stdSimplex.obj₀Equiv_symm_apply, tangentMap_chart, DomMulAct.symm_mk_pow, DomMulAct.smul_linearMap_apply, restrictPreimageFinset_symm_apply_coe, toDegLex_symm_eq, OrderIso.symm_mk, ofLex_symm_eq, OrderMonoidIso.withZero_symm_apply_symm_apply, SimpleGraph.walkLengthTwoEquivCommonNeighbors_symm_apply_coe, equivShrink_symm_div, CategoryTheory.conjugateEquiv_symm_id, CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_right_symm, CategoryTheory.Bicategory.conjugateEquiv_adjunction_id_symm, piCongrSigmaFiber_symm_apply, OrderMonoidIso.withZero_symm_apply_apply, DomAddAct.map_mk_symm_nhds, CategoryTheory.nerve.functorOfNerveMap_map, StandardEtalePair.homEquiv_symm_apply, SSet.opFunctorCompOpFunctorIso_inv_app_app, sumComm_symm, arrowCongr'_symm, CategoryTheory.Adjunction.adjunctionOfEquivRight_counit_app, MulEquiv.symm_mk, arrowCongr'_apply, OrderDual.toDual_symm_eq, PreQuasiregular.equiv_symm_apply, CategoryTheory.Limits.biproduct.whiskerEquiv_inv, MulHom.op_symm_apply_apply, FreeAddGroup.freeAddGroupCongr_symm, CategoryTheory.Limits.FormalCoproduct.isoOfComponents_inv_φ, Matrix.submatrix_vecMul_equiv, TwoSidedIdeal.equivMatrix_symm_apply_ringCon, RelIso.relHomCongr_symm_apply, Perm.perm_symm_on_of_perm_on_finset, divRight_symm_apply, PresheafOfModules.homEquivOfIsLocallyBijective_symm_apply, Additive.toMul_symm_eq, DomMulAct.coe_mkHomeomorph_symm, NumberField.mixedEmbedding.fundamentalCone.completeBasis_apply_of_ne, CategoryTheory.ParametrizedAdjunction.inr_arrowHomEquiv_symm_apply_left, equivShrink_symm_sub, CategoryTheory.ShiftedHom.opEquiv_symm_add, sigmaPreimageEquiv_symm_apply_snd_coe, CategoryTheory.Functor.partialLeftAdjointHomEquiv_comp_symm_assoc, CategoryTheory.ParametrizedAdjunction.inl_arrowHomEquiv_symm_apply_left, AddConstMap.conjNeg_symm, CategoryTheory.Limits.Types.isColimit_iff_coconeTypesIsColimit, VertexOperator.of_coeff_apply_coeff, DomMulAct.smul_mulDistribActionHom_apply, IndexedPartition.equivQuotient_symm_proj_apply, MeasureTheory.Measure.domSMul_apply, CategoryTheory.Limits.opCompYonedaSectionsEquiv_symm_apply_coe, Finset.Nat.sigmaAntidiagonalTupleEquivTuple_symm_apply_snd_coe, RelIso.coe_fn_symm_mk, CategoryTheory.Limits.yonedaCompLimIsoCocones_inv_app, coe_fn_symm_mk, CategoryTheory.Limits.WalkingPair.equivBool_symm_apply_true, CategoryTheory.nerve.homEquiv_edgeMk_map_nerveMap, genericPoints.equiv_symm_apply, Quiver.SingleObj.toPrefunctor_symm_id, BoundedOrderHom.symm_dual_id, SSet.S.equivElements_symm_apply_dim, pointReflection_symm, CategoryTheory.Limits.MulticospanIndex.sectionsEquiv_symm_apply_val, sigmaAssocProd_apply_snd_snd, ofDegLex_symm_eq, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_left_symm_assoc, toHomeomorphOfIsInducing_symm_apply, AddMonoidAlgebra.mapDomainAddEquiv_apply, CategoryTheory.Endofunctor.Adjunction.algebraCoalgebraEquiv_inverse_obj_str, permCongr_symm_apply, Pi.single_comp_equiv, MultilinearMap.domDomCongrEquiv_symm_apply, Orientation.map_symm, WithCStarModule.equiv_symm_snd, CategoryTheory.CoreSmallCategoryOfSet.smallCategoryOfSet_id, CategoryTheory.tensorRightHomEquiv_symm_naturality, CategoryTheory.yonedaMonObjIsoOfRepresentableBy_inv_app_hom_apply, PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal, Perm.subtypeCongr.symm, Matrix.kroneckerTMul_assoc', DomMulAct.mem_stabilizer_iff, CategoryTheory.conjugateEquiv_symm_comp_assoc, CategoryTheory.Limits.coyonedaCompLimIsoCones_inv_app, CategoryTheory.Subfunctor.ofSection_eq_range, symm_comp_self, SimpleGraph.Iso.connectedComponentEquiv_symm_apply, exists_congr_left, CategoryTheory.Localization.SmallHom.equiv_equiv_symm, CommRing.Pic.mul_eq_tensor, Perm.symm_mul, finsuppUnique_symm_apply_support_val, Multiset.Icc_eq, zpowersHom_symm_apply, GrpCat.SurjectiveOfEpiAuxs.τ_symm_apply_infinity, Sequential.isoEquivHomeo_symm_apply, optionProdEquiv_symm_inr, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_one, Set.rangeInl_symm_apply_coe, CategoryTheory.ForgetEnrichment.equivFunctor_map, HVertexOperator.coeff_isPWOsupport, contMDiff_equivTangentBundleProd_symm, finSuccEquiv_symm_none, InfTopHom.symm_dual_comp, SheafOfModules.unitHomEquiv_symm_comp, Matrix.mul_val_succ, CategoryTheory.Functor.partialRightAdjointHomEquiv_symm_comp, uliftMultiplesHom_symm_apply, Algebra.PreSubmersivePresentation.jacobiMatrix_reindex, FirstOrder.Language.Term.constantsVarsEquiv_symm_apply, DirectSum.lequivCongrLeft_apply, prodPiEquivSumPi_apply, CategoryTheory.Functor.Final.colimitCoconeOfComp_isColimit, CategoryTheory.Functor.equivCatHom_symm_apply, Flag.symm_map, NonUnitalStarAlgHom.prodEquiv_symm_apply, DomMulAct.smul_addMonoidHom_apply, Finset.map_consEquiv_filter_piFinset, RootPairing.Hom.comp_indexEquiv_symm_apply, FreeAddSemigroup.lift_symm_apply, Affine.Simplex.excenterWeightsUnnorm_reindex, CategoryTheory.InjectiveResolution.of_def, Perm.sumCongr_symm, piCongrRight_symm_apply, CategoryTheory.Coyoneda.objOpOp_inv_app, CategoryTheory.GradedObject.comapEquiv_counitIso, sigmaCongrRight_symm, CategoryTheory.Subobject.wideCospan_map_term, RingEquiv.piCongrLeft_symm_apply, traverse_def, CategoryTheory.ProjectiveResolution.extEquivCohomologyClass_symm_sub, CategoryTheory.sheafToPresheafCompYonedaCompWhiskeringLeftSheafToPresheaf_app_app, Finset.Nat.antidiagonalEquivFin_symm_apply_coe, subtypeEquivCodomain_symm_apply, left_inv', sumSigmaDistrib_symm_apply, MonoidHom.toHomPerm_apply_symm_apply, sigmaPUnit_symm_apply_snd, Finset.Nat.sigmaAntidiagonalTupleEquivTuple_symm_apply_fst, optionEquivSumPUnit_symm_inr, CategoryTheory.Pi.equivalenceOfEquiv_unitIso, CategoryTheory.WithInitial.isColimitEquiv_symm_apply_desc, eq_symm_comp, HomotopicalAlgebra.BifibrantObject.HoCat.homEquivLeft_symm_apply, RingEquiv.piCongrLeft'_symm, CategoryTheory.Limits.SingleObj.Types.colimitEquivQuotient_symm_apply, RingEquiv.piCongrLeft'_apply, Subsemigroup.equivOp_symm_apply_coe, CategoryTheory.Pi.equivalenceOfEquiv_counitIso, PresheafOfModules.freeYonedaEquiv_symm_app, CategoryTheory.Limits.biproduct.matrixEquiv_symm_apply, piCongrLeft_symm_apply, Finset.map_snocEquiv_filter_piFinset, Homeomorph.coe_symm_toEquiv, IsometryEquiv.piCongrLeft'_apply, Perm.sign_trans_trans_symm, CategoryTheory.FinCategory.categoryAsType_comp, SetTheory.PGame.relabel_moveRight, FirstOrder.Language.Term.constantsVarsEquivLeft_apply, CategoryTheory.leftAdjointOfStructuredArrowInitialsAux_symm_apply, CategoryTheory.Bicategory.conjugateEquiv_symm_id, uliftZPowersHom_symm_apply, divLeft_symm_apply, Unitization.starLift_symm_apply_apply, AlgebraicGeometry.Scheme.Hom.irreducibleComponentsEquiv_symm_apply_coe, ofBijective_apply_symm_apply, CategoryTheory.Adjunction.eq_homEquiv_apply, isRight_finSumNatEquiv_symm_apply, MonoidHom.fiberEquivKer_symm_apply, arrowCongr_apply, LinearEquiv.coe_symm_toEquiv, CategoryTheory.MonObj.ofRepresentableBy_one, Perm.sum_comp', Set.BijOn.equiv_symm, CategoryTheory.Limits.Concrete.productEquiv_symm_apply_π, Function.FromTypes.curry_two_eq_curry, mapMatrix_symm, CategoryTheory.FinCategory.asTypeToObjAsType_map, MonoidAlgebra.opRingEquiv_symm_apply, Matrix.updateCol_reindex, DomMulAct.map_mk_symm_nhds, transPartialEquiv_symm_apply, DomMulAct.comap_mk_nhds, CategoryTheory.Iso.homFromEquiv_symm_apply, SheafOfModules.unitHomEquiv_symm_freeHomEquiv_apply, prodUnique_symm_apply, Perm.sameCycle_inv_apply_left, CategoryTheory.uliftYonedaEquiv_symm_apply_app, AddMonoidHom.fiberEquiv_symm_apply, ofLeftInverse_symm_apply, symm_apply_apply, Filter.tendsto_prodAssoc_symm, Perm.sign_symm, ContinuousMap.piEquiv_symm_apply, CategoryTheory.conjugateEquiv_symm_apply_app, coe_equiv_lpPiLp_symm, MonoidAlgebra.symm_mapDomainAddEquiv, AlgEquiv.symm_toEquiv_eq_symm, BoundedLatticeHom.symm_dual_comp, finCongr_symm_apply_coe, Perm.SameCycle.symm_apply_right, CategoryTheory.Functor.LeibnizAdjunction.adj_counit_app_left, CategoryTheory.Functor.partialLeftAdjointHomEquiv_symm_comp, MonoidAlgebra.lift_symm_apply, Multiset.consEquiv_symm_some, DirectSum.decompose_symm_sub, DomAddAct.isOpenEmbedding_mk_symm, Function.Embedding.equiv_toEmbedding_trans_symm_toEmbedding, AlgebraicGeometry.AffineSpace.homOverEquiv_symm_apply_coe, Matrix.submatrix_single_equiv, sigmaFinsuppAddEquivDFinsupp_symm_apply, TotallyDisconnectedSpace.continuousMapEquivOfConnectedSpace_symm_apply_apply, optionSubtype_symm_apply_apply_coe, Perm.perm_inv_on_of_perm_on_finite, Affine.Simplex.excenterExists_reindex, sigmaAssocProd_apply_snd_fst, Quaternion.re_equivProd_symm_apply, AddMonoidAlgebra.opRingEquiv_symm_apply, Finset.map_symm_subset, WithCStarModule.equiv_symm_neg, LatticeHom.dual_symm_apply_toFun, Set.insert_symm_apply_inr, ContinuousAffineEquiv.toEquiv_symm, PartialEquiv.transEquiv_symm_apply, HahnSeries.SummableFamily.hsum_smul_module, Perm.decomposeFin_symm_apply_one, Submodule.coe_isComplEquivProj_symm_apply, Set.powersetCard.compl_symm, LieAlgebra.bracket_ofTwoCocycle, SheafOfModules.GeneratingSections.epi, mulLeft₀_symm_apply, MonoidHom.toAdditive_symm_apply_apply, symm_subLeft, CategoryTheory.Functor.partialRightAdjointHomEquiv_comp_symm_assoc, Matrix.updateRow_submatrix_equiv, PartialFun.Iso.mk_inv, AddAction.orbitZMultiplesEquiv_symm_apply', Complex.equivRealProd_symm_apply, toOrderIsoSet_symm_apply, Sym.coe_equivNatSum_symm_apply, sigmaAntidiagonalEquivProd_symm_apply_snd, CategoryTheory.Subfunctor.Subpresheaf.ofSection_eq_range', Rep.coinvariantsAdjunction_homEquiv_symm_apply_hom, sumPiEquivProdPi_symm_apply, leftInverse_symm, SSet.nonDegenerateEquivOfIso_symm_apply_coe, CategoryTheory.Subobject.symm_apply_mem_iff_mem_image, OrderAddMonoidIso.coe_toEquiv_symm, CategoryTheory.ParametrizedAdjunction.arrowHomEquiv_symm_apply_right, CategoryTheory.nerve.edgeMk_id, FirstOrder.Ring.lift_genericPolyMap, CategoryTheory.Limits.FormalCoproduct.inclHomEquiv_symm_apply_φ, OrthonormalBasis.coe_reindex, SimpleGraph.Iso.map_symm_apply, AddAction.orbitZMultiplesEquiv_symm_apply, equivCongr_apply_apply, ContinuousAlternatingMap.ofSubsingletonLIE_symm_apply, CategoryTheory.InjectiveResolution.extEquivCohomologyClass_symm_sub, sigmaSumDistrib_symm_apply, HahnSeries.SummableFamily.Equiv_toFun, CategoryTheory.Presieve.FamilyOfElements.singletonEquiv_symm_apply_self, AffineBasis.coord_reindex, Set.coe_biUnionEqSigmaOfDisjoint_symm_apply, powersHom_symm_apply, CategoryTheory.LiftLeftAdjoint.constructLeftAdjointEquiv_symm_apply, Matrix.mul_reindexLinearEquiv_one, Set.univ_symm_apply, MeasureTheory.LevyProkhorov.toMeasureEquiv_symm_apply_toMeasure, AddEquiv.symmEquiv_symm_apply_symm_apply, AddChar.toMonoidHomEquiv_symm_apply, SupHom.symm_dual_comp, CategoryTheory.Functor.curryingFlipEquiv_symm_apply_map_app, coe_ofInjective_symm, CategoryTheory.PreZeroHypercover.interLift_h₀, CategoryTheory.ProjectiveResolution.extEquivCohomologyClass_symm_mk_hom, CategoryTheory.uliftCoyonedaEquiv_symm_apply_app, LieAlgebra.Orthogonal.indefiniteDiagonal_assoc, toPartialEquiv_symm_apply, FirstOrder.Field.lift_genericMonicPoly, CategoryTheory.Functor.representableByUliftFunctorEquiv_symm_apply_homEquiv, Function.piCongrLeft'_symm_update, symm_symm_apply, CategoryTheory.Discrete.equivalence_unitIso, Fin.castLEquiv_symm_apply, Matrix.coe_ofAddEquiv_symm, CategoryTheory.Adjunction.leftAdjointCompNatTrans₀₁₃_eq_conjugateEquiv_symm, add_def, PiTensorProduct.reindex_comp_tprod, equivShrink_symm_bot, Function.OfArity.curryEquiv_symm_apply, SSet.opFunctor_map, pow_finEquivZPowers_symm_apply, Perm.perm_inv_mapsTo_of_mapsTo, Normal.algHomEquivAut_symm_apply, Homeomorph.continuousMapCongr_symm_apply, CategoryTheory.FunctorToTypes.binaryCoproductEquiv_symm_apply, Unitization.lift_symm_apply_apply, CategoryTheory.Bicategory.Adj.Hom₂.conjugateEquiv_symm_τr, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_symm_map, powCoprime_symm_apply, SSet.Subcomplex.ofSimplexProd_eq_range, CategoryTheory.nerve.homEquiv_edgeMk, FormalMultilinearSeries.changeOriginIndexEquiv_symm_apply_fst, CategoryTheory.Functor.homObjEquiv_symm_apply_app, HahnSeries.SummableFamily.smul_apply, CategoryTheory.Adjunction.mkOfHomEquiv_counit_app, MultilinearMap.curryFinFinset_apply, Module.Basis.map_equiv, Fin.revPerm_symm, RootPairing.Equiv.inv_indexEquiv, Set.congr_symm_apply, CategoryTheory.Functor.RepresentableBy.comp_homEquiv_symm, Matrix.uniqueEquiv_symm_apply, HomotopicalAlgebra.leftHomotopyClassEquivRightHomotopyClass_symm_mk, AlgEquiv.equivCongr_symm, SetTheory.PGame.toRightMovesNim_one_symm, divRight₀_symm_apply, Perm.decomposeFin.symm_sign, Sym.coe_equivNatSumOfFintype_symm_apply, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_right_symm, SSet.op_σ, Module.Relations.Solution.IsPresentation.linearMapEquiv_symm_apply, uliftPowersHom_symm_apply, CategoryTheory.Functor.Initial.limitConeComp_isLimit, IsDedekindDomain.HeightOneSpectrum.equivHeightOneSpectrum_symm_apply, CategoryTheory.Functor.FullyFaithful.homEquiv_symm_apply, equivPUnit_symm_apply, equivCongr_refl_right, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_symm_fst, CategoryTheory.Bicategory.mateEquiv_symm_apply', CategoryTheory.ProjectiveResolution.extAddEquivCohomologyClass_symm_apply, punitProd_symm_apply, EisensteinSeries.gammaSetDivGcdSigmaEquiv_symm_eq, Module.Ray.map_symm, Perm.extendDomain_apply_subtype, sigmaSigmaSubtype_symm_apply, AlgHom.opComm_symm_apply_apply, Homeomorph.sumProdDistrib_symm_apply, Topology.IsQuotientMap.liftEquiv_symm_apply_coe, DomMulAct.isInducing_mk_symm, QuaternionAlgebra.equivTuple_symm_apply, inv_symm, Delone.DeloneSet.mapIsometry_symm_apply_carrier, AffineBasis.coe_reindex, sub_def, forall_congr_left, funSplitAt_symm_apply, AddMonoidHom.toMultiplicativeRight_symm_apply_apply, Topology.WithUpperSet.toUpperSet_symm, CategoryTheory.nerve.nonempty_compStruct_iff, FormalMultilinearSeries.changeOriginIndexEquiv_symm_apply_snd_snd_coe, CategoryTheory.Limits.Types.Small.productIso_inv_comp_π, Finset.Nat.antidiagonalTuple_two, CategoryTheory.uliftYonedaEquiv_symm_map, TopHom.dual_symm_apply_apply, FreeAbelianGroup.liftAddEquiv_symm_apply, Finsupp.mapRange.equiv_symm, MulArchimedeanOrder.of_symm_eq, OrderIso.coe_symm_toEquiv, normalizedFactorsEquivOfQuotEquiv_symm, finRotate_succ_eq_decomposeFin, subtypeOrEquiv_symm_inl, withBotSubtypeNe_symm_apply_coe, CompHausLike.isoEquivHomeo_symm_apply, TrivSqZeroExt.invertibleEquivInvertibleFst_symm_apply_invOf, AlgEquiv.sumArrowEquivProdArrow_symm_apply_inr, optionSubtypeNe_symm_of_ne, Bundle.TotalSpace.toProd_symm_apply_proj, gelfandStarTransform_symm_apply, FintypeCat.equivEquivIso_symm_apply_symm_apply, CategoryTheory.FinCategory.categoryAsType_id, finSumNatEquiv_symm_apply_fin, AffineEquiv.toEquiv_symm, CategoryTheory.Functor.Final.colimitCoconeComp_isColimit, coe_subtypeEquivCodomain_symm, Perm.sameCycle_symm_apply_right, DomAddAct.continuous_mk_symm, CategoryTheory.Limits.IsColimit.ofCoconeEquiv_symm_apply_desc, embeddingCongr_symm, WithZero.expEquiv_symm, AlgebraicGeometry.coe_opensRestrict_symm_apply, Function.IsFixedPt.equiv_symm, TensorAlgebra.lift_symm_apply, Primrec.of_equiv_symm_iff, LatticeHom.symm_dual_id, DirectSum.decompose_symm_neg, LinearMap.polyCharpolyAux_map_eval, CommRing.Pic.inv_eq_dual, RootPairing.EmbeddedG2.indexEquivAllRoots_symm_apply, symm_toHomeomorph, Affine.Simplex.excenterWeights_reindex, CategoryTheory.LiftLeftAdjoint.constructLeftAdjointEquiv_apply, finSumNatEquiv_symm_apply_add_left, CategoryTheory.Limits.IsInitial.to_eq_descCoconeMorphism, piCongrLeft_symm_preimage_univ_pi, AlgHom.prodEquiv_symm_apply, CategoryTheory.tensorRightHomEquiv_symm_coevaluation_comp_whiskerRight, QuadraticAlgebra.equivProd_symm_apply, Function.update_comp_equiv, Algebra.GrothendieckGroup.lift_symm_apply, Topology.isGeneratedBy_iff, CategoryTheory.Oplax.StrongTrans.Modification.equivOplax_symm_apply, ContinuousAlternatingMap.piEquiv_symm_apply, Affine.Simplex.median_reindex, DyckWord.equivTreesOfNumNodesEq_symm_apply_coe, PartialEquiv.transEquiv_target, Fin.coe_of_injective_castSucc_symm, Matrix.submatrix_id_mul_right, Prod.Lex.sumLexProdLexDistrib_symm_apply, MulChar.equivToUnitHom_symm_coe, exteriorPower.presentation.relationsSolutionEquiv_symm_apply_var, NumberField.det_basisOfFractionalIdeal_eq_absNorm, Matrix.reindex_apply, Function.fromTypes_zero_equiv_symm_apply, dotProduct_comp_equiv_symm, CategoryTheory.Sieve.overEquiv_symm_iff, DihedralGroup.oddCommuteEquiv_symm_apply, MonoidHom.fiberEquiv_symm_apply, equivShrink_symm_zero, CategoryTheory.GrothendieckTopology.overEquiv_symm_mem_over, CategoryTheory.conjugateEquiv_counit_symm, CategoryTheory.Sieve.functorPushforward_over_map, DirectSum.decompose_symm_one, CommRing.Pic.mk_eq_iff, sumIsLeft_symm_apply_coe, Pi.mulSingle_comp_equiv, sumProdDistrib_symm_apply_right, sumArrowEquivProdArrow_symm_apply_inr, CategoryTheory.PreGaloisCategory.autIsoFibers_inv_app, CategoryTheory.Limits.Fork.IsLimit.homIso_symm_apply, CategoryTheory.Functor.natTransEquiv_symm_apply_app, CategoryTheory.Functor.homEquivOfIsLeftKanExtension_symm_apply, Orthonormal.equiv_symm, Module.Basis.equiv_symm, PiTensorProduct.reindex_trans, addMonoidAlgebraEquivDirectSum_symm_apply, biSup_comp, OrderIso.toEquiv_symm, ContinuousMultilinearMap.ofSubsingleton_symm_apply_apply, CommRing.Pic.mk_eq_self, HahnSeries.SummableFamily.smul_hsum, Matrix.updateRow_reindex, Topology.IsGeneratedBy.continuous_equiv_symm, right_inv', finSuccEquiv'_symm_coe_above, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_symm_naturality_left, CategoryTheory.Limits.FintypeCat.productEquiv_symm_comp_π_apply, optionSubtype_apply_symm_apply, MulAction.toPerm_symm_apply, Continuous.continuous_symm_of_equiv_compact_to_t2, InfHom.dual_symm_apply_toFun, List.getEquivOfForallCountEqOne_symm_apply_val, Multiplicative.ofAdd_symm_eq, CategoryTheory.InducedCategory.homEquiv_symm_apply_hom, AddMonoidHom.toMultiplicative_symm_apply_apply, CategoryTheory.Limits.FormalCoproduct.isoOfComponents_inv_f, CategoryTheory.Functor.coconeTypesEquiv_symm_apply_pt, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_one_assoc, ContinuousMultilinearMap.prodEquiv_symm_apply_fst, GradedAlgebra.proj_recompose, PadicInt.coe_adicCompletionIntegersEquiv_symm_apply, LieAlgebra.lieCharacterEquivLinearDual_symm_apply, SkewMonoidAlgebra.coeff_equivMapDomain, symm_comp_eq, sigmaEquivProd_symm_apply, PiTensorProduct.reindex_tprod, DomAddAct.symm_mk_neg, CategoryTheory.conjugateEquiv_symm_iso, toPartialEquivOfImageEq_symm_apply, DomMulAct.continuous_mk_symm, HahnSeries.of_symm_smul_of_eq_mul, LinearEquiv.isAdjointPair_symm_iff, HahnSeries.SummableFamily.smul_toFun, PowerBasis.liftEquiv_symm_apply, AddEquiv.op_symm_apply_symm_apply, LinearIsometryEquiv.coe_prodAssoc_symm, CategoryTheory.Subfunctor.ofSection_eq_range', bijOn_symm_image, Abelianization.coe_lift_symm, DomMulAct.smul_mulActionHom_apply, MulEquiv.arrowCongr_apply, Perm.sameCycle_inv_apply_right, OrderHom.symm_dual_id, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_symm_naturality_right, trans_cancel_right, CategoryTheory.PreGaloisCategory.fiberBinaryProductEquiv_symm_fst_apply, SetTheory.PGame.Relabelling.mk'_rightMovesEquiv, symm_apply_eq, symm_symm, CategoryTheory.Limits.IsLimit.homEquiv_symm_π_app, AddUnits.addLeft_symm, Matrix.submatrix_updateCol_equiv, finSuccAboveEquiv_symm_apply_ne_last, CommRing.Pic.mapAlgebra_apply, NonUnitalAlgHom.prodEquiv_symm_apply, CategoryTheory.tensorLeftHomEquiv_tensor, Topology.WithScott.toScott_symm_eq, pnatEquivNat_symm_apply, CommRing.Pic.ext_iff, zero_def, sumSumSumComm_symm, IsPrimitiveRoot.primitiveRootsPowEquiv_symm_apply_coe, CategoryTheory.enrichedFunctorTypeEquivFunctor_symm_apply_obj, DomAddAct.symm_mk_add, CategoryTheory.FinCategory.asTypeToObjAsType_obj, sigmaFinsuppEquivDFinsupp_symm_apply, CategoryTheory.Limits.CofanTypes.equivOfIsColimit_symm_apply, optionProdEquiv_symm_inl, Set.opEquiv_self_symm_apply_coe, Delone.DeloneSet.mapIsometry_symm_apply_coveringRadius, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_three, SheafOfModules.freeHomEquiv_symm_comp, Affine.Simplex.exsphere_reindex, CategoryTheory.CosimplicialObject.cechConerveEquiv_symm_apply, SetTheory.PGame.relabel_moveLeft, symm_trans_apply, Matrix.submatrix_mulVec_equiv, ContinuousMap.val_unitsLift_symm_apply_apply, CategoryTheory.Functor.RepresentableBy.equivOfIsoObj_symm_apply, CategoryTheory.FunctorToTypes.binaryProductEquiv_symm_apply, SimpleGraph.comap_symm, symm_swap, CategoryTheory.GradedObject.comapEquiv_unitIso, CommRing.Pic.instFreeAsModuleOfNat, SimpleGraph.Walk.IsHamiltonian.getVertEquiv_symm_apply, Module.Relations.Solution.directSumEquiv_symm_apply_var, FreeGroup.lift_symm_apply, Finset.univ_perm_option, AddConstEquiv.coe_symm_toEquiv, BoundedOrderHom.symm_dual_comp, piCongrLeft_symm_preimage_pi, BialgEquiv.toEquiv_symm, Real.sinhEquiv_symm_apply, SetTheory.PGame.moveRight_nim, Module.Basis.det_reindex_symm, sSupHom.dual_symm_apply_toFun, AddCircle.continuous_equivIoc_symm, AddEquiv.toEquiv_symm, Finsupp.equivFunOnFinite_symm_apply_support, Perm.signAux3_symm_trans_trans, Quiver.SingleObj.pathEquivList_symm_cons, Fin.cycleIcc_to_cycleRange, CochainComplex.HomComplex.CohomologyClass.equivOfIsKInjective_symm_apply, UniqueFactorizationMonoid.normalizedFactorsEquiv_symm_apply, Set.univPi_symm_apply_coe, MeasureTheory.diracProba_comp_diracProbaEquiv_symm_eq_val, CategoryTheory.Limits.IsLimit.equivIsoLimit_symm_apply, AffineBasis.basisOf_reindex, FintypeCat.equivEquivIso_apply_inv, finSumFinEquiv_symm_apply_castAdd, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_symm_snd, plift_symm_apply, SSet.stdSimplex.map_apply, Matrix.comp_toSquareBlock, CategoryTheory.CartesianClosed.homEquiv_symm_apply_eq, toHomeomorphOfContinuousOpen_symm_apply, DomMulAct.smul_aeeqFun_aeeq, NonUnitalRingHom.op_symm_apply_apply, Perm.signAux_eq_signAux2, CategoryTheory.bijection_symm_apply_id, SSet.ι₀_snd, ComplexShape.symmetryEquiv_symm_apply_coe, Matrix.cramer_reindex, Rep.resIndAdjunction_homEquiv_symm_apply, Function.Embedding.equiv_symm_toEmbedding_trans_toEmbedding, Subgroup.quotientEquivProdOfLE_symm_apply, DomMulAct.isQuotientMap_mk_symm, CategoryTheory.ShortComplex.Homotopy.equivSubZero_symm_apply, CategoryTheory.Limits.whiskeringLimYonedaIsoCones_inv_app_app, CategoryTheory.Presheaf.freeYonedaHomEquiv_symm_comp, CategoryTheory.ParametrizedAdjunction.inr_arrowHomEquiv_symm_apply_left_assoc, ContinuousMap.sigmaEquiv_symm_apply, AddEquiv.add_submonoid_map_symm_apply, toPEquiv_symm, CategoryTheory.ProjectiveResolution.of_def, PerfectRing.liftEquiv_symm_apply, WithCStarModule.equiv_symm_smul, FreeNonUnitalNonAssocAlgebra.lift_symm_apply, FreeSemigroup.lift_symm_apply, pow_def, AddHom.op_symm_apply_apply, Shrink.ext_iff, subtypeEquivRight_symm_apply, CategoryTheory.Sieve.overEquiv_symm_pullback, DomAddAct.comap_mk.symm_nhds, OrthonormalBasis.reindex_apply, Fin.insertNthEquiv_symm_apply, Zsqrtd.lift_symm_apply_coe, Finset.union_symm_left, RootPairing.flipEquiv_symm_apply, CategoryTheory.unit_conjugateEquiv_symm, EquivFunctor.mapEquiv_symm, PiTensorProduct.reindex_symm, addRight_symm_apply, Set.sumCompl_symm_apply_of_mem, CategoryTheory.Grpd.freeForgetAdjunction_homEquiv_symm_apply, Nat.divModEquiv_symm_apply, Matrix.toLin'_reindex, HahnModule.coeff_smul_order_add_order, HahnModule.of_symm_smul, ofBoolAlg_symm_eq, EquivLike.coe_symm_apply_apply, Topology.WithLowerSet.toLowerSet_symm, AddEquiv.arrowCongr_apply, DomAddAct.symm_mk_zsmul, Monoid.PushoutI.homEquiv_symm_apply, equivShrink_symm_top, Set.sumCompl_symm_apply, CategoryTheory.Functor.LeibnizAdjunction.adj_counit_app_right, PartENat.withTopEquiv_symm_ofNat, CategoryTheory.Groupoid.isoEquivHom_symm_apply_hom, CategoryTheory.FunctorToTypes.shrink_map, CategoryTheory.Limits.FormalCoproduct.cofanHomEquiv_symm_apply_φ, CategoryTheory.LocallyDiscrete.locallyDiscreteEquiv_symm_apply_as, sumPiEquivProdPi_symm_preimage_univ_pi, Function.Embedding.congr_apply, CategoryTheory.Localization.homEquiv_symm_apply, SimpleGraph.incidenceSetEquivNeighborSet_symm_apply_coe, conj_symm, Unitization.val_unitsFstOne_mulEquiv_quasiregular_symm_apply_coe, CategoryTheory.CountableCategory.instLocallySmallObjAsType, Finsupp.lcongr_symm, Matrix.compRingEquiv_symm_apply, Finsupp.sumFinsuppEquivProdFinsupp_symm_inr, BoundedLatticeHom.dual_symm_apply_toFun, finCycle_symm_apply, DomMulAct.symm_mk_one, CategoryTheory.FinCategory.objAsTypeToAsType_map, Function.fromTypes_succ_equiv_symm_apply, CategoryTheory.Functor.flippingEquiv_symm_apply_map_app, Perm.set_support_inv_eq, MultilinearMap.freeDFinsuppEquiv_def, CategoryTheory.tensorRightHomEquiv_symm_coevaluation_comp_whiskerLeft, Affine.Simplex.exradius_reindex, UniformEquiv.coe_symm_toEquiv, CategoryTheory.Limits.SingleObj.Types.limitEquivFixedPoints_symm_apply, Function.FromTypes.curryEquiv_symm_apply, TopologicalSpace.Compacts.equiv_symm_apply, withTopSubtypeNe_symm_apply_coe, Unitization.val_inv_unitsFstOne_mulEquiv_quasiregular_symm_apply_coe, Subgroup.quotientEquivProdOfLE'_symm_apply, AlgebraicGeometry.AffineSpace.homOfVector_toSpecMvPoly_assoc, CategoryTheory.ComposableArrows.arrowEquiv_symm_apply, refl_symm, AddEquiv.mulOp_symm_apply, RootPairing.Hom.coroot_coweightMap, MaximalSpectrum.equivSubtype_symm_apply_asIdeal, CategoryTheory.Limits.Types.Small.productIso_inv_comp_π_apply, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_functor_obj, QuotientAddGroup.equivIcoMod_symm_apply, CompleteLatticeHom.symm_dual_id, SSet.ι₁_snd_assoc, PresheafOfModules.toSheaf_map_sheafificationHomEquiv_symm, finTwoArrowEquiv_symm_apply, CategoryTheory.Limits.compCoyonedaSectionsEquiv_symm_apply_coe, CategoryTheory.Enriched.Functor.functorHom_whiskerLeft_natTransEquiv_symm_app, CategoryTheory.Functor.flippingEquiv_symm_apply_obj_obj, Units.mulLeft_symm, FreeAddMonoid.ofList_symm, finEquivMultiples_symm_apply, CStarMatrix.toCLM_apply_single, graph_inv, Finset.equivBitIndices_symm_apply, piCongr'_symm_apply_symm_apply, DirectSum.decompose_symm_algebraMap, eq_comp_symm, AddEquiv.toMultiplicativeLeft_symm_apply_symm_apply, Finset.union_symm_right, Fin.cycleRange_symm_zero, MulEquiv.symm_monoidHomCongrRightEquiv, Computable.symm, Matroid.mapEquiv_dep_iff, Set.powersetCard.ofFinEmbEquiv_symm_apply, Matrix.det_reindex, CategoryTheory.TransportEnrichment.forgetEnrichmentEquivInverse_map, SSet.OneTruncation₂.nerveEquiv_symm_apply_map, Multiset.uIcc_eq, multiplesHom_symm_apply, MvPolynomial.universalFactorizationMapPresentation_map, isLeft_finSumNatEquiv_symm_apply, ulift_symm_apply, DirectSum.decompose_symm_add, IsArtinianRing.primeSpectrumEquivMaximalSpectrum_symm_apply_asIdeal, Delone.DeloneSet.mapIsometry_symm, MeasureTheory.VectorMeasure.equivMeasure_symm_apply, WithLp.equiv_symm_apply, Affine.Simplex.touchpointWeights_reindex, Fin.coe_of_injective_castLE_symm, DomMulAct.isClosedEmbedding_mk_symm, CategoryTheory.InjectiveResolution.extEquivCohomologyClass_symm_zero, Topology.WithUpper.ofUpper_symm, AddChar.toAddMonoidHomEquiv_symm_zero, coe_vaddConst_symm, Module.Basis.coe_reindex, MeasurableEquiv.measurable_invFun, toBoolAlg_symm_eq, CStarMatrix.toCLM_apply_eq_sum, DFinsupp.linearEquivFunOnFintype_symm_apply, Perm.decomposeOption_symm_of_none_apply, LieAlgebra.LieEquiv.ofCoboundary_toFun, PadicInt.coe_adicCompletionIntegersEquiv_apply, invOn, DFinsupp.equivCongrLeft_apply, Matrix.reindexLieEquiv_symm, AlgEquiv.piCongrLeft'_apply, Subgroup.IsComplement.equiv_symm_apply, pprodCongr_symm_apply, SSet.ι₁_snd, Orthonormal.map_equiv, DyckWord.equivTree_symm_apply, Homeomorph.piCongrLeft_apply, QuaternionAlgebra.lift_symm_apply, ContinuousMultilinearMap.prodEquiv_symm_apply, Continuous.homeoOfEquivCompactToT2.t1_counterexample, AddMonoidHom.op_symm_apply_apply, AddConstEquiv.toEquiv_symm, Tropical.tropEquiv_symm_coe_fn, Monoid.CoprodI.Word.equivPair_symm, DFA.symm_reindex, AddEquiv.toMultiplicativeRight_symm_apply_apply, CategoryTheory.GrothendieckTopology.yonedaEquiv_symm_map, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_functor_map, optionSubtypeNe_symm_apply, CategoryTheory.PreOneHypercover.inter_p₂, exteriorPower.ιMultiDual_apply_ιMulti, PiTensorProduct.lift_comp_reindex, CategoryTheory.WithTerminal.isLimitEquiv_symm_apply_lift, FreeGroupBasis.reindex_apply, CategoryTheory.Sheaf.ΓObjEquivSections_naturality_symm, Matrix.vecMul_permMatrix, PrimeSpectrum.primeSpectrumProd_symm_inr, arrowCongr_symm, mulRight_symm, CategoryTheory.Limits.Types.limitEquivSections_symm_apply, Finsupp.domCongr_symm, Perm.perm_symm_mapsTo_iff_mapsTo, CategoryTheory.ProjectiveResolution.extEquivCohomologyClass_symm_zero, CategoryTheory.Pseudofunctor.isPrestackFor_iff_isSheafFor, sigmaUnique_symm_apply, Set.sumDiffSubset_symm_apply_of_notMem, ofBijective_symm_apply_apply, Shrink.algEquiv_apply, CategoryTheory.Bicategory.conjugateEquiv_symm_comm, Setoid.prodQuotientEquiv_symm_apply, FreeMonoid.ofList_symm, CategoryTheory.Bicategory.mateEquiv_symm_apply, ModuleCat.ExtendRestrictScalarsAdj.homEquiv_symm_apply, CategoryTheory.Limits.Concrete.prodEquiv_symm_apply_snd, optionSubtype_symm_apply_apply_none, tangentBundle_model_space_coe_chartAt_symm, CategoryTheory.PreZeroHypercover.interSnd_h₀, coe_piCongr', CategoryTheory.conjugateEquiv_symm_comp, Affine.Simplex.range_faceOpposite_reindex, Affine.Simplex.altitudeFoot_reindex, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_symm_app_apply, finSumFinEquiv_symm_last, Matrix.cons_mul, MulOpposite.opEquiv_symm_apply, SupHom.symm_dual_id, image_eq_preimage, CategoryTheory.Sheaf.ΓObjEquivHom_naturality_symm, subLeft_symm_apply, MonoidHom.op_symm_apply_apply, RootPairing.Equiv.comp_indexEquiv_symm_apply, ofInjective_symm_apply, subtypeEquivRight_symm_apply_coe, SimpleGraph.map_symm, Function.piCongrLeft'_update, piCongrLeft_sumInl, equivTangentBundleProd_symm_apply_snd, option_symm_apply_none_iff, smul_def, AddMonoidAlgebra.lift_symm_apply, Matrix.one_submatrix_mul, Perm.coe_inv, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy_homEquiv_symm_apply, SimpleGraph.Walk.IsHamiltonian.supportGetEquiv_symm_apply_val, circleEquivGen_symm_apply, Quiver.Hom.opEquiv_symm_apply, Matrix.reindex_updateRow, WeakDual.CharacterSpace.compContinuousMap_apply, CategoryTheory.tensorRightHomEquiv_tensor, prodProdProdComm_symm, AddEquiv.symm_addMonoidHomCongrLeftEquiv, DomAddAct.vadd_aeeqFun_aeeq, CategoryTheory.Limits.FormalCoproduct.homPullbackEquiv_symm_apply_f_coe, CStarMatrix.toCLM_apply, AddEquiv.symm_map_add, CategoryTheory.Limits.WalkingPair.swap_symm_apply_tt, BoundedLatticeHom.symm_dual_id, Finsupp.equivFunOnFinite_symm_coe, AddEquiv.addMonoidHomCongrRightEquiv_symm_apply, finSumFinEquiv_symm_apply_natAdd, LinearEquiv.piCongrLeft'_symm_apply, AddChar.toAddMonoidHomEquiv_symm_apply, CategoryTheory.Functor.corepresentableByUliftFunctorEquiv_symm_apply_homEquiv, CategoryTheory.Functor.RepresentableBy.equivUliftYonedaIso_symm_apply_homEquiv, finsuppUnique_symm_apply_apply, MulEquiv.symm_monoidHomCongrLeftEquiv, CategoryTheory.discreteEquiv_symm_apply_as, Encodable.decode_ofEquiv, Topology.IsGeneratedBy.equiv_symm_comp_continuous_iff, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_app_app, DomAddAct.vadd_Lp_ae_eq, FintypeCat.equivEquivIso_symm_apply_apply, Sym2.coe_lift_symm_apply, AlgebraicGeometry.IsOpenImmersion.opensEquiv_symm_apply, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_two, CategoryTheory.Limits.IsColimit.homEquiv_symm_naturality, CategoryTheory.Equivalence.induced_inverse_map, CategoryTheory.Presieve.FamilyOfElements.singletonEquiv_symm_apply, existsUnique_congr_left, AddChar.toMonoidHomEquiv_symm_one, WithCStarModule.linearEquiv_symm_apply, CategoryTheory.Functor.coconeTypesEquiv_symm_apply_ι, SetTheory.PGame.moveRight_neg, Function.Involutive.symm_eq_self_of_involutive, Matrix.submatrix_updateRow_equiv, induced_symm, val_unitsEquivProdSubtype_symm_apply, Set.image_symm_preimage, TwoSidedIdeal.coe_equivMatrix_symm_apply, Fin.consEquiv_symm_apply, sumCompl_apply_symm_of_neg, subtypeEquivCodomain_symm_apply_eq, derivationToSquareZeroEquivLift_symm_apply_apply_coe, Delone.DeloneSet.mapIsometry_symm_apply_packingRadius, CategoryTheory.Enriched.Functor.natTransEquiv_symm_app_app_apply, CategoryTheory.Functor.curryingFlipEquiv_symm_apply_obj_map, Denumerable.ofEquiv_ofNat, FirstOrder.Language.Term.constantsVarsEquivLeft_symm_apply, CliffordAlgebra.lift_symm_apply, inducedStructure_funMap, Matrix.submatrix_id_mul_left, sigmaAssocProd_symm_apply_snd, Rep.diagonalOneIsoLeftRegular_inv_hom, Perm.one_symm, Set.rangeSplittingImageEquiv_symm_apply_coe, ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty_symm_apply, Nat.pairEquiv_symm_apply, Matrix.compLinearEquiv_symm_apply, Filter.Realizer.ofEquiv_F, CategoryTheory.mateEquiv_symm_apply, QuadraticAlgebra.lift_symm_apply_coe, ofFiberEquiv_apply, AlgEquiv.op_symm_apply_symm_apply, UniversalEnvelopingAlgebra.lift_symm_apply, OnePoint.equivProjectivization_symm_apply_mk, HomotopicalAlgebra.BifibrantObject.HoCat.homEquivRight_symm_apply, Perm.mul_symm, CategoryTheory.Functor.curryingEquiv_symm_apply_map_app, coe_finRotate_symm_of_ne_zero, FormalMultilinearSeries.changeOriginIndexEquiv_symm_apply_snd_fst, CategoryTheory.Presheaf.uliftYonedaAdjunction_homEquiv_app, CategoryTheory.Bicategory.conjugateEquiv_symm_comp, CommBialgCat.isoEquivBialgEquiv_symm_apply, HahnModule.ext_iff, CategoryTheory.Limits.Types.Small.limitCone_pt_ext_iff, RingEquiv.sumArrowEquivProdArrow_symm_apply, CategoryTheory.Limits.Types.Small.limitCone_π_app, CategoryTheory.PreGaloisCategory.fiberBinaryProductEquiv_symm_snd_apply, Finsupp.sumFinsuppEquivProdFinsupp_symm_apply, natSumNatEquivNat_symm_apply, permCongr_apply, Fintype.coe_finsetOrderIsoSet_symm, Matrix.col_eq_transpose, IsometryEquiv.piCongrLeft_symm_apply, CategoryTheory.CoreSmallCategoryOfSet.smallCategoryOfSet_comp, CategoryTheory.ReflQuiv.adj.homEquiv_naturality_left_symm, CategoryTheory.Adjunction.homEquiv_naturality_right_square_iff, Homeomorph.continuous_sumAssoc_symm, CategoryTheory.Functor.flippingEquiv_symm_apply_obj_map, WithCStarModule.norm_single, MultilinearMap.iteratedFDeriv_aux, Matrix.reindexAlgEquiv_symm, HahnModule.of_symm_add, CategoryTheory.Adjunction.compPreadditiveYonedaIso_hom_app_app_apply, sumCongr_symm, DFinsupp.subtypeSupportEqEquiv_symm_apply_coe, MulRingNorm.mulRingNormEquivAbsoluteValue_symm_apply, Unitization.lift_symm_apply, WithCStarModule.equiv_symm_sub, CategoryTheory.Adjunction.homAddEquiv_symm_zero, AddEquiv.addMonoidHomCongrLeftEquiv_symm_apply, Perm.decomposeFin_symm_apply_succ, Perm.sign_symm_trans_trans, AlgebraicGeometry.coprodSpec_coprodMk, CStarMatrix.mul_entry_mul_eq_inner_toCLM, CoxeterMatrix.reindexGroupEquiv_apply_simple, Finset.mem_map_equiv, MulEquiv.symm_map_mul, prodComm_symm, CategoryTheory.Limits.Multifork.IsLimit.sectionsEquiv_symm_apply_val, MulAction.orbitZPowersEquiv_symm_apply, PadicInt.continuousAddCharEquiv_of_norm_mul_symm_apply, MulChar.ofUnitHom_eq, PartENat.withTopEquiv_symm_one, Sym2.sortEquiv_symm_apply, permCongr_symm, CStarMatrix.ofMatrix_symm_apply, CategoryTheory.CommSq.left_adjoint_hasLift_iff, WithBot.ofDual_symm_apply, AddEquiv.symmEquiv_symm_apply_apply, sigmaSigmaSubtype_apply, toLex_symm_eq, permCongrHom_symm, IsFreeGroup.lift_symm_apply, QuaternionAlgebra.re_equivProd_symm_apply, ContinuousLinearEquiv.unitsEquivAut_symm_apply, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_counitIso, AddUnits.addRight_symm, Invertible.mulLeft_symm_apply, Metric.Snowflaking.symm_toSnowflaking, mul_def, ContinuousMultilinearMap.prodEquiv_symm_apply_snd, CategoryTheory.ShiftedHom.opEquiv'_symm_apply, Circle.argEquiv_symm_apply, Valuation.toAddValuation_symm_eq, sigmaAssocProd_symm_apply_fst, LinearEquiv.toEquiv_symm, CategoryTheory.Monad.MonadicityInternal.comparisonLeftAdjointHomEquiv_symm_apply, CategoryTheory.CommSq.LiftStruct.opEquiv_symm_apply, CategoryTheory.Bicategory.mateEquiv_eq_iff, CategoryTheory.Arrow.equivSigma_symm_apply_hom, preimage_symm_preimage, Submodule.isIdempotentElemEquiv_symm_apply_coe, DFinsupp.sigmaCurryLEquiv_symm_apply, piComm_symm, FirstOrder.Language.Formula.realize_equivSentence_symm_con, MonoidHom.toAdditiveRight_symm_apply_apply, symm_trans_swap_trans, AddMonoidHom.toMultiplicativeLeft_symm_apply_apply, Fin.succFunEquiv_symm_apply, WithCStarModule.inner_single_right, FreeAddMonoid.freeAddMonoidCongr_symm_of, CategoryTheory.Limits.Cotrident.IsColimit.homIso_symm_apply, AddEquiv.coe_toEquiv_symm, HahnModule.coeff_smul, FreeMonoid.toList_symm, CategoryTheory.Limits.Concrete.prodEquiv_symm_apply_fst, mulLeft_symm_apply, LatticeHom.symm_dual_comp, Finsupp.equivMapDomain_symm_apply, finSuccEquiv_symm_some, Sym2.coe_lift₂_symm_apply, Fin.equivSubtype_symm_trans_valEmbedding, MulEquiv.submonoidMap_symm_apply, Matrix.kronecker_assoc', CategoryTheory.yonedaGrpObjIsoOfRepresentableBy_inv, Cubic.equiv_symm_apply_b, LinearEquiv.sumPiEquivProdPi_symm_apply, AddEquiv.coe_prodAssoc_symm, WithTop.toDual_symm_apply, SheafOfModules.Presentation.map_relations_I, Multiplicative.toAdd_symm_eq, DomMulAct.comap_mk.symm_nhds, MultilinearMap.domDomCongrLinearEquiv'_apply, finEquivZMultiples_symm_apply, ComplexShape.Embedding.homEquiv_symm_apply, prodSumDistrib_symm_apply_right, trans_swap_trans_symm, HahnModule.coeff_single_zero_smul, Shrink.toEquiv_homeomorph, SSet.stdSimplex.objEquiv_symm_mem_nonDegenerate_iff_mono, Affine.Simplex.altitude_reindex, AddEquiv.toMultiplicativeRight_symm_apply_symm_apply, RingHom.op_symm_apply_apply, finSuccEquiv'_symm_some_below, DualNumber.coe_lift_symm_apply, Finset.sigmaAntidiagonalEquivProd_symm_apply_snd_coe, CategoryTheory.Functor.isIso_lanAdjunction_homEquiv_symm_iff, Matrix.cramer_submatrix_equiv, CategoryTheory.Limits.PullbackCone.IsLimit.equivPullbackObj_symm_apply_fst, Matrix.mulVecLin_submatrix, QuaternionAlgebra.imK_equivProd_symm_apply, piEquivPiSubtypeProd_symm_apply, CategoryTheory.OverPresheafAux.unitAuxAux_inv_app_fst, finRotate_succ_symm_apply, sigmaSubtype_symm_apply_coe_snd, Matrix.comp_symm_diagonal, CategoryTheory.OverPresheafAux.unitAuxAux_inv_app_snd_coe, CategoryTheory.Functor.sectionsEquivHom_naturality_symm, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_two_assoc, equivCongr_symm, CategoryTheory.Equalizer.Presieve.Arrows.compatible_iff_of_small, Bifunctor.mapEquiv_symm_apply, CategoryTheory.Adjunction.homAddEquiv_symm_neg, OrderIso.conj_symm_apply, CoalgEquiv.coe_symm_toEquiv, FirstOrder.Language.LEquiv.onBoundedFormula_symm_apply, Diffeomorph.contMDiff_invFun, CoalgEquiv.coe_toEquiv_symm, MulAction.orbitEquivQuotientStabilizer_symm_apply, CategoryTheory.Functor.RepresentableBy.uniqueUpToIso_hom, toHomeomorphOfContinuousClosed_symm_apply, Partition.partscopyEquiv_symm_apply_coe, Diffeomorph.symm_toEquiv, Perm.decomposeOption_symm_sign, Topology.WithGeneratedByTopology.continuous_from_iff, SetTheory.PGame.Domineering.shiftRight_symm_apply, Ideal.associatesEquivIsPrincipal_symm_apply, CategoryTheory.Limits.Pi.whiskerEquiv_hom, CategoryTheory.unit_mateEquiv_symm, AddEquiv.toMultiplicative_symm_apply_apply, symm_simpleGraph, Matrix.mul_submatrix_one, IsSimpleOrder.equivBool_symm_apply, finsetSubtypeComm_symm_apply, image_eq_preimage_symm, MulEquiv.toAdditive_symm_apply_apply, CategoryTheory.Functor.CorepresentableBy.equivUliftCoyonedaIso_symm_apply_homEquiv, sumProdDistrib_symm_apply_left, Finset.univ_perm_fin_succ, withBotCongr_symm, Topology.WithLawson.to_Lawson_symm_eq, measurable_equivCurry_symm, uniqueSigma_symm_apply, CategoryTheory.WithTerminal.widePullbackShapeEquiv_inverse_obj, CategoryTheory.Sheaf.ΓHomEquiv_naturality_right_symm, CategoryTheory.Sieve.overEquiv_symm_generate, SSet.stdSimplex.obj₀Equiv_symm_mem_face_iff, CategoryTheory.Limits.FormalCoproduct.inclHomEquiv_symm_apply_f, CategoryTheory.Equivalence.induced_unitIso, LinearMap.polyCharpolyAux_map_aeval, sInfHom.dual_symm_apply_toFun, embeddingEquivOfFinite_symm_apply, WithTop.toDual_symm, CStarMatrix.reindexₐ_symm, DomAddAct.isQuotientMap_mk_symm, pprodProd_symm_apply, SSet.stdSimplex.objEquiv_symm_comp, nsmulCoprime_symm_apply, finProdFinEquiv_symm_apply, symmEquiv_symm_apply_apply, AddEquiv.addSubmonoidMap_symm_apply, NumberField.inverse_basisMatrix_mulVec_eq_repr, MonoidAlgebra.liftMagma_symm_apply, IntermediateField.algHomAdjoinIntegralEquiv_symm_apply_gen, AddMonoidAlgebra.liftMagma_symm_apply, Affine.Simplex.faceOppositeCentroid_reindex, permCongr_def, CategoryTheory.Equivalence.induced_inverse_obj, MulEquiv.coe_prodAssoc_symm, removeNone_symm, trans_cancel_left, CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_left_symm, CategoryTheory.Functor.IsRepresentedBy.uliftYonedaIso_hom, SetTheory.PGame.Domineering.shiftUp_symm_apply, CategoryTheory.Bicategory.conjugateIsoEquiv_symm_apply_hom, SimpleGraph.Iso.boxProdSumDistrib_symm_apply, AddAction.toPermHom_apply_symm_apply, CategoryTheory.MonObj.ofRepresentableBy_mul, ofColex_symm_eq, Matrix.compAlgEquiv_symm_apply, CategoryTheory.PreGaloisCategory.fiberPullbackEquiv_symm_fst_apply, CategoryTheory.Limits.limitCompYonedaIsoCocone_inv, CategoryTheory.SimplicialObject.cechNerveEquiv_symm_apply, QuotientAddGroup.equivIocMod_symm_apply, CategoryTheory.Bicategory.iterated_mateEquiv_conjugateEquiv_symm, FirstOrder.Ring.MvPolynomialSupportLEEquiv_symm_apply_coeff, finTwoArrowEquiv'_symm_apply, Hamming.toHamming_symm_eq, withTopCongr_symm, MvPolynomial.renameSymmetricSubalgebra_symm_apply_coe, AddAction.toPerm_symm_apply, Finsupp.lcongr_symm_single, Perm.perm_symm_mapsTo_of_mapsTo, prodCongrLeft_symm, Set.coe_unionEqSigmaOfDisjoint_symm_apply, UniformSpace.Completion.uniformContinuous_completionSeparationQuotientEquiv_symm, Prefunctor.costar_conj_star, Monoid.PushoutI.NormalWord.summand_smul_def', CategoryTheory.forgetAdjToOver.homEquiv_symm, FinEnum.down_equiv_symm, self_comp_symm, Finsupp.equivMapDomain_apply, Finsupp.domLCongr_symm, RingEquiv.piCongrLeft_apply, ChainComplex.fromSingle₀Equiv_symm_apply_f_zero, AddSubsemigroup.equivOp_symm_apply_coe, DirectSum.decomposeLinearEquiv_symm_apply, sumAssoc_symm_apply_inr_inl, Affine.Simplex.touchpoint_reindex, PresheafOfModules.comp_toPresheaf_map_sheafifyHomEquiv'_symm_hom, AddHom.mulOp_symm_apply_apply, CategoryTheory.Discrete.equivalence_counitIso, OrthonormalBasis.equiv_symm, AddAction.orbitEquivQuotientStabilizer_symm_apply, AlgebraicGeometry.Scheme.IsLocallyDirected.glueDataι_naturality, Finset.map_insertNthEquiv_filter_piFinset, LinearEquiv.symmEquiv_symm_apply_apply, IsPGroup.powEquiv_symm_apply, IsometryEquiv.piCongrLeft_apply, ContinuousMultilinearMap.iteratedFDerivComponent_apply, EquivLike.apply_coe_symm_apply, Matrix.updateCol_submatrix_equiv, ulift_symm_down, Topology.WithScott.ofScott_symm_eq, CFilter.ofEquiv_val, Setoid.piQuotientEquiv_symm_apply, RingEquiv.op_symm_apply_symm_apply, Perm.perm_inv_on_of_perm_on_finset, CategoryTheory.ShiftedHom.opEquiv_symm_apply_comp, CategoryTheory.Monad.MonadicityInternal.comparisonAdjunction_counit, BotHom.dual_symm_apply_apply, ArchimedeanOrder.val_symm_eq, Quiver.starEquivCostar_symm_apply, prodPiEquivSumPi_symm_apply, MulEquiv.symmEquiv_symm_apply_apply, CategoryTheory.PreGaloisCategory.fiberPullbackEquiv_symm_snd_apply, Perm.extendDomain_symm, emptySum_symm_apply, AddOpposite.opEquiv_symm_apply, CategoryTheory.isoOpEquiv_symm_apply, subtypePreimage_symm_apply_coe_pos, LieAlgebra.LieEquiv.ofCoboundary_invFun, Matrix.piEquiv_symm_apply, CochainComplex.toSingle₀Equiv_symm_apply_f_succ, toBoolRing_symm_eq, coinduced_symm, Rep.coindResAdjunction_homEquiv_symm_apply, RingEquiv.coe_toEquiv_symm, FirstOrder.Language.LEquiv.onBoundedFormula_symm, LocallyConstant.congrLeft_symm_apply, smulRight_symm_apply, prodCongr_symm, SupBotHom.symm_dual_id, CategoryTheory.Localization.SmallShiftedHom.postcompEquiv_symm_apply, piCongrLeft'_symm, finSumFinEquiv_symm_apply_castSucc, CategoryTheory.Limits.WalkingPair.equivBool_symm_apply_false, ContinuousMap.val_addUnitsLift_symm_apply_apply, MulAction.orbitZPowersEquiv_symm_apply', Affine.Simplex.reindex_points, sigmaEquivProd_sigmaCongrRight, CategoryTheory.typeEquiv_counitIso_hom_app_val_app, Sym.equivCongr_symm_apply, Perm.inv_def, FreeMagma.lift_symm_apply, CategoryTheory.CategoryOfElements.toCostructuredArrow_obj, CategoryTheory.Limits.FormalCoproduct.cofanHomEquiv_symm_apply_f, CategoryTheory.Presheaf.uliftYonedaAdjunction_unit_app_app, Affine.Simplex.range_face_reindex, CategoryTheory.ShiftedHom.opEquiv'_symm_add, AddMonoidHom.fiberEquivKer_symm_apply, CliffordAlgebra.even.lift_symm_apply_bilin, ofRightInverseOfCardLE_symm_apply, CategoryTheory.PreZeroHypercover.inter_def, SSet.stdSimplex.yonedaEquiv_map, Finmap.keysLookupEquiv_symm_apply_keys, sigmaAntidiagonalEquivProd_symm_apply_fst, optionEquivSumPUnit_symm_inl, Pointed.Iso.mk_inv_toFun, CategoryTheory.Enriched.Functor.natTransEquiv_symm_whiskerRight_functorHom_app, CategoryTheory.Localization.SmallShiftedHom.precompEquiv_symm_apply, Bundle.TotalSpace.toProd_symm_apply_snd, CategoryTheory.CommSq.left_adjoint, Set.insert_symm_apply_inl, ChainComplex.toSingle₀Equiv_symm_apply_f_zero, DFinsupp.equivProdDFinsupp_symm_apply, HahnModule.orderTop_vAdd_le_orderTop_smul, CategoryTheory.Limits.limitCompCoyonedaIsoCone_inv, CategoryTheory.ShiftedHom.opEquiv'_zero_add_symm, finCongr_symm_apply, CategoryTheory.ShiftedHom.opEquiv'_symm_op_opShiftFunctorEquivalence_counitIso_inv_app_op_shift, Cubic.equiv_symm_apply_c, MonoidAlgebra.uniqueRingEquiv_symm_apply, CategoryTheory.MonoidalClosed.homEquiv_symm_apply_eq, QuaternionAlgebra.coe_symm_addEquivProd, PiTensorProduct.map_comp_reindex_symm, Fin.cycleRange_symm_succ, image_symm_image, piCongrLeft_apply_eq_cast, Prefunctor.symmetrifyStar, Set.rangeInr_symm_apply_coe, WType.equivSigma_symm_apply, mkFactorOrderIsoOfFactorDvdEquiv_symm_apply_coe, LightCondensed.ihomPoints_symm_apply, Complex.lift_symm_apply_coe, FreeAlgebra.lift_symm_apply, CategoryTheory.Adjunction.leftAdjointOfEquiv_map, Perm.decomposeFin_symm_of_one, DomMulAct.smul_apply, Matrix.reindex_symm, ContinuousMap.equivFnOfDiscrete_symm_apply_apply, Primrec.of_equiv_symm, MeasurableEquiv.coe_sumPiEquivProdPi_symm, ZSpan.quotientEquiv.symm_apply, emultiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_emultiplicity, arrowProdEquivProdArrow_symm_apply, CategoryTheory.Limits.IsLimit.homEquiv_symm_π_app_assoc, WittVector.liftEquiv_symm_apply_coe, CategoryTheory.Limits.WalkingPair.swap_symm_apply_ff, Quaternion.imJ_equivProd_symm_apply, Matrix.reindexLinearEquiv_symm, Finsupp.coe_equivFunOnFinite_symm, TopHom.symm_dual_comp, InverseSystem.piLTLim_symm_apply, prodPUnit_symm_apply, FreeAddGroup.lift_symm_apply, LocallyConstant.congrRightRingEquiv_symm_apply_apply, ContinuousAffineEquiv.coe_symm_toEquiv, piCongrFiberwise_symm_apply, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_symm_naturality_left, AddMonoidAlgebra.uniqueRingEquiv_symm_apply, CategoryTheory.nerve.σ_zero_nerveEquiv_symm, CategoryTheory.Functor.homEquivOfIsRightKanExtension_symm_apply, OnePoint.smul_infty_def, Set.piCongrLeft_comp_restrict, Finsupp.mapDomain_equiv_apply, Function.Embedding.optionEmbeddingEquiv_symm_apply, Affine.Simplex.eulerPoint_reindex, CommAlgCat.isoEquivAlgEquiv_symm_apply, BoundedOrderHom.dual_symm_apply_toOrderHom, Perm.sigmaCongrRight_symm, ContinuousLinearEquiv.arrowCongrEquiv_symm_apply, subtypeQuotientEquivQuotientSubtype_symm_mk, DomMulAct.stabilizerMulEquiv_apply, prodEquivPiFinTwo_symm_apply, CategoryTheory.Limits.SingleObj.colimitTypeRelEquivOrbitRelQuotient_symm_apply, AlternatingMap.ofSubsingleton_symm_apply_apply, FinEnum.up_equiv_symm, mulLeft_symm, coe_embeddingFinSucc_symm, Matrix.tail_transpose, CategoryTheory.equivYoneda_inv_app, CategoryTheory.uliftCoyonedaEquiv_symm_map_assoc, NumberField.mixedEmbedding.fundamentalCone.idealSetEquiv_symm_apply, PartENat.withTopEquiv_symm_zero, toHomeomorph_symm, AddEquiv.toMultiplicative_symm_apply_symm_apply, BialgEquiv.coe_toEquiv_symm, SetTheory.PGame.toLeftMovesNim_symm_lt, Algebra.Generators.Hom.equivAlgHom_symm_apply_val, MultilinearMap.ofSubsingleton_symm_apply_apply, symm_bijective, Matrix.row_eq_self, CategoryTheory.Adjunction.homAddEquiv_symm_apply, CategoryTheory.Comonad.ComonadicityInternal.comparisonRightAdjointHomEquiv_symm_apply_f, CompactlyGenerated.isoEquivHomeo_symm_apply, tangentBundleModelSpaceHomeomorph_coe_symm, Finset.sum_map_equiv, CategoryTheory.FunctorToTypes.naturality_symm, sumCompl_apply_symm_of_pos, CategoryTheory.Limits.Trident.IsLimit.homIso_symm_apply, LieAlgebra.of_symm_nsmul, CategoryTheory.codiscreteEquiv_symm_apply_as, IsometryEquiv.piCongrLeft'_symm_apply, finSuccEquiv'_symm_none, Set.sumDiffSubset_symm_apply_of_mem, ContinuousMap.equivFnOfDiscrete_symm_apply, CategoryTheory.Groupoid.invEquiv_symm_apply, trans_eq_refl_iff_symm_eq, DirectSum.decompose_symm_mul, FreeGroupBasis.lift_symm_apply, sumSumSumComm_apply, FreeMonoid.lift_symm_apply, CategoryTheory.LiftRightAdjoint.constructRightAdjointEquiv_apply, CategoryTheory.Adjunction.homAddEquiv_symm_add, Functor.mapEquiv_symm_apply, MulEquiv.symmEquiv_symm_apply_symm_apply, finSuccAboveEquiv_symm_apply_last, CategoryTheory.conjugateEquiv_symm_comm, OrderIso.equivClosureOperator_symm_apply, MulEquiv.toEquiv_symm, CategoryTheory.Functor.CorepresentableBy.uniqueUpToIso_hom, CategoryTheory.ParametrizedAdjunction.inl_arrowHomEquiv_symm_apply_left_assoc, CoxeterMatrix.reindex_apply, Quiver.SingleObj.toHom_symm_apply, piCongrLeft_sumInr, WithBot.ofDual_symm, Affine.Simplex.signedInfDist_reindex, piCongr'_apply, Set.image_symm_apply, Filter.prod_assoc_symm, Matrix.diagonalInvertibleEquivInvertible_symm_apply, CategoryTheory.CountableCategory.instCountableHomObjAsType, NumberField.Ideal.primesOverSpanEquivMonicFactorsMod_symm_apply_eq_span, boolProdEquivSum_symm_apply, DomMulAct.smul_Lp_ae_eq, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_app_app, DirectSum.equivCongrLeft_apply, AddCircle.continuous_equivIco_symm, Fin.insertNth_comp_cycleRange_symm, Topology.WithGeneratedByTopology.isOpen_iff, prodAssoc_symm_apply, DomAddAct.coe_mkHomeomorph_symm, subtypeUnivEquiv_symm_apply, CategoryTheory.Functor.RepresentableBy.uniqueUpToIso_inv, Perm.prod_comp', equivShrink_symm_add, CompactlySupportedContinuousMap.continuousMapEquiv_symm_apply, setOf_apply_symm_eq_image_setOf, AddEquiv.finsuppUnique_symm_apply_support_val, LightCondensed.ihom_map_val_app, Perm.IsCycle.zpowersEquivSupport_symm_apply, CategoryTheory.Iso.toEquiv_symm_fun, sigmaProdDistrib_symm_apply, Matrix.of_symm_single, CategoryTheory.Adjunction.homEquiv_symm_id, Perm.VectorsProdEqOne.vectorEquiv_symm_apply, FreeAddMonoid.map_symm_apply_map_eq, piCongr_symm_apply, symmEquiv_apply_symm_apply, CategoryTheory.ShiftedHom.opEquiv'_symm_comp, inducedStructure_RelMap, Function.Embedding.toEquivRange_symm_apply_self, CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.comp_homEquiv_symm, addRight_symm, RingEquiv.piCongrLeft'_symm_apply, CategoryTheory.Adjunction.homEquiv_naturality_right_symm, eq_image_iff_symm_image_eq, FirstOrder.Language.LEquiv.onFormula_symm, InfTopHom.dual_symm_apply_toFun, ofPreimageEquiv_apply, Matroid.mapEquiv_indep_iff, OrderMonoidIso.coe_toEquiv_symm, CategoryTheory.FinCategory.objAsTypeToAsType_obj, CategoryTheory.Bicategory.conjugateEquiv_symm_iso, finSuccEquivLast_symm_none, OrderHom.symm_dual_comp, OnePoint.not_continuous_cofiniteTopology_of_symm, Units.mulRight_symm, CategoryTheory.MonoidalClosed.FunctorCategory.homEquiv_naturality_two_symm, CategoryTheory.FreeGroupoid.functorEquiv_symm_apply, CategoryTheory.Functor.IsRepresentedBy.iff_isIso_uliftYonedaEquiv, Finsupp.equivCongrLeft_symm, eq_conj, sumCompl_symm_apply_neg, Matrix.toLin'_submatrix, MulEquiv.toAdditive_symm_apply_symm_apply, Multiset.cast_symm_apply_snd, finSumNatEquiv_symm_apply_add_right, BotHom.symm_dual_id, HomologicalComplex.evalCompCoyonedaCorepresentableByDoubleId_homEquiv_symm_apply, continuous_symm_iff, ContinuousMap.equivBoundedOfCompact_symm_apply, RingEquiv.op_symm_apply_apply, sInfHom.symm_dual_comp, CochainComplex.HomComplex.CohomologyClass.equivOfIsKProjective_symm_apply, CategoryTheory.Limits.IsColimit.equivIsoColimit_symm_apply, FirstOrder.Language.Term.relabelEquiv_symm_apply, DirectSum.liftRingHom_symm_apply_coe, Set.preimage_equiv_eq_image_symm, SSet.OneTruncation₂.ofNerve₂.natIso_inv_app_map, Module.Basis.toMatrix_reindex, LieAlgebra.of_symm_zero, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_symm_apply, OrderDual.ofDual_symm_eq, finSuccEquiv'_symm_coe_below, AddChar.toMonoidHomEquiv_symm_mul, MeasureTheory.tendsto_diracProbaEquivSymm_iff_tendsto, finCongr_symm, Prefunctor.symmetrifyCostar, sigmaSigmaSubtypeEq_symm_apply, Finsupp.restrictSupportEquiv_symm_single, bijOn_symm, CategoryTheory.Adjunction.equivHomsetLeftOfNatIso_symm_apply, sSupHom.symm_dual_comp, boolProdNatEquivNat_symm_apply, SimpleGraph.Iso.mapNeighborSet_symm_apply_coe, CategoryTheory.Iso.homCongr_symm_apply, Hamming.ofHamming_symm_eq, finsuppEquivDFinsupp_symm_apply, IsometryEquiv.coe_symm_toEquiv, coe_piCongr_symm, Perm.equivUnitsEnd_symm_apply_symm_apply, CategoryTheory.Functor.partialRightAdjointHomEquiv_symm_comp_assoc, polarCoord_apply, prod_assoc_symm_image, TrivSqZeroExt.liftEquivOfComm_symm_apply_coe, WithCStarModule.inner_single_left, MeasurableEquiv.coe_toEquiv_symm, PartENat.withTopEquiv_symm_le, HNNExtension.NormalWord.unitsSMulEquiv_symm_apply, Polynomial.toMatrix_sylvesterMap, CategoryTheory.MonoidalClosed.curryHomEquiv'_symm_apply, Shrink.continuousLinearEquiv_apply, CategoryTheory.rightAdjointOfCostructuredArrowTerminalsAux_symm_apply, Matrix.of_symm_apply, CategoryTheory.tensorLeftHomEquiv_symm_coevaluation_comp_whiskerRight, AlgEquiv.opComm_symm_apply_apply, CategoryTheory.PreGaloisCategory.evaluationEquivOfIsGalois_symm_fiber, CategoryTheory.Functor.partialLeftAdjointHomEquiv_symm_comp_assoc, SheafOfModules.pullbackPushforwardAdjunction_homEquiv_symm_unitToPushforwardObjUnit, orderIsoShrink_symm_apply, BialgEquiv.coe_symm_toEquiv, CategoryTheory.Functor.FullyFaithful.isoEquiv_symm_apply, mulRight₀_symm_apply, ContinuousMultilinearMap.piEquiv_symm_apply, StarAlgHom.prodEquiv_symm_apply, FreeAddMagma.lift_symm_apply, CategoryTheory.Localization.structuredArrowEquiv_symm_apply, Nat.factorizationEquiv_inv_apply, FreeLieAlgebra.lift_symm_apply, AlgEquiv.opComm_symm_apply_symm_apply, CategoryTheory.Limits.Sigma.whiskerEquiv_inv, CategoryTheory.CommSq.instHasLift_1, DirectSum.decomposeAddEquiv_symm_apply, DomAddAct.symm_mk_nsmul, Monoid.CoprodI.lift_symm_apply, sumAssoc_symm_apply_inr_inr, Topology.WithUpper.toUpper_symm, neg_symm, self_trans_symm, CategoryTheory.Sieve.overEquiv_symm_top, Rep.homEquiv_symm_apply_hom, CoxeterSystem.reindex_simple, Quiver.SingleObj.pathEquivList_symm_nil, DFA.reindex_apply_step, CategoryTheory.Iso.homToEquiv_symm_apply, CategoryTheory.unitCompPartialBijective_symm_natural, Finsupp.sumFinsuppEquivProdFinsupp_symm_inl, DirectSum.decompose_symm_zero, SSet.OneTruncation₂.nerveEquiv_symm_apply_obj, ArchimedeanOrder.of_symm_eq, AddEquiv.symm_addMonoidHomCongrRightEquiv, measurable_piEquivPiSubtypeProd_symm, removeNone_aux_inv, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_inv_app_app, AffineBasis.reindex_apply, Multiset.coeEquiv_symm_apply_fst, EquivLike.self_comp_coe_symm, subtypeOrEquiv_symm_inr, algebraMap_def, PresentedGroup.equivPresentedGroup_symm_apply_of, CochainComplex.fromSingle₀Equiv_symm_apply_f_zero, HomologicalComplex.evalCompCoyonedaCorepresentableBySingle_homEquiv_symm_apply, CategoryTheory.Iso.homCongr_symm, AlgEquiv.piCongrLeft_symm_apply, NumberField.Ideal.ramificationIdx_primesOverSpanEquivMonicFactorsMod_symm_apply, measurable_piCurry_symm, WithCStarModule.equiv_symm_add, AlgebraicGeometry.AffineSpace.homOfVector_toSpecMvPoly, Finsupp.curryEquiv_symm_apply, CategoryTheory.conjugateIsoEquiv_symm_apply_inv, Matrix.comp_symm_transpose, SSet.stdSimplex.spineId_vertex, Specialization.toEquiv_symm, divLeft₀_symm_apply, Fin.equivSubtype_symm_apply, HahnModule.support_smul_subset_vadd_support, Finset.subset_map_symm, DFA.evalFrom_reindex, CategoryTheory.Adjunction.homEquiv_naturality_left_symm, comp_symm_eq, InfHom.symm_dual_id, CategoryTheory.Groupoid.isoEquivHom_symm_apply_inv, piCongrLeft'_symm_apply, AlgebraicGeometry.SpecToEquivOfLocalRing_symm_apply, CategoryTheory.ShiftedHom.opEquiv_symm_apply, FirstOrder.Language.Theory.CompleteType.mem_typeOf, Subgroup.IsComplement'.QuotientMulEquiv_symm_apply, CategoryTheory.Presheaf.restrictedULiftYonedaHomEquiv'_symm_app_naturality_left, AlgebraicGeometry.affineOpensRestrict_symm_apply_coe, CategoryTheory.Adjunction.homEquiv_counit, Affine.Simplex.mongePlane_reindex, addLeft_symm_apply, SSet.S.equivElements_symm_apply_simplex, TopologicalSpace.IrreducibleCloseds.equivSubtype_symm_apply, KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map, PiTensorProduct.map_reindex, CategoryTheory.Adjunction.leftAdjointCompNatTrans₀₂₃_eq_conjugateEquiv_symm, MvPolynomial.finSuccEquiv_rename_finSuccEquiv, symmEquiv_symm_apply_symm_apply, AddSubgroup.quotientEquivProdOfLE'_symm_apply, RootPairing.Hom.coroot_coweightMap_apply, PartENat.withTopEquiv_symm_coe, simpleGraph_apply, WithTop.ofDual_symm_apply, CategoryTheory.Presheaf.coconeOfRepresentable_ι_app, Topology.WithUpperSet.ofUpperSet_symm, LinearAlgebra.FreeProduct.lift_symm_apply, DomMulAct.isOpenEmbedding_mk_symm, CategoryTheory.Comonad.ComonadicityInternal.comparisonAdjunction_counit, Action.diagonalSuccIsoTensorDiagonal_hom_hom, CategoryTheory.Limits.IsColimit.ι_app_homEquiv_symm_assoc, HahnModule.of_symm_zero, QuaternionAlgebra.imJ_equivProd_symm_apply, CategoryTheory.Functor.curryingFlipEquiv_symm_apply_obj_obj, CategoryTheory.conjugateEquiv_adjunction_id_symm, CategoryTheory.LiftRightAdjoint.constructRightAdjointEquiv_symm_apply, finRotate_symm_lt_iff_ne_zero, MulEquiv.withZero_symm_apply_apply, OrderHom.dual_symm_apply_coe, CategoryTheory.ProjectiveResolution.extEquivCohomologyClass_symm_add, MulEquiv.op_symm_apply_symm_apply, AlgEquiv.arrowCongr_symm, AddEquiv.toMultiplicativeLeft_symm_apply_apply, CategoryTheory.ShrinkHoms.comp_def, finSumNatEquiv_symm_apply_of_ge, WithBot.toDual_symm_apply, Matrix.mulVecLin_reindex, AlgEquiv.op_symm_apply_apply, SSet.op_map, DomMulAct.symm_mk_inv, FirstOrder.Language.BoundedFormula.mapTermRelEquiv_symm_apply, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_symm_app_apply, Perm.subtypeEquivSubtypePerm_symm_apply, MeasureTheory.continuous_diracProbaEquivSymm, piCurry_symm_apply, AlternatingMap.domDomCongrₗ_symm_apply, FreeAddMonoid.map_apply_map_symm_eq, PadicInt.continuousAddCharEquiv_symm_apply, optionSubtype_symm_apply_apply_some, ContinuousMultilinearMap.domDomCongrEquiv_symm_apply, FintypeCat.uSwitchEquiv_symm_naturality, Quaternion.imI_equivProd_symm_apply, CategoryTheory.Discrete.equivOfEquivalence_symm_apply, CategoryTheory.ShrinkHoms.inverse_map, Algebra.TensorProduct.liftEquiv_symm_apply_coe, funUnique_symm_apply, Ordinal.to_leftMoves_one_toPGame_symm, CategoryTheory.Meq.equiv_symm_eq_apply, BotHom.symm_dual_comp, zmultiplesHom_symm_apply, Rack.act'_symm_apply, PEquiv.transpose_toMatrix_toPEquiv_apply, finTwoArrowEquiv'_sum_eq, Cubic.equiv_symm_apply_d, DomMulAct.symm_mk_mul, one_def, HVertexOperator.coeff_apply_apply, Perm.decomposeOption_symm_apply, prodCongrRight_symm, WithCStarModule.equiv_symm_fst, QuaternionAlgebra.imI_equivProd_symm_apply, Homeomorph.homeomorph_mk_coe_symm, vadd_def, AddMonoidHom.mulOp_symm_apply_apply, Fin.appendEquiv_symm_apply, eq_symm_iff_trans_eq_refl, Finset.symInsertEquiv_symm_apply_coe, CategoryTheory.coyonedaEquiv_symm_map, CategoryTheory.PreZeroHypercover.interFst_h₀, CategoryTheory.Adjunction.homEquiv_naturality_right_square, PowerBasis.liftEquiv'_symm_apply_apply, PiTensorProduct.lift_comp_reindex_symm, CategoryTheory.PreOneHypercover.inter_p₁, LinearEquiv.coe_toEquiv_symm, Finsupp.optionEquiv_symm_apply, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_symm_map, Quiver.starEquivCostar_symm_apply_snd, Finsupp.equivFunOnFinite_symm_single, finSuccEquiv'_symm_some, subset_symm_image, SymAlg.unsym_symm, div_def, PrimeSpectrum.primeSpectrumProd_symm_inl_asIdeal, FreeAddMonoid.toList_symm, preimage_piEquivPiSubtypeProd_symm_pi, DomAddAct.vadd_apply, HahnModule.coeff_smul_left, psigmaCongrRight_symm, CategoryTheory.Presheaf.restrictedULiftYonedaHomEquiv'_symm_naturality_right_assoc, cast_symm, Finsupp.equivFunOnFinite_symm_sum, SSet.Truncated.HomotopyCategory.homToNerveMk_app_zero, sigmaPreimageEquiv_symm_apply_fst, IsArtinianRing.primeSpectrumEquivMaximalSpectrum_symm_comp_asIdeal, Matroid.mapEquiv_isBasis_iff, MulEquiv.coe_toEquiv_symm, AddValuation.toValuation_symm_eq, ofLeftInverseOfCardLE_symm_apply, piCongrLeft'_symm_apply_apply, CategoryTheory.Pretriangulated.preadditiveYoneda_shiftMap_apply, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_symm_naturality_right, biInf_comp, stdSimplexEquivIcc_symm_apply_coe, pprodEquivProdPLift_symm_apply, LinearEquiv.symmEquiv_symm_apply_symm_apply, CategoryTheory.evalEquiv_symm_apply, CategoryTheory.Adjunction.representableBy_homEquiv, WithCStarModule.equivL_symm_apply, sigmaFiberEquiv_symm_apply_fst, CategoryTheory.Functor.CorepresentableBy.homEquiv_symm_comp, ContinuousMap.unitsLift_symm_apply_apply_inv', MulEquiv.op_symm_apply_apply, DomMulAct.isEmbedding_mk_symm, coe_notMemRangeEquiv_symm, YoungDiagram.equivListRowLens_symm_apply, CategoryTheory.GrothendieckTopology.yonedaEquiv_symm_naturality_left, CategoryTheory.HomOrthogonal.matrixDecomposition_symm_apply, Abelianization.lift_symm_apply, Setoid.quotientKerEquivOfRightInverse_symm_apply, PiTensorProduct.map_comp_reindex_eq, WeakDual.CharacterSpace.equivAlgHom_symm_coe, CategoryTheory.Localization.Construction.objEquiv_symm_apply, Multiset.consEquiv_symm_none, CategoryTheory.Endofunctor.Adjunction.Coalgebra.toAlgebraOf_obj_str, CategoryTheory.Limits.IsLimit.homEquiv_symm_naturality, self_comp_ofInjective_symm, CategoryTheory.conjugateIsoEquiv_symm_apply_hom, Subgroup.quotientEquivSigmaZMod_symm_apply, MonoidHom.toAdditiveLeft_symm_apply_apply, addEquiv_symm_apply, Finmap.keysLookupEquiv_symm_apply_lookup, Finsupp.mapDomain.linearEquiv_symm, Cubic.equiv_symm_apply_a, MvPolynomial.degreeOf_eq_natDegree, prodPProd_symm_apply, UniformEquiv.uniformEquiv_mk_coe_symm, subtypeEquivCodomain_symm_apply_ne, functionSwap_symm_apply, AddValuation.ofValuation_symm_eq, CategoryTheory.Adjunction.homEquiv_naturality_right_square_assoc, CategoryTheory.Under.postAdjunctionRight_counit_app_right, Specialization.ofEquiv_symm, CochainComplex.HomComplex.Cochain.equivHomotopy_symm_apply_hom, Matrix.coe_ofLinearEquiv_symm, CategoryTheory.unitCompPartialBijectiveAux_symm_apply, WithConv.symm_equiv_apply, MonoidAlgebra.mapDomainAddEquiv_apply, CategoryTheory.Presheaf.restrictedULiftYonedaHomEquiv'_symm_app_naturality_left_assoc, CategoryTheory.PreGaloisCategory.fiberEqualizerEquiv_symm_ι_apply, subtypeSubtypeEquivSubtypeInter_symm_apply_coe_coe, WithBot.toDual_symm, Multiset.coeEquiv_symm_apply_snd_val, Perfection.lift_symm_apply, CategoryTheory.Bicategory.Adjunction.homEquiv₁_symm_apply, CategoryTheory.Discrete.equivalence_inverse, CategoryTheory.Limits.IsColimit.ι_app_homEquiv_symm, extend_apply, isOpenMap_symm_iff, NumberField.canonicalEmbedding_eq_basisMatrix_mulVec, sigmaFiberEquiv_symm_apply_snd_coe, InverseSystem.invLimEquiv_symm_apply_coe, CategoryTheory.Limits.Cofork.IsColimit.homIso_symm_apply, CategoryTheory.Limits.SingleObj.Types.sections.equivFixedPoints_symm_apply_coe, OrderHom.equivFunctor_symm_apply, Function.OfArity.curry_two_eq_curry, symm_eq_iff_trans_eq_refl, sumArrowEquivProdArrow_symm_apply_inl, LieAlgebra.of_symm_add, Combinatorics.Subspace.reindex_isMono, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_right_symm_assoc, CategoryTheory.opEquiv_symm_apply, Fin.revPerm_symm_apply, Quaternion.equivTuple_symm_apply, Unitization.starLift_symm_apply, equivTangentBundleProd_symm_apply_proj, CategoryTheory.Adjunction.homEquiv_symm_rightAdjointUniq_hom_app, CategoryTheory.Abelian.Ext.homEquiv₀_symm_apply, CategoryTheory.Bicategory.conjugateEquiv_symm_apply', subRight_symm_apply, Matrix.comp_symm_apply, RootPairing.Equiv.id_indexEquiv_symm_apply, SupHom.dual_symm_apply_toFun, CategoryTheory.nerve.functorOfNerveMap_obj, CategoryTheory.Adjunction.homAddEquiv_symm_sub, CategoryTheory.Endofunctor.Adjunction.Coalgebra.homEquiv_naturality_str_symm, MulEquiv.monoidHomCongrRightEquiv_symm_apply, MulEquiv.subgroupMap_symm_apply, Quiver.SingleObj.toPrefunctor_symm_comp, mulRight_symm_apply, subtypeSubtypeEquivSubtype_symm_apply_coe_coe, sumIsRight_symm_apply_coe, FirstOrder.Language.LEquiv.onSentence_symm_apply, image_symm_apply_coe, FreeMonoid.freeMonoidCongr_symm_of, Matrix.invertibleEquivDetInvertible_symm_apply, CategoryTheory.sheafHomSectionsEquiv_symm_apply_coe_apply, Perm.SameCycle.symm_apply_left, TrivSqZeroExt.liftEquiv_symm_apply_coe, boolNot_symm_apply, Trunc.finChoiceEquiv_symm_apply, Matrix.submatrixEquivInvertibleEquivInvertible_symm_apply, MvPolynomial.renameEquiv_symm, finSumNatEquiv_symm_apply_of_lt, mulEquiv_symm_apply, Set.powersetCard.mem_range_ofFinEmbEquiv_symm_iff_mem, CategoryTheory.map_shrinkYonedaEquiv, PEquiv.mul_toMatrix_toPEquiv, EquivFunctor.mapEquiv_symm_apply, Quiver.SingleObj.toPrefunctor_symm_apply, Perm.perm_inv_mapsTo_iff_mapsTo, Perm.perm_symm_on_of_perm_on_finite, Perm.sameCycle_symm_apply_left, Finsupp.equivFunOnFinite_symm_eq_sum, AlgEquiv.funUnique_symm_apply, conj_apply, CategoryTheory.Adjunction.homEquiv_apply_eq, CategoryTheory.yonedaEquiv_symm_map, FreeMonoid.map_symm_apply_map_eq, psigmaEquivSigmaPLift_symm_apply, Magma.AssocQuotient.lift_symm_apply, rightInverse_symm, LinearEquiv.piCongrLeft'_apply, AddEquiv.symm_mk, ofUnique_symm_apply, WithCStarModule.equiv_symm_pi_apply, prodSumDistrib_symm_apply_left, optionSubtype_symm_apply_symm_apply, finSuccAboveOrderIso_symm_apply_ne_last, NumberField.Units.regulator_eq_det', CategoryTheory.uliftCoyonedaEquiv_symm_map, CategoryTheory.TwoSquare.lanBaseChange_app, CategoryTheory.yonedaEquiv_symm_naturality_left, Invertible.mulRight_symm_apply, Affine.Simplex.height_reindex, CategoryTheory.Pseudofunctor.presheafHomObjHomEquiv_symm_apply, Metric.PiNatEmbed.toPiNatEquiv_symm_apply, CategoryTheory.PreZeroHypercover.isoMk_inv_h₀, DirectSum.decompose_symm_of, Finset.orderIsoColex_symm_apply, Preorder.piCongrLeft_comp_restrictLe, SetTheory.PGame.Relabelling.mk'_leftMovesEquiv, sigmaPUnit_symm_apply_fst, ExteriorAlgebra.invertibleAlgebraMapEquiv_symm_apply_invOf_toQuot, GrpCat.SurjectiveOfEpiAuxs.τ_symm_apply_fromCoset, CategoryTheory.Pseudofunctor.isPrestackFor_iff_isSheafFor', Diffeomorph.toEquiv_coe_symm, CategoryTheory.yonedaEquiv_symm_naturality_right, RelIso.preimage_symm_apply, Matroid.mapSetEquiv_indep_iff, FreeAbelianGroup.liftMonoid_symm_coe, CoalgEquiv.toEquiv_symm, Ctop.ofEquiv_val, subtypeEquiv_symm, subtypePreimage_symm_apply_coe_neg, DomMulAct.symm_mk_zpow, LinearIsometryEquiv.piLpCongrLeft_symm, CategoryTheory.PreZeroHypercover.isoMk_inv_s₀, Matrix.comp_symm_single, Matrix.head_transpose, Ctop.Realizer.ofEquiv_F, CategoryTheory.Functor.partialLeftAdjointHomEquiv_comp_symm, Set.image_equiv_eq_preimage_symm, prodShear_symm_apply, AddEquiv.op_symm_apply_apply, apply_ofInjective_symm, SSet.stdSimplex.objEquiv_symm_apply, CompleteLatticeHom.symm_dual_comp, AlgebraicGeometry.pointEquivClosedPoint_symm_apply_coe, FormalMultilinearSeries.changeOriginSeriesTerm_changeOriginIndexEquiv_symm, Ordinal.toLeftMovesToPGame_symm_lt, symm_image_image, FreeGroup.freeGroupCongr_symm, Topology.WithLower.toLower_symm, CategoryTheory.Limits.biproduct.whiskerEquiv_hom_eq_lift, CategoryTheory.Functor.CoconeTypes.IsColimit.equiv_symm_ι_apply, CategoryTheory.Limits.PullbackCone.IsLimit.equivPullbackObj_symm_apply_snd, ringEquiv_symm_apply, equivShrink_symm_mul, subtypeSubtypeEquivSubtypeExists_symm_apply_coe_coe, CategoryTheory.uliftYonedaEquiv_symm_map_assoc, Finset.prod_map_equiv, SetTheory.PGame.moveLeft_nim, CategoryTheory.Functor.curryingEquiv_symm_apply_obj_map, CategoryTheory.Comonad.ComonadicityInternal.comparisonAdjunction_counit_f_aux, Set.Finite.subtypeEquivToFinset_symm_apply_coe, AlgEquiv.prodCongr_symm_apply, Algebra.GrothendieckAddGroup.lift_symm_apply, DomAddAct.isEmbedding_mk_symm, equivEquivIso_inv, Set.union_symm_apply_right, Set.matrix_eq_pi, rootsOfUnityEquivNthRoots_symm_apply, AlgHom.op_symm_apply_apply, symm_toPartialEquiv, Monoid.PushoutI.NormalWord.summand_smul_def, AddMonoidAlgebra.symm_mapDomainAddEquiv, SemidirectProduct.equivProd_symm_apply_left, Perm.val_inv_equivUnitsEnd_apply, Function.update_apply_equiv_apply, SSet.stdSimplex.nonDegenerateEquiv_symm_apply_coe, CategoryTheory.Adjunction.rightAdjointOfEquiv_map, Opposite.equivToOpposite_symm_coe, LieAlgebra.of_symm_smul, SymAlg.sym_symm, MvPolynomial.mapEquivMonic_symm_map_algebraMap, Matrix.reindex_updateCol, symm_divLeft, CategoryTheory.Bicategory.conjugateEquiv_symm_apply, DirectSum.decompose_symm_sum, List.Nodup.getEquiv_symm_apply_val, sumCompl_symm_apply_of_neg, NumberField.Units.regOfFamily_eq_det', Affine.Simplex.excenter_reindex, CochainComplex.toSingle₀Equiv_symm_apply_f_zero, ContinuousMultilinearMap.curryFinFinset_apply, DomAddAct.isClosedEmbedding_mk_symm, Filter.map_equiv_symm, Set.sumCompl_symm_apply_compl, subtypePreimage_symm_apply_coe, image_symm_eq_preimage, UniformEquiv.piCongrLeft_apply, ModuleCat.freeHomEquiv_symm_apply, finsetCongr_symm, equivShrink_symm_inv, SeparationQuotient.liftNormedAddGroupHomEquiv_symm_apply_coe, Orientation.reindex_symm, SimpleGraph.Iso.connectedComponentEquiv_symm, AlgEquiv.piCongrLeft'_symm_apply, ChainComplex.fromSingle₀Equiv_symm_apply_f_succ, PartENat.withTopEquiv_symm_lt, WithLp.congr_symm, ContinuousMap.addUnitsLift_symm_apply_apply_neg', Affine.Simplex.reindex_symm_reindex, curry_symm_apply, CategoryTheory.Presieve.compatibleEquivGenerateSieveCompatible_symm_apply_coe, piFinTwoEquiv_symm_apply, sigmaSubtype_symm_apply_coe_fst, PartENat.withTopEquiv_symm_top, symm_image_subset, optionSubtypeNe_symm_self, Quotient.finChoiceEquiv_symm_apply, CategoryTheory.InjectiveResolution.extEquivCohomologyClass_symm_mk_hom, CategoryTheory.Bicategory.conjugateIsoEquiv_symm_apply_inv, CategoryTheory.Subfunctor.Subpresheaf.ofSection_eq_range, Combinatorics.Subspace.reindex_apply, Finsupp.restrictSupportEquiv_symm_apply_coe, equivShrink_symm_one, CategoryTheory.Pi.equivalenceOfEquiv_functor, MulEquiv.monoidHomCongrLeftEquiv_symm_apply, Shrink.linearEquiv_apply, piUnique_symm_apply, CategoryTheory.Limits.opHomCompWhiskeringLimYonedaIsoCocones_inv_app_app, CategoryTheory.Presheaf.freeYonedaHomEquiv_symm_comp_assoc, finEquivPowers_symm_apply, CategoryTheory.Limits.Types.colimitEquivColimitType_symm_apply, Valuation.ofAddValuation_symm_eq, List.Nodup.getEquivOfForallMemList_symm_apply_val, CategoryTheory.GradedObject.comapEquiv_functor, algEquiv_symm_apply, OrthonormalBasis.repr_reindex, NormedAddGroupHom.Equalizer.liftEquiv_symm_apply_coe, TopologicalSpace.IrreducibleCloseds.equivSubtype'_symm_apply, CategoryTheory.Limits.IsTerminal.from_eq_liftConeMorphism, EquivLike.coe_symm_comp_self, DFinsupp.equivFunOnFintype_symm_single, prod_assoc_symm_preimage, Topology.WithLawson.of_Lawson_symm_eq, Module.Basis.reindex_apply, WType.NatαEquivPUnitSumPUnit_symm_apply, NumberField.Ideal.primesOverSpanEquivMonicFactorsMod_symm_apply, Module.Basis.repr_reindex_apply, OrderMonoidIso.toEquiv_symm, setCongr_symm_apply, HahnSeries.iterateEquiv_symm_apply, Matroid.mapEquiv_isBase_iff, CategoryTheory.Presheaf.restrictedULiftYonedaHomEquiv'_symm_naturality_right, NumberField.house.basis_repr_norm_le_const_mul_house, comp_equiv_symm_dotProduct, CategoryTheory.Limits.Multifork.toSections_fac, equivShrink_symm_neg, WithCStarModule.equiv_symm_zero, NumberField.mixedEmbedding.fundamentalCone.expMap_basis_of_ne, finFunctionFinEquiv_symm_apply_val, ofFiberEquiv_symm_apply, finSuccEquiv'_symm_some_above, Additive.ofMul_symm_eq, SimpleGraph.Iso.mapEdgeSet_symm_apply, symm_trans_self, AlternatingMap.domDomCongrEquiv_symm_apply, SSet.Subcomplex.PairingCore.pairing_p_symm_equivI, Algebra.PreSubmersivePresentation.reindex_map, CategoryTheory.Cat.Hom.equivFunctor_symm_apply, ofBoolRing_symm_eq, DomAddAct.comap_mk_nhds, RingHom.equivRatAlgHom_symm_apply, Set.sumCompl_symm_apply_of_notMem, finSuccEquivLast_symm_some, Perm.set_support_symm_eq, MvPolynomial.mapEquivMonic_symm_map, toFun_inducedStructureEquiv_Symm, CategoryTheory.CartesianMonoidalCategory.homEquivToProd_symm_apply, DomAddAct.symm_mk_zero, WithLp.equiv_symm_apply_ofLp, equivShrink_symm_smul, SSet.stdSimplex.face_singleton_compl, Matrix.compAddEquiv_symm_apply, Fintype.finsetEquivSet_symm_apply, Fin.snocEquiv_symm_apply, CategoryTheory.InjectiveResolution.extAddEquivCohomologyClass_symm_apply, FirstOrder.Language.Formula.realize_equivSentence_symm, ofPreimageEquiv_symm_apply, ContinuousAlternatingMap.ofSubsingleton_symm_apply_apply, Topology.WithGeneratedByTopology.isClosed_iff, Finset.sigmaAntidiagonalEquivProd_symm_apply_fst, CategoryTheory.Pseudofunctor.StrongTrans.Modification.equivOplax_symm_apply, AddSubmonoid.equivOp_symm_apply_coe, RootPairing.equiv_of_mapsTo_symm_apply, MultilinearMap.ofSubsingletonₗ_symm_apply, CategoryTheory.shrinkYonedaEquiv_symm_map, toIso_inv, Filter.comap_equiv_symm, CategoryTheory.Adjunction.homEquiv_symm_apply, CategoryTheory.Functor.Initial.limitConeOfComp_isLimit, GradedRing.proj_recompose, TopologicalSpace.Compacts.equiv_symm, Perm.decomposeFin_symm_apply_zero, AlgebraicGeometry.Spec.homEquiv_symm_apply, SimpleGraph.Iso.comap_symm_apply, Affine.Simplex.reindex_reindex_symm, SetTheory.PGame.toRightMovesNim_symm_lt, QuaternionAlgebra.coe_symm_addEquivTuple, AddChar.coe_toMonoidHomEquiv_symm, CategoryTheory.CategoryOfElements.toCostructuredArrow_map, Int.divModEquiv_symm_apply, CategoryTheory.GrothendieckTopology.yonedaEquiv_symm_naturality_right, DirectSum.IsInternal.subordinateOrthonormalBasisIndex_def, Matrix.isUnit_comp_symm_iff, eq_symm_apply, RingEquiv.prodCongr_symm_apply, setSubtypeComm_symm_apply, CategoryTheory.nerve.edgeMk_surjective, CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.comp_homEquiv_symm_assoc, DomAddAct.isInducing_mk_symm, CategoryTheory.Over.postAdjunctionLeft_counit_app_left, Set.opEquiv_symm_apply, CategoryTheory.Adjunction.equivHomsetRightOfNatIso_symm_apply, optionCongr_symm, trans_eq_refl_iff_eq_symm, Fin.insertNth_apply_cycleRange_symm, Finsupp.lcongr_apply_apply, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_left_symm, SemidirectProduct.equivProd_symm_apply_right, sumCompl_symm_apply_of_pos, some_removeNone_iff, Topology.WithLowerSet.ofLowerSet_symm, CategoryTheory.shrinkYonedaEquiv_shrinkYoneda_map, CategoryTheory.Adjunction.CoreHomEquivUnitCounit.homEquiv_counit, AffineEquiv.ofBijective.symm_eq, Finsupp.equivFunOnFinite_symm_apply_apply, addLeft_symm, SetTheory.PGame.toLeftMovesNim_one_symm, listUniqueEquiv_symm_apply, HahnModule.of_symm_sub, CategoryTheory.ShiftedHom.opEquiv'_add_symm, Polynomial.UniversalFactorizationRing.fromTensor_comp_universalFactorizationMap, CStarMatrix.toCLM_apply_single_apply, CategoryTheory.Functor.functorHomEquiv_symm_apply_app_app, piCongrLeft'_apply, MulEquiv.withZero_symm_apply_symm_apply, CategoryTheory.Equivalence.induced_counitIso, AddChar.coe_toAddMonoidHomEquiv_symm, AddSubgroup.equivOp_symm_apply_coe, invertibleEquivOfLeftInverse_symm_apply, CategoryTheory.Limits.Types.isLimitEquivSections_symm_apply, DFinsupp.domLCongr_apply, Polynomial.UniversalFactorizationRing.fromTensor_comp_universalFactorizationMap', LinearEquiv.funCongrLeft_symm, RootPairing.Hom.id_indexEquiv_symm_apply, CategoryTheory.enrichedFunctorTypeEquivFunctor_symm_apply_map, pprodEquivProd_symm_apply, Fin2.equivFin_symm_apply, CategoryTheory.yonedaEquiv_symm_app_apply, CategoryTheory.Arrow.equivSigma_symm_apply_right, sInfHom.symm_dual_id, CategoryTheory.tensorLeftHomEquiv_symm_naturality, DFinsupp.sigmaFinsetFunEquiv_apply_snd_coe, CategoryTheory.nerve.homEquiv_symm_apply, Set.mem_image_equiv, Algebra.discr_reindex, MeasurableEquiv.piFinSuccAbove_apply, Ordinal.toPGame_moveLeft', CategoryTheory.ProjectiveResolution.extEquivCohomologyClass_symm_neg, LocallyConstant.congrRight_symm_apply, CategoryTheory.ShiftedHom.opEquiv_symm_comp, SkewMonoidAlgebra.lift_symm_apply, CategoryTheory.Limits.compYonedaSectionsEquiv_symm_apply_coe, LinearMap.IsSymmetric.eigenvectorBasis_def, piCongrSet_symm_apply, apply_eq_iff_eq_symm_apply, KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk_symm_apply_eq_span, CategoryTheory.unitCompPartialBijective_symm_apply, Function.fromTypes_nil_equiv_symm_apply, MeasureTheory.SignedMeasure.toJordanDecompositionEquiv_symm_apply, uliftZMultiplesHom_symm_apply, SetTheory.PGame.moveLeft_neg, psigmaEquivSigma_symm_apply

Equiv.Computable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	Equiv.Computable	Equiv.symm	—	—

Equiv.Perm.Disjoint

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Equiv.Perm.Disjoint	—	—	—

Equiv.Perm.SameCycle

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Equiv.Perm.SameCycle	—	—	zpow_neg Equiv.symm_apply_apply

Equiv.Perm.subtypeCongr

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	Equiv.symm Equiv.Perm.subtypeCongr	—	—

Equiv.subsingleton

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	—	—	—	Function.Injective.subsingleton Equiv.injective

Ergodic

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	Ergodic DFunLike.coe MeasurableEquiv EquivLike.toFunLike MeasurableEquiv.instEquivLike	MeasurableEquiv.symm	—	MeasureTheory.MeasurePreserving.symm toMeasurePreserving PreErgodic.aeconst_set toPreErgodic MeasurableEquiv.image_eq_preimage_symm MeasurableEquiv.preimage_image

EuclideanGeometry.Sphere.IsDiameter

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	EuclideanGeometry.Sphere.IsDiameter	—	—	right_mem RCLike.charZero_rclike midpoint_eq_center midpoint_comm

EuclideanGeometry.Sphere.IsExtTangent

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	EuclideanGeometry.Sphere.IsExtTangent	—	—	EuclideanGeometry.Sphere.IsExtTangentAt.symm

EuclideanGeometry.Sphere.IsExtTangentAt

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	EuclideanGeometry.Sphere.IsExtTangentAt	—	—	mem_right mem_left Wbtw.symm Real.instIsOrderedRing wbtw

Filter.EventuallyEq

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Filter.EventuallyEq	—	—	Filter.Eventually.mono

Finmap.Disjoint

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Finmap.Disjoint	—	—	—

FirstOrder.Language.ElementarilyEquivalent

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	FirstOrder.Language.ElementarilyEquivalent	—	—	—

FirstOrder.Language.Equiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	15 math math: symm_apply_apply, self_comp_symm_toHom, self_comp_symm, symm_bijective, symm_comp_self, apply_symm_apply, symm_comp_self_toHom, comp_symm, FirstOrder.Language.DirectLimit.Equiv_isup_symm_inclusion, self_comp_symm_toEmbedding, symm_symm, FirstOrder.Language.PartialEquiv.symm_apply, FirstOrder.Language.DirectLimit.Equiv_isup_symm_inclusion_apply, symm_comp_self_toEmbedding, Equiv.toFun_inducedStructureEquiv_Symm

FirstOrder.Language.FGEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	1 math math: symm_coe

FirstOrder.Language.LEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	5 math math: FirstOrder.Language.addEmptyConstants_symm_isExpansionOn, symm_toLHom, symm_invLHom, onBoundedFormula_symm, onFormula_symm

FirstOrder.Language.PartialEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	7 math math: monotone_symm, symm_symm, symm_bijective, symm_le_iff, symm_le_symm, symm_apply, FirstOrder.Language.FGEquiv.symm_coe

FirstOrder.Language.Theory.Iff

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	FirstOrder.Language.Theory.Iff	—	—	FirstOrder.Language.BoundedFormula.realize_iff

Function.Commute

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Function.Commute	—	—	—

GenLoop.Homotopic

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	GenLoop.Homotopic	—	ContinuousMap.HomotopicRel.symm

Graph.Adj

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Graph.Adj	—	—	Graph.IsLink.symm

Graph.IsLink

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Graph.IsLink	—	—	Graph.isLink_symm edge_mem

GroupExtension.Equiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	4 math math: symm_symm_apply, coe_symm, refl_symm_apply, trans_symm_apply

GroupExtension.IsConj

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	GroupExtension.IsConj	—	—	map_inv MonoidHom.instMonoidHomClass neg_add_cancel zpow_zero one_mul mul_one neg_neg zpow_one Mathlib.Tactic.Group.zpow_trick_one'

Homeomorph

Definitions

Name	Category	Theorems
symm 📖	CompOp	180 math math: smulOfNeZero_symm_apply, TopCat.isoOfHomeo_inv, Trivialization.preimageHomeomorph_symm_apply, Topology.IsGeneratedBy.homeomorph_symm_coe, AlgebraicGeometry.coprodSpec_apply, finTwoArrow_symm_apply, IsCoveringMapOn.homeomorph_comp_iff, vadd_symm_apply, stdSimplexHomeomorphUnitInterval_symm_apply_coe, Sequential.homeoOfIso_symm_apply, LocallyConstant.congrLeftRingEquiv_apply_apply, TopCat.uliftFunctorObjHomeo_symm_naturality_apply, OpenPartialHomeomorph.transHomeomorph_symm_apply, DomMulAct.coe_mkHomeomorph_symm, LinearIsometryEquiv.coe_symm_toHomeomorph, OpenPartialHomeomorph.transHomeomorph_target, shearAddRight_symm_coe, piCongrLeft_symm_apply, Equiv.toHomeomorphOfIsInducing_symm_apply, LinearIsometryEquiv.toHomeomorph_symm, Topology.IsQuotientMap.lift_apply, AddCircle.homeomorphCircle'_symm_apply, funSplitAt_symm_apply, opensCongr_apply, coe_symm_toEquiv, NumberField.mixedEmbedding.polarSpaceCoord_target, self_trans_symm, Topology.IsQuotientMap.homeomorph_symm_apply, affineHomeomorph_symm_apply, DilationEquiv.coe_symm_toHomeomorph, OpenPartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv_symm_apply, OpenPartialHomeomorph.homeomorphOfImageSubsetSource_symm_apply, continuousMapCongr_symm_apply, AlgebraicGeometry.Scheme.homeoOfIso_symm, continuousMapCongr_apply, Real.sinhHomeomorph_symm_apply, CompHausLike.homeoOfIso_symm_apply, sumProdDistrib_symm_apply, subtype_symm_apply_coe, LocallyConstant.congrLeftₐ_apply_apply, symm_toHomotopyEquiv, isBigOWith_congr, smul_symm_apply, symm_inv, Equiv.symm_toHomeomorph, Set.univ_symm_apply_coe, piSplitAt_symm_apply, subLeft_symm_apply, ContinuousMap.homeoFnOfDiscrete_symm_apply, sumArrowHomeomorphProdArrow_symm_apply, Nonneg.val_unitsHomeomorphPos_symm_apply_coe, EReal.expHomeomorph_symm, addRight_symm, Sequential.isoOfHomeo_inv, Equiv.toHomeomorphOfContinuousOpen_symm_apply, preimage_symm, TopologicalSpace.Compacts.coe_equiv_apply_eq_preimage, divRight_symm_apply, Nonneg.val_inv_unitsHomeomorphPos_symm_apply_coe, CommRingCat.HomTopology.precompHomeomorph_symm_apply, DilationEquiv.toHomeomorph_symm, Bundle.Trivial.homeomorphProd_symm_apply_snd, contDiff_symm, TopologicalSpace.Compacts.equiv_symm_apply, CompactlyGenerated.isoOfHomeo_inv, comap_nhds_eq, continuous_symm, CommRingCat.HomTopology.mvPolynomialHomeomorph_symm_apply_hom, image_symm, NumberField.mixedEmbedding.volume_preserving_mixedSpaceToRealMixedSpace_symm, continuousMapOfUnique_symm_apply, unitInterval.symmHomeomorph_symm_apply, homeomorphUnitSphereProd_symm_apply_coe, vaddConst_symm_apply, onePointCongr_symm_apply, sumComm_symm, mulLeft₀_symm_apply, isBigO_congr, homeomorphSphereProd_symm_apply_coe, opensCongr_symm, ContinuousLinearEquiv.coe_symm_toHomeomorph, prodUnique_symm_apply_snd, symm_neg, shearMulRight_symm_coe, addLeft_symm, CompactlyGenerated.homeoOfIso_symm_apply, piCongrRight_symm, image_symm_apply_coe, GenLoop.loopHomeo_symm_apply, NumberField.mixedEmbedding.polarCoord_symm_eq, Equiv.toHomeomorphOfContinuousClosed_symm_apply, Set.prod_symm_apply_coe, Function.RightInverse.homeomorph_symm_apply, homeomorphOfUnique_symm_apply, AlexDisc.Iso.mk_inv, IsHomeomorph.homeomorph_symm_apply, uniqueProd_symm_apply_snd, piCurry_symm_apply, image_eq_preimage_symm, sigmaProdDistrib_symm_apply, toContinuousMap_comp_symm, ContinuousMap.sigmaCodHomeomorph_symm_apply, subRight_symm_apply, Bundle.Trivial.homeomorphProd_symm_apply_proj, self_comp_symm, iccHomeoI_symm_apply_coe, Equiv.toHomeomorph_symm, tangentBundleModelSpaceHomeomorph_coe_symm, LocallyConstant.congrLeftₗ_apply_apply, Trivialization.preimageSingletonHomeomorph_symm_apply, DomAddAct.coe_mkHomeomorph_symm, toMeasurableEquiv_symm_coe, piUnique_symm_apply, piEquivPiSubtypeProd_symm_apply, ENNReal.logHomeomorph_symm, MulOpposite.opHomeomorph_symm_apply, Diffeomorph.symm_toHomeomorph, divLeft_symm_apply, toOpenPartialHomeomorphOfImageEq_symm_apply, ContinuousLinearEquiv.toHomeomorph_symm, sumSumSumComm_symm, Trivialization.sourceHomeomorphBaseSetProd_symm_apply, symm_toPartialHomeomorph, transOpenPartialHomeomorph_symm_apply, mulRight₀_symm_apply, toOpenPartialHomeomorph_symm_apply, TopCat.homeoOfIso_symm_apply, symm_apply_eq, CompHausLike.isoOfHomeo_inv_hom_hom_apply, prodProdProdComm_symm, ContinuousMap.homeoFnOfDiscrete_symm_apply_apply, Fin.appendHomeomorph_symm_apply, sumCongr_symm, PrimeSpectrum.coe_preimageHomeomorphFiber_symm_apply_coe_asIdeal, AffineEquiv.coe_toHomeomorphOfFiniteDimensional_symm, homeomorph_mk_coe_symm, contDiff_symm_deriv, apply_symm_apply, symm_trans_self, NumberField.mixedEmbedding.homeoRealMixedSpacePolarSpace_symm_apply, contMDiff_tangentBundleModelSpaceHomeomorph_symm, symm_trans_apply, Metric.Snowflaking.homeomorph_symm_apply, compStarAlgEquiv'_symm_apply, refl_symm, Metric.PiNatEmbed.toPiNatHomeo_symm_apply, symm_apply_apply, ContinuousMulEquiv.coe_toHomeomorph_symm, Topology.IsEmbedding.toHomeomorph_symm_apply, UniformEquiv.toHomeomorph_symm_apply, AffineIsometryEquiv.toHomeomorph_symm, prodCongr_symm, ContinuousAlgEquiv.symm_toHomeomorph, symm_map_nhds_eq, funUnique_symm_apply, symm_bijective, prodComm_symm, mulLeft_symm, symm_comp_self, piFinTwo_symm_apply, symm_comp_toContinuousMap, AlgebraicGeometry.Scheme.Hom.fiberι_fiberHomeo_symm, OrderIso.coe_toHomeomorph_symm, symm_toOpenPartialHomeomorph, isLittleO_congr, AffineIsometryEquiv.coe_symm_toHomeomorph, IsometryEquiv.coe_toHomeomorph_symm, IsCoveringMapOn.homeomorph_comp, mulRight_symm, Diffeomorph.coe_toHomeomorph_symm, AddOpposite.opHomeomorph_symm_apply, TopologicalSpace.Compacts.equiv_symm, LocallyConstant.congrLeft_apply, OpenPartialHomeomorph.toHomeomorphSourceTarget_symm_apply_coe, ModelWithCorners.toHomeomorph_symm_apply, symm_symm, AddCircle.homeomorphAddCircle_symm_apply_mk, ContinuousAddEquiv.coe_toHomeomorph_symm, polynomialFunctions.comap_compRightAlgHom_iccHomeoI, unitBall_symm_apply

HomotopicalAlgebra.Cylinder

Definitions

Name	Category	Theorems
symm 📖	CompOp	8 math math: instIsGoodSymmOfRespectsIsoCofibrations, symm_i₁, symm_i, symm_π, symm_i₀, instIsVeryGoodSymmOfRespectsIsoCofibrations, symm_i_assoc, symm_I

HomotopicalAlgebra.Cylinder.LeftHomotopy

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

HomotopicalAlgebra.LeftHomotopyRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	HomotopicalAlgebra.LeftHomotopyRel	—	—	HomotopicalAlgebra.Cylinder.LeftHomotopy.leftHomotopyRel

HomotopicalAlgebra.PathObject

Definitions

Name	Category	Theorems
symm 📖	CompOp	8 math math: symm_p₁, symm_p₀, symm_p_assoc, symm_p, instIsVeryGoodSymmOfRespectsIsoFibrations, symm_ι, symm_P, instIsGoodSymmOfRespectsIsoFibrations

HomotopicalAlgebra.PathObject.RightHomotopy

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

HomotopicalAlgebra.Precylinder

Definitions

Name	Category	Theorems
symm 📖	CompOp	7 math math: symm_i₀, symm_i_assoc, symm_π, symm_i₁, LeftHomotopy.symm_h, symm_I, symm_i

HomotopicalAlgebra.Precylinder.LeftHomotopy

Definitions

Name	Category	Theorems
symm 📖	CompOp	1 math math: symm_h

HomotopicalAlgebra.PrepathObject

Definitions

Name	Category	Theorems
symm 📖	CompOp	7 math math: symm_p, symm_ι, symm_p₁, symm_P, symm_p_assoc, symm_p₀, RightHomotopy.symm_h

HomotopicalAlgebra.PrepathObject.RightHomotopy

Definitions

Name	Category	Theorems
symm 📖	CompOp	1 math math: symm_h

HomotopicalAlgebra.RightHomotopyRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	HomotopicalAlgebra.RightHomotopyRel	—	—	HomotopicalAlgebra.PathObject.RightHomotopy.rightHomotopyRel

Homotopy

Definitions

Name	Category	Theorems
symm 📖	CompOp	1 math math: symm_hom

HomotopyEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

IncompRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IncompRel	—	—	—

Inseparable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Inseparable	—	—	—

Int.ModEq

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Int.ModEq	—	—	—

IntermediateField.LinearDisjoint

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IntermediateField.LinearDisjoint IntermediateField SetLike.instMembership IntermediateField.instSetLike IntermediateField.toField IntermediateField.algebra' Semifield.toCommSemiring Field.toSemifield Algebra.toSMul CommSemiring.toSemiring Algebra.id IsScalarTower.left MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring DistribMulAction.toMulAction AddCommMonoid.toAddMonoid NonUnitalNonAssocSemiring.toAddCommMonoid NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing EuclideanDomain.toCommRing Field.toEuclideanDomain Module.toDistribMulAction Algebra.toModule IntermediateField.instAlgebraSubtypeMem IntermediateField.isScalarTower_mid'	—	—	IsScalarTower.left IntermediateField.isScalarTower_mid' IntermediateField.linearDisjoint_iff' Subalgebra.LinearDisjoint.symm

IntervalIntegrable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IntervalIntegrable	—	—	—

IsBoundedBilinearMap

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsBoundedBilinearMap	—	—	add_right smul_right add_left smul_left bound LE.le.trans_eq Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.mul_congr Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.add_mul Mathlib.Tactic.Ring.mul_add Mathlib.Tactic.Ring.mul_pf_left Mathlib.Tactic.Ring.mul_pf_right Mathlib.Tactic.Ring.one_mul Mathlib.Tactic.Ring.mul_zero Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.zero_mul

IsBrauerEquivalent

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsBrauerEquivalent	—	—	—

IsChain

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsChain	—	—	Set.Pairwise.mono'

IsCompl

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsCompl	—	—	Disjoint.symm disjoint Codisjoint.symm codisjoint

IsConj

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsConj	—	—	SemiconjBy.units_inv_symm_left

IsConjRoot

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsConjRoot	—	—	—

IsCoprime

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsCoprime	—	—	add_comm

IsLocalRing.ResidueField.mapEquiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	RingEquiv.symm IsLocalRing.ResidueField Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing IsLocalRing.instCommRingResidueField Distrib.toAdd IsLocalRing.ResidueField.mapEquiv	—	—

IsMaxAntichain

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsMaxAntichain	—	—	IsAntichain.flip isAntichain

IsMaxChain

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsMaxChain	—	—	IsChain.symm isChain

IsMoritaEquivalent

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsMoritaEquivalent	—	—	Nonempty.map cond

IsRelPrime

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsRelPrime CommMonoid.toMonoid	—	—	—

IsSl2Triple

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	IsSl2Triple	NegZeroClass.toNeg SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid LieRing.toAddCommGroup	—	h_ne_zero neg_eq_iff_eq_neg lie_skew lie_e_f neg_lie lie_h_f_nsmul lie_h_e_nsmul

IsVisible

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	IsVisible	—	—	isVisible_comm

IsometryEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	58 math math: LinearIsometryEquiv.coe_symm_toIsometryEquiv, preimage_sphere, toRealLinearIsometryEquiv_symm_apply, apply_symm_apply, withLpUniqueProd_symm_apply, subLeft_symm_apply, toDilationEquiv_symm, preimage_emetric_closedBall, constVSub_symm_apply, addRight_symm, divRight_symm, symm_apply_apply, Fin.appendIsometry_symm_apply, constVAdd_symm, Delone.DeloneSet.mapIsometry_symm_apply_carrier, preimage_emetric_ball, addLeft_symm, divLeft_symm_apply, preimage_closedEBall, Delone.DeloneSet.mapIsometry_symm, subRight_symm, inv_symm, piFinTwo_symm_apply, sumArrowIsometryEquivProdArrow_symm_apply, piCongrLeft_symm_apply, symm_symm, piCongrLeft_apply, Fin.appendIsometryOfEq_symm_apply, withLpProdUnique_symm_apply, piCongrLeft'_symm_apply, symm_apply_eq, constSMul_symm, funUnique_symm_apply, mulRight_symm, coe_symm_toEquiv, eq_symm_apply, image_symm, coe_toRealLinearIsometryEquivOfMapZero_symm, vaddConst_symm_apply, withLpProdComm_symm, neg_symm, symm_bijective, mulLeft_symm, coe_symm_toDilationEquiv, symm_comp_self, withLpProdAssoc_symm_apply, ContinuousMap.isometryEquivBoundedOfCompact_symm_apply, LinearIsometryEquiv.toIsometryEquiv_symm, AffineIsometryEquiv.coe_symm_toIsometryEquiv, withLpProdCongr_symm_apply, preimage_symm, preimage_closedBall, preimage_ball, AffineIsometryEquiv.toIsometryEquiv_symm, coe_toHomeomorph_symm, symm_trans_apply, preimage_eball, self_comp_symm

Joined

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	Joined	—	—

JoinedIn

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	JoinedIn	—	—	mem Joined.symm

LieEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	22 math math: skewAdjointLieSubalgebraEquiv_symm_apply, LieAlgebra.Extension.toKer_bracket, refl_symm, symm_trans, LieAlgebra.Extension.ringModuleOf_bracket, Matrix.reindexLieEquiv_symm, AlgEquiv.toLieEquiv_symm_apply, Matrix.lieConj_symm_apply, apply_symm_apply, symm_trans_self, symm_apply_apply, LieAlgebra.Extension.bracket, symm_bijective, lieEquivMatrix'_symm_apply, ofSubalgebras_symm_apply, LinearEquiv.lieConj_symm, LieAlgebra.Extension.ringModuleOf_bracket_proj, self_trans_symm, symm_symm, LieAlgebra.Extension.twoCocycleOf_coe_coe, LieAlgebra.Extension.oneCochainOfTwoSplitting_apply, LieModule.toEnd_matrix

LieModuleEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	9 math math: symm_trans, apply_eq_iff_eq_symm_apply, symm_apply_apply, symm_symm, LieModule.maxTrivEquiv_of_equiv_symm_eq_symm, symm_bijective, apply_symm_apply, symm_trans_self, self_trans_symm

LinearEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	766 math math: DirectSum.IsInternal.ofBijective_coeLinearMap_same, multilinearMapCongrLeft_symm_apply, MultilinearMap.domDomCongrLinearEquiv'_symm_apply, LinearMap.lTensor_ker_subtype_tensorKerEquiv_symm, Rep.resCoindHomEquiv_symm_apply_hom, IsLocalizedModule.iso_symm_apply', Polynomial.degreeLT.addLinearEquiv_symm_apply_inr, TensorProduct.congr_symm, LinearMap.apply_symm_toPerfPair_self, LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff, Submodule.rTensorOne_symm_apply, iSupIndep.linearEquiv_symm_apply, Submodule.topEquiv_symm_apply_coe, PiTensorProduct.lift_reindex, Matrix.toLin'_symm, QuaternionAlgebra.coe_linearEquivTuple_symm, Matrix.kroneckerTMulStarAlgEquiv_symm_apply, Algebra.Presentation.differentials.comm₁₂_single, Pretrivialization.linearEquivAt_symm_apply, Module.Basis.map_equivFun, Submodule.comap_equiv_eq_map_symm, Rep.MonoidalClosed.linearHomEquiv_symm_hom, ofBijective_symm_apply_apply, TensorProduct.AlgebraTensorModule.rid_symm_apply, BilinForm.toMatrix_symm, symm_mem_transvections_iff, AlternatingMap.constLinearEquivOfIsEmpty_symm_apply, PiTensorProduct.tmulEquiv_symm_apply, ContinuousLinearEquiv.toLinearEquiv_symm, dotProductEquiv_symm_apply, addMonoidHomLequivNat_symm_apply, TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_symm, PiTensorProduct.dualDistribEquivOfBasis_symm_apply, Representation.ofMulActionSelfAsModuleEquiv_symm_apply, Module.Basis.equivFun_symm_apply, KaehlerDifferential.polynomialEquiv_symm, TensorProduct.equivFinsuppOfBasisLeft_symm_apply, Module.DualBases.basis_repr_symm_apply, congrRight_symm, Module.Basis.constr_symm_apply, KaehlerDifferential.tensorKaehlerEquiv_symm_D_tmul', Finsupp.supportedEquivFinsupp_symm_single, PiTensorProduct.lift_reindex_symm, kroneckerTMulAlgEquiv_symm_single_tmul, mk_coe', PiTensorProduct.map_reindex_symm, LieAlgebra.LoopAlgebra.twoCochainOfBilinear_apply_apply, LocalizedModule.restrictScalars_map_eq, Submodule.prodEquivOfIsCompl_symm_apply_snd_eq_zero, ZMod.invDFT_apply, PiTensorProduct.subsingletonEquiv_symm_apply, MulOpposite.coe_opLinearEquiv_symm, Submodule.toLinearEquiv_orthogonalDecomposition, RootPairing.Equiv.coweightEquiv_symm_coweightMap, TensorProduct.piScalarRight_symm_algebraMap, LinearMap.BilinForm.congr_apply, TensorProduct.AlgebraTensorModule.congr_symm, RootPairing.rootSpan_dualAnnihilator_le_ker_rootForm, restrictScalars_symm_apply, KaehlerDifferential.tensorKaehlerEquiv_symm_D_tmul, SpecialLinearGroup.toLinearEquiv_symm_to_linearMap, Matrix.transposeLinearEquiv_symm, domMulActCongrRight_symm_apply, congrQuadraticMap_symm, conj_apply_apply, skewSwap_symm_apply, TensorProduct.AlgebraTensorModule.congr_symm_tmul, Algebra.Presentation.differentials.comm₂₃, conjAlgEquiv_symm_apply_apply, starLinearEquiv_symm_apply, Rep.diagonalHomEquiv_symm_apply, Finsupp.linearEquivFunOnFinite_symm_coe, symm_comp, Module.Dual.eval_comp_comp_evalEquiv_eq, symm_mem_dilatransvections_iff, LinearMap.BilinForm.toMatrix'_symm, Coalgebra.coassoc_symm_apply, LinearMap.baseChange_baseChange, Shrink.linearEquiv_symm_apply, Orientation.map_symm, ULift.moduleEquiv_symm_apply, contractLeft_assoc_coevaluation, cast_symm_apply, ContinuousMultilinearMap.piLinearEquiv_symm_apply, Module.AEval.of_symm_smul, Module.Basis.ofZLatticeComap_apply, IsLocalizedModule.iso_symm_apply, TensorProduct.AlgebraTensorModule.rightComm_symm_tmul, toLinearEquiv_toContinuousLinearEquiv_symm, Algebra.IsPushout.cancelBaseChange_symm_tmul, piFinTwo_symm_apply, comp_toLinearMap_symm_eq, coe_inv, star_dotProduct_toMatrix₂_mulVec, LocallyConstant.congrRightₗ_symm_apply_apply, TensorProduct.quotientTensorQuotientEquiv_symm_apply_mk_tmul, skewAdjointLieSubalgebraEquiv_symm_apply, LinearMap.ringLmapEquivSelf_symm_apply, Matrix.toBilin_symm, comp_symm_cancel_left, Submodule.dualCopairing_eq, KaehlerDifferential.quotKerTotalEquiv_symm_comp_D, Submodule.quotEquivOfEqBot_symm_apply, finsuppTensorFinsupp'_symm_single_eq_tmul_single_one, Module.Relations.Solution.IsPresentation.linearEquiv_symm_var, MulOpposite.coe_opLinearEquiv_symm_addEquiv, SemimoduleCat.hom_inv_rightUnitor, Rep.coinvariantsTensorFreeLEquiv_symm_apply, toModuleIso_inv, Algebra.Extension.lTensor_cotangentComplex_eq_cotangentComplexBaseChange, Finsupp.curryLinearEquiv_symm_apply_apply, coe_symm_toEquiv, LinearMap.toMatrixₛₗ₂'_symm, DFinsupp.mapRange.linearEquiv_symm, congrLeft_symm_apply, conjAlgEquiv_apply, exteriorPower.alternatingMapLinearEquiv_symm_map, symm_apply_eq, Submodule.equivSubtypeMap_symm_apply, LinearMap.toMatrix_symm, TensorProduct.gradedMul_def, WithConv.symm_linearEquiv_apply, Finsupp.linearCombination_one_tmul, Matrix.toLinearEquivRight'OfInv_symm_apply, eq_symm_apply, Module.Basis.map_repr, MultilinearMap.fromDirectSumEquiv_symm_apply, LinearMap.trace_eq_contract_of_basis', LinearMap.BilinForm.congr_comp, CliffordAlgebra.changeFormEquiv_symm, Finsupp.LinearEquiv.finsuppUnique_symm_apply, MultilinearMap.domDomCongrLinearEquiv_symm_apply, LinearIsometryEquiv.toLinearEquiv_symm, image_symm_eq_preimage, PiTensorProduct.reindex_comp_tprod, symm_toLexLinearEquiv, Matrix.SpecialLinearGroup.toLin'_symm_to_linearMap, LocalizedModule.equivTensorProduct_symm_apply_tmul_one, DistribMulAction.toLinearEquiv_symm_apply, ModuleCat.homLinearEquiv_symm_apply, Finsupp.sigmaFinsuppLEquivPiFinsupp_symm_apply, Module.reflection_symm, Module.Basis.toMatrix_map, PiLp.sumPiLpEquivProdLpPiLp_symm_apply_ofLp, LinearMap.tensorKerEquivOfSurjective_symm_tmul, TensorProduct.equivFinsuppOfBasisLeft_symm, Module.Basis.coe_toOrthonormalBasis_repr, ExteriorAlgebra.liftAlternatingEquiv_symm_apply, Submodule.map_unop_mul, ModularGroup.lcRow0Extend_symm_apply, TensorPower.cast_symm, LocalizedModule.equivTensorProduct_symm_apply_tmul, arrowCongr_symm_apply, TensorProduct.rightComm_symm, Module.FinitePresentation.linearEquivMap_symm_apply, Module.AEval'.of_symm_X_smul, zero_symm, LieAlgebra.LoopAlgebra.toFinsupp_symm_single, Matrix.toLinearEquiv'_symm_apply, Module.Basis.repr_smul, Matrix.toLin_symm, Module.Ray.map_symm, QuadraticForm.tensorRId_symm_apply, Rep.coindVEquiv_symm_apply_coe, extendScalarsOfSurjective_symm, TensorProduct.quotientTensorEquiv_symm_apply_mk_tmul, PiTensorProduct.ofDirectSumEquiv_symm_lof_tprod, Module.Invertible.linearEquivOfRightInverse_symm_apply, IsLocalizedModule.iso_symm_comp, TensorProduct.AlgebraTensorModule.rTensor_tensor, funUnique_symm_apply, Matrix.toLinearMapₛₗ₂'_symm, Submodule.Quotient.equiv_symm, continuous_decomposeProdAdjoint_symm, RootPairing.Equiv.weightMap_weightEquiv_symm, finsuppTensorFinsuppLid_symm_single_smul, PiTensorProduct.congr_symm_tprod, dualTensorHomEquivOfBasis_symm_cancel_right, Matrix.toLpLin_symm_pow, TensorProduct.tensorTensorTensorComm_symm, LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton', mapMatrix_symm, finsuppTensorFinsupp'_symm_single_mul, ofLeftInverse_symm_apply, isOpenMap_toWeakSpace_symm, LinearMap.polyCharpolyAux_map_eval, det_symm_mul_det, LinearMap.IsPerfectCompl.isCompl_right, ZLattice.comap_equiv_apply, comp_symm_eq, Submodule.botEquivPUnit_symm_apply, TensorProduct.map_map_assoc_symm, KaehlerDifferential.tensorKaehlerEquivBase_symm_apply, Finsupp.sumFinsuppLEquivProdFinsupp_symm_apply, TensorProduct.comm_symm, GradedTensorProduct.auxEquiv_symm_one, TensorProduct.finsuppLeft_symm_apply_single, Finsupp.linearEquivFunOnFinite_symm_apply, BilinForm.dotProduct_toMatrix_mulVec, multilinearMapCongrRight_symm_apply, piUnique_symm_apply, MultilinearMap.coe_currySumEquiv_symm, eq_symm_comp, LinearMap.IntrinsicStar.starLinearEquiv_eq_arrowCongr, IsTensorProduct.equiv_symm_apply, symm_conjAlgEquiv, MultilinearMap.curryFinFinset_symm_apply_piecewise_const, PiTensorProduct.ofFinsuppEquiv_symm_single_tprod, ofSubmodule'_symm_apply, SpecialLinearGroup.congr_linearEquiv_symm, TensorProduct.prodLeft_symm_tmul, LinearMap.IsSymmetric.toLinearMap_symm, Module.Basis.equiv_symm, KaehlerDifferential.linearMapEquivDerivation_symm_apply, invFun_eq_symm, coe_toContinuousLinearEquiv_symm, Algebra.Generators.CotangentSpace.compEquiv_symm_zero, TensorProduct.dualDistribEquivOfBasis_symm_apply, SpecialLinearGroup.congr_linearEquiv_coe_apply, TensorProduct.leftComm_def, Module.DirectLimit.congr_symm_apply_of, sumArrowLequivProdArrow_symm_apply_inr, LinearMap.rTensor_lTensor_comp_assoc_symm, NumberField.mixedEmbedding.fundamentalCone.prod_deriv_expMap_single, LinearMap.liftBaseChangeEquiv_symm_apply, conj_exact_iff_exact, Finsupp.sumFinsuppLEquivProdFinsupp_symm_inl, Algebra.TensorProduct.opAlgEquiv_apply, Representation.asModuleEquiv_symm_map_smul, AffineEquiv.linear_symm, symm_neg, Algebra.TensorProduct.equivFinsuppOfBasis_symm_apply, LinearMap.prodEquiv_symm_apply, ModuleCat.hom_inv_associator, LinearMap.trace_eq_contract', AlgEquiv.toLinearEquiv_symm, symm_flip, Submodule.dualCoannihilator_map_linearEquiv_flip, eq_toLinearMap_symm_comp, LinearMap.BilinForm.dotProduct_toMatrix_mulVec, CharacterModule.intSpanEquivQuotAddOrderOf_symm_apply_coe, CategoryTheory.Iso.toLinearEquiv_symm, Matrix.toBilin'_symm, StarModule.decomposeProdAdjoint_symm_apply, RootPairing.corootSpan_dualAnnihilator_map_eq, coe_toAddEquiv_symm, Module.Basis.coe_toOrthonormalBasis_repr_symm, comp_symm_assoc, Submodule.coe_quotEquivOfEqBot_symm, symm_apply_apply, CharacterModule.intSpanEquivQuotAddOrderOf_apply, Submodule.prodEquivOfIsCompl_symm_apply_fst_eq_zero, Algebra.Generators.cotangentCompLocalizationAwayEquiv_symm_comp_inl, extendOfIsometry_symm_eq, TensorProduct.AlgebraTensorModule.leftComm_symm_tmul, QuadraticForm.dualProdIsometry_invFun, Function.Exact.split_tfae, TensorProduct.AlgebraTensorModule.assoc_symm_tmul, TensorProduct.lid_symm_apply, Submodule.map_unop_pow, Submodule.orderIsoMapComap_apply', Rep.resIndAdjunction_homEquiv_symm_apply, PiTensorProduct.liftEquiv_symm_apply, LocalizedModule.coe_map_eq, Module.Basis.equiv'_symm_apply, comp_symm, SemimoduleCat.hom_inv_leftUnitor, Module.subsingletonEquiv_symm_apply, extendScalarsOfIsLocalization_symm_apply, piRing_symm_apply, refl_symm, TensorProduct.piRight_symm_apply, DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem_ne, AdicCompletion.ofTensorProductEquivOfFiniteNoetherian_symm_of, Matrix.conjTransposeLinearEquiv_symm, symm_comp_cancel_right, finsuppTensorFinsupp'_symm_single_eq_single_one_tmul, extendOfIsometry_symm_apply, PiTensorProduct.reindex_symm, det_symm, ContinuousLinearMap.equivRange_symm_toLinearEquiv, Polynomial.mkDerivationEquiv_symm_apply, Finsupp.mapRange.linearEquiv_symm, symm_comp_assoc, Finsupp.lcongr_symm, CategoryTheory.HomOrthogonal.matrixDecompositionLinearEquiv_symm_apply, symm_trans_self, AlgEquiv.coe_symm_toLinearEquiv, Module.Basis.repr_symm_single, kroneckerLinearEquiv_symm_kronecker, eq_comp_toLinearMap_symm, finsuppTensorFinsuppRid_symm_single_smul, TensorProduct.prodRight_symm_tmul, conj_apply, ofSubmodules_symm_apply, TensorProduct.gradedComm_symm, Submodule.restrictScalarsEquiv_symm_apply, toLinearMap_symm_comp_eq, ZMod.invDFT_def', isPositive_symm_iff, Polynomial.toFinsuppIsoLinear_symm_apply_toFinsupp, TensorProduct.piRight_symm_single, FourierTransform.fourierEquiv_symm_apply, LinearIsometryEquiv.coe_symm_toLinearEquiv, IsLocalizedModule.mapEquiv_symm_apply, LinearMap.toMatrix'_symm, Module.symm_dualMap_evalEquiv, Module.Invertible.linearEquivOfLeftInverse_symm_apply, Algebra.Extension.cotangentComplexBaseChange_eq_lTensor_cotangentComplex, coe_toContinuousLinearEquiv_symm', tensorKaehlerQuotKerSqEquiv_symm_tmul_D, Matrix.kroneckerTMulAlgEquiv_symm_apply, MulOpposite.counit_def, DFinsupp.linearEquivFunOnFintype_symm_apply, arrowCongr_apply, Submodule.prodEquivOfIsCompl_symm_apply, skewProd_symm_apply, Submodule.Quotient.restrictScalarsEquiv_symm_mk, Algebra.Presentation.differentials.comm₂₃', PiTensorProduct.lift_comp_reindex, SemimoduleCat.hom_inv_associator, conjRingEquiv_apply_apply, Matrix.toLpLin_symm_id, LieModule.toLinearMap_maxTrivLinearMapEquivLieModuleHom_symm, Rep.leftRegularHomEquiv_symm_apply, symm_trans_apply, Module.Basis.det_map', Submodule.mulRightMap_eq_mulMap_comp, kroneckerTMulLinearEquiv_symm_kroneckerTMul, MultilinearMap.freeFinsuppEquiv_def, ZMod.invDFT_apply', LinearMap.GeneralLinearGroup.congrLinearEquiv_apply, image_eq_preimage_symm, ContinuousLinearEquiv.toSpanNonzeroSingleton_symm_apply, LinearIsometryEquiv.withLpProdCongr_symm_apply, conj_symm_conj, coe_curry_symm, IsBaseChange.equiv_symm_apply, TensorProduct.congr_symm_tmul, Submodule.orderIsoMapComap_symm_apply', exteriorPower.oneEquiv_symm_apply, QuadraticForm.tensorLId_symm_apply, LinearMap.BilinForm.toMatrix_symm, Rep.ofMulActionSubsingletonIsoTrivial_inv_hom, symm_symm, piCongrLeft'_symm_apply, piCongrRight_symm, TensorProduct.finsuppScalarRight_symm_apply_single, symm_smul, rTensor_symm_tmul, Complex.equivRealProdLm_symm_apply, det_conj, LinearMap.apply_toPerfPair_flip, WithCStarModule.linearEquiv_symm_apply, Complex.equivRealProdLm_symm_apply_im, AddEquiv.toIntLinearEquiv_symm, symm_comp_eq, IsLocalizedModule.linearEquiv_symm_apply, Representation.coind'_apply_apply, GradedTensorProduct.mulHom_apply, Module.DirectLimit.linearEquiv_symm_mk, Module.Basis.repr_symm_single_one, ModuleCat.hom_inv_rightUnitor, Equiv.tensorProductAssoc_def, Matrix.compLinearEquiv_symm_apply, Submodule.prodEquivOfIsCompl_symm_apply_left, transvection.symm_eq, DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem, dualMap_symm, uniqueProd_symm_apply, Matrix.toLinOfInv_symm_apply, Polynomial.degreeLT.addLinearEquiv_symm_apply_inl, Polynomial.degreeLT.addLinearEquiv_symm_apply_inr_basis, Submodule.map_equiv_eq_comap_symm, RootPairing.Equiv.coweightMap_coweightEquiv_symm, Algebra.IsPushout.cancelBaseChangeAux_symm_tmul, Representation.asModuleEquiv_symm_map_rho, LinearMap.isPairSelfAdjoint_equiv, LinearMap.toMatrix₂_symm, AdjoinRoot.powerBasisAux'_repr_symm_apply, Rep.coindIso_inv_hom_hom, TensorProduct.directSumLeft_symm_lof_tmul, Representation.ofModule_asModule_act, LinearMap.quotKerEquivRange_symm_apply_image, LieAlgebra.IsKilling.lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg, TensorProduct.tensorQuotEquivQuotSMul_symm_mk, Submodule.Quotient.equiv_symm_apply, QuadraticForm.dualProdIsometry_toFun, QuadraticForm.tensorAssoc_symm_apply, PiTensorProduct.isEmptyEquiv_symm_apply, PiTensorProduct.subsingletonEquiv_symm_apply', Matrix.toLinearMap₂_symm, Algebra.Extension.H1Cotangent.equivOfFormallySmooth_symm, comp_symm_cancel_right, symm_bijective, CharacterModule.homEquiv_symm_apply_apply_apply, starL'_symm_apply, PiTensorProduct.tmulEquivDep_symm_apply, Submodule.toLinearMap_prodEquivOfIsCompl_symm, LieAlgebra.IsKilling.lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg_aux, finTwoArrow_symm_apply, MultilinearMap.curryFinFinset_symm_apply_const, toEquiv_symm, coe_symm_mk, Pi.orthonormalBasis.toBasis, LinearIndependent.linearCombinationEquiv_symm_apply, SpecialLinearGroup.congr_linearEquiv_apply_apply, DFinsupp.sigmaCurryLEquiv_symm_apply, Finsupp.llift_symm_apply, Representation.linHom.invariantsEquivRepHom_symm_apply_coe, toModuleIsoₛ_inv, AddEquiv.coe_toLinearEquiv_symm, CochainComplex.HomComplex.Cochain.leftShiftLinearEquiv_symm_apply, Module.AEval.of_symm_X_smul, sumPiEquivProdPi_symm_apply, Finsupp.supportedEquivFinsupp_symm_apply_coe_support_val, symmEquiv_apply_apply, Function.Exact.split_tfae', Algebra.TensorProduct.linearEquivIncludeRange_symm_tmul, symm_conj_apply, DFinsupp.lsum_symm_apply, conjAlgEquiv_apply_apply, algEquivOfRing_symm_apply, Submodule.quotientEquivOfIsCompl_symm_apply, Module.Relations.Solution.IsPresentation.uniq_symm_var, conj_conj_symm, AdicCompletion.congr_symm_apply, Polynomial.degreeLT.addLinearEquiv_symm_apply, eq_comp_symm, AffineMap.toConstProdLinearMap_symm_apply, LocallyConstant.congrLeftₗ_symm_apply_apply, Matrix.toLin'OfInv_symm_apply, Finsupp.linearCombination_restrict, Algebra.Presentation.differentials.comm₁₂, Submodule.map_symm_eq_iff, RootPairing.ker_rootForm_eq_dualAnnihilator, QuadraticAlgebra.linearEquivTuple_symm_apply, DirectSum.decomposeLinearEquiv_symm_comp_lof, IsLocalizedModule.iso_symm_apply_aux, ofInjectiveEndo_right_inv, IsBaseChange.endHom_apply, Module.Basis.parallelepiped_eq_map, TensorProduct.piScalarRight_symm_single, ContinuousAlternatingMap.piLinearEquiv_symm_apply, LinearMap.polyCharpolyAux_map_aeval, prodUnique_symm_apply, TensorProduct.quotTensorEquivQuotSMul_symm_mk, AdicCompletion.of_ofLinearEquiv_symm, Trivialization.coordChangeL_symm_apply, SpecialLinearGroup.toLinearEquiv_symm_apply, Submodule.quotientPi_symm_apply, LinearMap.BilinForm.comp_congr, prodCongr_symm, LinearMap.kerComplementEquivRange_symm_apply, PiLp.basisFun_eq_pi_basisFun, Module.Basis.end_repr_symm_apply, PeriodPair.latticeEquiv_symm_apply, MvPolynomial.rTensor_symm_apply_single, Complex.equivRealProdLm_symm_apply_re, Finsupp.lcongr_symm_single, sigmaFinsuppLequivDFinsupp_symm_apply, TensorProduct.leftComm_symm_tmul, Finsupp.domLCongr_symm, ofEq_symm, Rep.finsuppTensorRight_inv_hom, DirectSum.decomposeLinearEquiv_symm_apply, isSymmetric_symm_iff, RingEquiv.toSemilinearEquiv_symm_apply, symmEquiv_symm_apply_apply, TensorProduct.assoc_tensor'', Matrix.toLinearMapₛₗ₂_symm, Trivialization.coe_coordChangeL', WithCStarModule.map_top_submodule, AlgHom.toLinearMap_fromOpposite, Module.Basis.coe_ofRepr, AdicCompletion.tensor_map_id_left_eq_map, CoalgEquiv.symm_toLinearEquiv, Module.End.mem_invtSubmodule_symm_iff_le_map, CharacterModule.dual_rTensor_conj_homEquiv, CoalgEquiv.symm_toCoalgHom, Submodule.mulLeftMap_eq_mulMap_comp, symm_mk, TensorProduct.directSum_symm_lof_tmul, Rep.coindResAdjunction_homEquiv_symm_apply, continuous_symm, LinearMap.IsProj.eq_conj_prod_map', submoduleMap_symm_apply, Polynomial.degreeLT.addLinearEquiv_symm_apply', toSpanNonzeroSingleton_symm_apply_smul, CoalgEquiv.coe_symm_toLinearEquiv, congrQuadraticMap_symm_apply, BialgEquiv.trans_symm_apply, Finsupp.sumFinsuppLEquivProdFinsupp_symm_inr, Module.Dual.baseChange_baseChange, symm_trans_cancel_right, PiTensorProduct.map_comp_reindex_symm, RootPairing.polarizationEquiv_symm_apply_coroot, coe_rTensor_symm, trans_symm_cancel_left, MeasureTheory.ComplexMeasure.equivSignedMeasureₗ_symm_apply, piCurry_symm_apply, Submodule.map_unop_one, symmEquiv_apply_symm_apply, Representation.coinvariantsTprodLeftRegularLEquiv_symm_apply, Module.apply_evalEquiv_symm_apply, Matrix.reindexLinearEquiv_symm, Module.Basis.mulOpposite_repr_eq, Module.Basis.coe_repr_symm, Rep.finsuppTensorLeft_inv_hom, CliffordAlgebra.reverseEquiv_symm_apply, Finsupp.linearEquivFunOnFinite_symm_single, TensorProduct.rid_symm_apply, CategoryTheory.Preadditive.homSelfLinearEquivEndMulOpposite_symm_apply, Submodule.prodEquivOfIsCompl_symm_apply_right, LieAlgebra.IsKilling.cartanEquivDual_symm_apply_mem_corootSpace, Rep.leftRegularHomEquiv_symm_single, Algebra.Generators.CotangentSpace.compEquiv_symm_inr, prodProdProdComm_symm, AlgEquiv.ofLinearEquiv_symm, Matrix.kroneckerAlgEquiv_symm_apply, ofTop_symm_apply, Rep.leftRegularTensorTrivialIsoFree_inv_hom, Algebra.TensorProduct.basis_repr_symm_apply, Ideal.pi_mkQ_rTensor, KaehlerDifferential.quotKerTotalEquiv_symm_apply, LinearMap.quotKerEquivOfSurjective_symm_apply, TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_symm_tmul, smulOfNeZero_symm_apply, apply_symm_apply, LinearMap.GeneralLinearGroup.congrLinearEquiv_symm, dualTensorHomEquivOfBasis_symm_cancel_left, Module.Invertible.rTensorEquiv_symm_apply_apply, Complex.selfAdjointEquiv_symm_apply, withLpCongr_symm, AddEquiv.coe_symm_toNatLinearEquiv, TensorProduct.AlgebraTensorModule.cancelBaseChange_symm_tmul, TensorProduct.lift.equiv_symm_apply, det_mul_det_symm, CategoryTheory.Linear.homCongr_symm_apply, ModuleCat.uliftFunctor_map, Rep.linearization_δ_hom, conjRingEquiv_symm_apply_apply, Matrix.uniqueLinearEquiv_symm_apply, Algebra.TensorProduct.linearEquivIncludeRange_symm_toLinearMap, RootPairing.rootSpan_dualAnnihilator_map_eq_iInf_ker_root', finsuppTensorFinsupp_symm_single, addMonoidHomLequivInt_symm_apply, LinearMap.det_conj, Submodule.lTensorOne_symm_apply, RootPairing.Equiv.weightEquiv_symm_weightMap, multilinearCurryLeftEquiv_symm_apply, TensorProduct.toLinearMap_symm_lid, ContinuousLinearEquiv.homothety_inverse, ofSubsingleton_symm_apply, Finsupp.linearCombination_eq_fintype_linearCombination, Finsupp.supportedEquivFinsupp_symm_apply_coe, coe_lTensor_symm, MultilinearMap.curryMidLinearEquiv_symm_apply, finsuppLequivDFinsupp_symm_apply, Algebra.TensorProduct.equivPiOfFiniteBasis_symm_apply, TensorProduct.finsuppRight_symm_apply_single, Trivialization.coe_coordChangeL, CoalgEquiv.trans_symm_apply, LieModule.coe_maxTrivLinearMapEquivLieModuleHom_symm, KaehlerDifferential.tensorKaehlerEquivOfFormallyEtale_symm_D_algebraMap, LinearMap.BilinForm.apply_toDual_symm_apply, Module.Basis.linearMap_repr_symm_apply, lieConj_symm, Submodule.comap_unop_pow, sumArrowLequivProdArrow_symm_apply_inl, MeasureTheory.Measure.addHaar_preimage_linearEquiv, GradedTensorProduct.of_symm_of, Matrix.toLpLin_symm_comp, TensorProduct.tensorTensorTensorAssoc_symm_tmul, coe_inv_det, Module.Relations.Solution.IsPresentation.postcomp_uniq_symm, LinearMap.toMatrix₂_symm', ZMod.invDFT_def, Representation.coinvariantsFinsuppLEquiv_symm_apply, Subalgebra.linearEquivOp_symm_apply_coe, TensorProduct.equivFinsuppOfBasisRight_symm, NumberField.mixedEmbedding.fundamentalCone.expMapBasis_apply, ContinuousLinearMap.toSpanSingletonLE_symm_apply, MulOpposite.comul_def, Coalgebra.coassoc_symm, exteriorPower.alternatingMapLinearEquiv_symm_apply, TensorProduct.lid'_symm_apply, ofInjectiveEndo_left_inv, Function.Exact.linearEquivOfSurjective_symm_apply, LieAlgebra.IsKilling.traceForm_coroot, Module.Basis.mapCoeffs_repr, Ideal.cotangentEquivIdeal_symm_apply, Polynomial.taylorLinearEquiv_symm, DirectSum.IsInternal.ofBijective_coeLinearMap_of_ne, lTensor_symm_tmul, MulOpposite.opLinearEquiv_symm_toAddEquiv, Polynomial.degreeLT.addLinearEquiv_symm_apply_inl_basis, ModuleCat.hom_inv_leftUnitor, AlternatingMap.domDomCongrₗ_symm_apply, TensorProduct.directLimitLeft_symm_of_tmul, Orientation.map_apply, RootPairing.corootSpan_dualAnnihilator_le_ker_rootForm, LinearMap.toMvPolynomial_eval_eq_apply, KaehlerDifferential.mvPolynomialBasis_repr_symm_single, arrowCongrAddEquiv_apply, Rep.freeLift_hom, coe_ofTop_symm_apply, baseChange_symm, Submodule.fstEquiv_symm_apply_coe, det_coe_symm, trans_symm, DirectSum.linearEquivFunOnFintype_symm_single, symm_smul_apply, fromModuleCatToModuleCatLinearEquiv_symm_apply_coe, PiTensorProduct.lift_comp_reindex_symm, PiTensorProduct.ofDFinsuppEquiv_symm_single_tprod, coe_toEquiv_symm, Submodule.equivOpposite_symm_apply, Basis.equivFun_symm_single, TensorProduct.map_map_comp_assoc_symm_eq, MvPolynomial.scalarRTensor_symm_apply_single, symm_trans_cancel_left, MultilinearMap.constLinearEquivOfIsEmpty_symm_apply, RootPairing.rootSpan_dualAnnihilator_map_eq, symmEquiv_symm_apply_symm_apply, RootPairing.Equiv.inv_coweightMap, AlternatingMap.domLCongr_apply, Algebra.Generators.cotangentCompLocalizationAwayEquiv_symm_inl, Algebra.Generators.cotangentCompLocalizationAwayEquiv_symm_inr, Submodule.comap_unop_one, CategoryTheory.Abelian.Ext.linearEquiv₀_symm_apply, LinearMap.quotientInfEquivSupQuotient_symm_apply_left, LinearMap.lflip_symm, LinearMap.coe_toContinuousLinearMap_symm, Submodule.mulMap_op, AddEquiv.toNatLinearEquiv_symm, ofIsUnitDet_symm_apply, Finsupp.mapDomain.linearEquiv_symm, Matrix.piLinearEquiv_symm_apply, LinearMap.extendScalarsOfIsLocalizationEquiv_symm_apply, Rep.freeLiftLEquiv_symm_apply, ContinuousLinearMap.coprodEquiv_symm_apply, Matrix.coe_ofLinearEquiv_symm, Rep.MonoidalClosed.linearHomEquivComm_symm_hom, AffineBasis.coord_smul, IsBaseChange.linearMapLeftRightHom_apply, CochainComplex.HomComplex.Cochain.rightShiftLinearEquiv_symm_apply, LinearMap.coprodEquiv_symm_apply, arrowCongrAddEquiv_symm_apply, map_mem_invtSubmodule_iff, Matrix.SpecialLinearGroup.toLin'_symm_apply, MulOpposite.coe_opLinearEquiv_symm_toLinearMap, TensorProduct.assoc_tensor, Submodule.dualAnnihilator_map_linearEquiv_flip_symm, LinearMap.transvection.congr, Module.Basis.repr_symm_apply, Module.Dual.congr_apply_apply, coeFn_toContinuousLinearEquivOfContinuous_symm, Module.Basis.coe_ofEquivFun, ofLinear_symm_apply, alternatingMapCongrRight_symm_apply_apply, WithLp.linearEquiv_symm_apply, MultilinearMap.curryFinFinset_symm_apply, Module.Invertible.rightCancelEquiv_comp_rTensor_comp_symm, WithLp.coe_symm_linearEquiv, TensorProduct.finsuppScalarLeft_symm_apply_single, CategoryTheory.Abelian.Ext.homLinearEquiv_symm_apply, Module.compHom.toLinearEquiv_symm_apply, Equiv.tensorProductComm_def, Submodule.map_dualCoannihilator_linearEquiv_flip, Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso_inv_f_hom_apply, TensorProduct.toLinearMap_symm_rid, ContinuousLinearMap.toUniformConvergenceCLM_symm_apply, TensorProduct.assoc_tensor', Finsupp.linearCombination_eq_fintype_linearCombination_apply, transvection.symm_eq', Submodule.dualQuotEquivDualAnnihilator_symm_apply_mk, ofInjective_symm_apply, Submodule.mem_map_equiv, LinearMap.BilinForm.congr_symm, TensorProduct.tensorQuotientEquiv_symm_apply_tmul_mk, coe_symm_mk', Algebra.kerTensorProductMapIdToAlgHomEquiv_symm_apply, GradedTensorProduct.symm_of_of, RootPairing.corootSpan_dualAnnihilator_map_eq_iInf_ker_coroot', Submodule.map_equivMapOfInjective_symm_apply, Finsupp.supportedEquivFinsupp_symm_apply_coe_apply, RootPairing.invtSubmodule_reflection_of_invtSubmodule_coreflection, SemimoduleCat.homLinearEquiv_symm_apply, Module.Basis.det_map, LieModule.shiftedGenWeightSpace.shift_symm_apply, ofLinear_symm_toLinearMap, TensorProduct.directLimitRight_symm_of_tmul, Finsupp.finsuppProdLEquiv_symm_apply_apply, ModuleCat.monoidalClosed_pre_app, TensorProduct.assoc_symm_tmul, AdicCompletion.sumEquivOfFintype_symm_apply, CategoryTheory.InducedCategory.homLinearEquiv_symm_apply_hom, trans_dualMap_symm_flip, TensorProduct.comm_symm_tmul, LinearMap.quotientInfEquivSupQuotient_symm_apply_right, exteriorPower.zeroEquiv_symm_apply, Module.AEval.mem_mapSubmodule_apply, TensorProduct.equivFinsuppOfBasisRight_symm_apply, self_trans_symm, Matrix.toLpLinAlgEquiv_symm_apply, LinearMap.lTensor_tensor, LieEquiv.ofBijective_invFun, Submodule.sndEquiv_symm_apply_coe, AlternatingMap.domLCongr_symm, Submodule.comapSubtypeEquivOfLe_symm_apply, TensorProduct.AlgebraTensorModule.rightComm_symm, toFGModuleCatIso_inv, RootPairing.ker_corootForm_eq_dualAnnihilator, LinearMap.lsum_symm_apply, Module.Basis.coord_equivFun_symm, TensorProduct.quotTensorEquivQuotSMul_symm_comp_mkQ, dotProduct_toMatrix₂_mulVec, Matrix.entryLinearMap_eq_comp, trans_symm_cancel_right, LinearMap.IsPerfectCompl.isCompl_left, symm_comp_cancel_left, TensorProduct.rightComm_def, Submodule.map_dualAnnihilator_linearEquiv_flip_symm, TensorProduct.directSumRight_symm_lof_tmul, ContinuousLinearEquiv.arrowCongrSL_toLinearEquiv_symm_apply, symm_prodComm, MultilinearMap.fromDFinsuppEquiv_symm_apply, AdicCompletion.ofLinearEquiv_symm_of, InnerProductSpace.symm_toEuclideanLin_rankOne, contractLeft_assoc_coevaluation', DirectSum.lequivProdDirectSum_symm_apply, Matrix.kroneckerStarAlgEquiv_symm_apply, Module.dualProdDualEquivDual_symm_apply, TensorProduct.assocIsometry_symm_apply, Fin.consLinearEquiv_symm_apply, Submodule.comap_unop_mul, LinearMap.lTensor_eqLocus_subtype_tensoreqLocusEquiv_symm, LinearMap.IsPositive.toLinearMap_symm, apply_ofBijective_symm_apply, TensorProduct.lid_tensor, ContinuousLinearEquiv.coe_symm_toLinearEquiv, DirectSum.linearEquivFunOnFintype_symm_coe, MultilinearMap.ofSubsingletonₗ_symm_apply, MultilinearMap.currySumEquiv_symm_apply, GradedTensorProduct.of_symm_one, Submodule.coe_prodEquivOfClosedCompl_symm, Rep.indResHomEquiv_symm_apply_hom, QuadraticForm.dualProdProdIsometry_invFun, RootPairing.Equiv.inv_weightMap, finsuppLEquivDirectSum_symm_lof, Trivialization.linearEquivAt_symm_apply, Finsupp.curryLinearEquiv_symm_apply, Module.Basis.coord_toDualEquiv_symm_apply, Finsupp.lsum_symm_apply, Module.Invertible.leftCancelEquiv_comp_lTensor_comp_symm, Module.Basis.coord_repr_symm, TensorProduct.tensorQuotEquivQuotSMul_symm_comp_mkQ, Module.Dual.congr_symm_apply_apply, LinearMap.rTensor_tensor, funCongrLeft_symm, Module.FinitePresentation.linearEquivMapExtendScalars_symm_apply, symm_ofLexLinearEquiv, AddEquiv.coe_symm_toIntLinearEquiv, IsNoetherian.equivPUnitOfProdInjective_symm_apply, PiTensorProduct.lift_symm

LinearIsometryEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	154 math math: DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply, coe_symm_toIsometryEquiv, hasFDerivAt_iff_hasGradientAt, coe_ofLinearIsometry_symm, Submodule.orthogonalDecomposition_symm_apply, IsometryEquiv.toRealLinearIsometryEquiv_symm_apply, piLpCongrRight_symm, Submodule.toLinearEquiv_orthogonalDecomposition_symm, symm_bijective, withLpProdComm_symm, iteratedFDerivWithin_succ_eq_comp_right, InnerProductSpace.toDual_symm_apply, IsHilbertSum.linearIsometryEquiv_symm_apply, coe_symm_toHomeomorph, MeasureTheory.charFun_toDual_symm_eq_charFunDual, Submodule.reflection_symm, iteratedFDerivWithin_succ_eq_comp_left, conjStarAlgEquiv_apply, toHomeomorph_symm, apply_symm_apply, Orientation.volumeForm_map, HasFTaylorSeriesUpToOn.zero_eq', image_eq_preimage_symm, adjoint_eq_symm, IsHilbertSum.hasSum_linearIsometryEquiv_symm, Submodule.coe_orthogonalDecomposition_symm, inner_map_eq_flip, Quaternion.linearIsometryEquivTuple_symm_apply, ContinuousAlternatingMap.ofSubsingletonLIE_symm_apply, toLinearEquiv_symm, adjoint_toLinearMap_eq_symm, PiLp.sumPiLpEquivProdLpPiLp_symm_apply_ofLp, ContinuousMultilinearMap.curryFinFinset_symm_apply_const, comp_hasFDerivWithinAt_iff', symm_prodComm, OrthonormalBasis.coe_ofRepr, Orientation.rotation_symm, self_trans_symm, Orientation.rotation_map, ofTop_symm_apply_coe, HasFDerivWithinAt.hasGradientWithinAt, Orthonormal.equiv_symm, LinearMap.IsSymmetric.diagonalization_symm_apply, coe_prodAssoc_symm, ContinuousMultilinearMap.ofSubsingletonₗᵢ_symm_apply, preimage_ball, continuousMultilinearCurryFin0_symm_apply, Orientation.rightAngleRotation_symm, continuousMultilinearCurryRightEquiv_symm_apply, MeasureTheory.lpMeasToLpTrimLie_symm_indicator, ContinuousMap.linearIsometryBoundedOfCompact_symm_apply, Module.Basis.coe_toOrthonormalBasis_repr_symm, ContinuousLinearMap.adjointAux_apply, LinearMap.isSymmetric_linearIsometryEquiv_conj_iff, ContinuousAlternatingMap.constOfIsEmptyLIE_symm_apply, LinearEquiv.extendOfIsometry_symm_eq, Orientation.rightAngleRotation_map', LinearEquiv.extendOfIsometry_symm_apply, coe_symm_toLinearEquiv, LinearMap.isPositive_linearIsometryEquiv_conj_iff, Orthonormal.linearIsometryEquiv_symm_apply_single_one, toContinuousLinearEquiv_symm, Submodule.starProjection_map_apply, symm_trans_self, withLpProdCongr_symm_apply, RCLike.realLinearIsometryEquiv_symm_apply, Complex.orthonormalBasisOneI_repr_symm_apply, symm_comp_self, iteratedDeriv_eq_equiv_comp, RCLike.complexLinearIsometryEquiv_symm_apply, OrthonormalBasis.sum_repr_symm, Submodule.reflection_map, ContinuousMultilinearMap.curryFinFinset_symm_apply_piecewise_const, FormalMultilinearSeries.leftInv_coeff_one, HilbertBasis.repr_symm_single, MulOpposite.opLinearIsometryEquiv_symm_apply, ContinuousAlternatingMap.piLIE_symm_apply_apply, OrthonormalBasis.measurePreserving_repr_symm, piLpCurry_symm_apply, coe_inv, symm_conjStarAlgEquiv_apply_apply, symm_apply_apply, submoduleMap_symm_apply_coe, symm_trans, symm_rTensor, TensorProduct.commIsometry_symm, OrthonormalBasis.equiv_symm, PiTensorProduct.liftIsometry_symm_apply, iteratedFDeriv_succ_eq_comp_right, coe_lpBCFₗᵢ_symm, coe_lpPiLpₗᵢ_symm, ContinuousMultilinearMap.piₗᵢ_symm_apply, prodComm_symm_apply, symm_conjStarAlgEquiv, coe_symm_toContinuousLinearEquiv, OrthonormalBasis.repr_symm_single, Orientation.oangle_map, Submodule.reflection_map_apply, withLpProdUnique_symm_apply, self_comp_symm, toMeasurableEquiv_symm, Orientation.kahler_map, ContinuousMultilinearMap.prodL_symm_apply, IsHilbertSum.linearIsometryEquiv_symm_apply_dfinsupp_sum_single, Orientation.areaForm_map, iteratedFDeriv_zero_eq_comp, continuousMultilinearCurryRightEquiv_symm_apply', TensorProduct.congrIsometry_symm, iteratedDerivWithin_eq_equiv_comp, hasFDerivWithinAt_iff_hasGradientWithinAt, comp_hasFDerivAt_iff', iteratedFDerivWithin_zero_eq_comp, IsometryEquiv.coe_toRealLinearIsometryEquivOfMapZero_symm, symm_units_smul, IsHilbertSum.linearIsometryEquiv_symm_apply_single, symm_smul_apply, star_eq_symm, coe_symm_trans, Orientation.rightAngleRotation_map, ContinuousLinearMap.flipₗᵢ'_symm, HasFDerivAt.hasGradientAt, rotation_symm, continuousMultilinearCurryLeftEquiv_symm_apply, iteratedFDeriv_succ_eq_comp_left, toIsometryEquiv_symm, continuousMultilinearCurryFin1_symm_apply, PadicInt.mahlerEquiv_symm_apply, HasFTaylorSeriesUpTo.zero_eq', ContinuousMultilinearMap.curryFinFinset_symm_apply, Complex.isometryOfOrthonormal_symm_apply, inv_def, symm_symm, piLpCongrLeft_symm, ContinuousAlternatingMap.prodLIE_symm_apply, Orientation.rotation_symm_apply, ContinuousLinearMap.flipₗᵢ_symm, FormalMultilinearSeries.rightInv_coeff_one, symm_lTensor, ContinuousLinearMap.fpowerSeries_apply_one, MeasureTheory.lpMeasToLpTrimLie_symm_toLp, conjStarAlgEquiv_apply_apply, withLpUniqueProd_symm_apply, ContinuousLinearMap.fpowerSeriesBilinear_apply_one, TensorProduct.assocIsometry_symm_apply, TensorProduct.lidIsometry_symm_apply, HilbertBasis.hasSum_repr_symm, withLpProdAssoc_symm_apply, ofEq_symm, preimage_sphere, symm_neg, ContinuousMultilinearMap.curryMidEquiv_symm_apply, coe_symm_toMeasurableEquiv, preimage_closedBall, Complex.conjLIE_symm

LinearPMap.IsFormalAdjoint

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	LinearPMap.IsFormalAdjoint	—	—	inner_conj_symm

List.Disjoint

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	—	—	—	—

List.IsRotated

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	List.IsRotated	—	—	List.rotate_rotate List.rotate_length_mul

Lists.Equiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Lists.Equiv	—	—	—

ManyOneEquiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	ManyOneEquiv	—	—	—

Mathlib.Tactic.LibraryRewrite.Rewrite

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

Mathlib.Tactic.LibraryRewrite.RewriteInterface

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

Mathlib.Tactic.LibraryRewrite.RewriteLemma

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

Matrix.IsAdjMatrix

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	Matrix.IsAdjMatrix	Matrix.IsSymm	—	—

MeasurableEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	82 math math: preimage_symm, symm_symm, MeasureTheory.MeasurePreserving.preErgodic_conjugate_iff, symm_mulRight₀, symm_bijective, symm_addLeft, piFinTwo_symm_apply, withDensity_ofReal_map_symm_apply_eq_integral_abs_det_fderiv_mul, self_comp_symm, MeasureTheory.MeasurePreserving.symm, comap_apply, toLp_symm_apply, symm_inv, funUnique_symm_apply, map_symm_map, symm_mulLeft₀, quasiMeasurePreserving_symm, coe_IicProdIoc_symm, symm_preimage_preimage, map_apply_eq_iff_map_symm_apply_eq, ProbabilityTheory.Kernel.prodAssoc_symm_prod, comap_symm, curry_symm_apply, ofInvolutive_symm, symm_mulRight, unitInterval.symmMeasurableEquiv_symm_apply, apply_symm_apply, withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul, piUnique_symm_apply, symm_mulLeft, Complex.measurableEquivRealProd_symm_polarCoord_symm_apply, MeasureTheory.Measure.infinitePi_map_curry_symm, piEquivPiSubtypeProd_symm_apply, map_map_symm, symm_apply_apply, Complex.measurableEquivPi_symm_apply, image_symm, EuclideanSpace.volume_preserving_symm_measurableEquiv_toLp, symm_addRight, MeasureTheory.Measure.compProd_assoc, piCurry_symm_apply, coe_curry_symm, coe_sumPiEquivProdPi_symm, symm_trans_self, MeasureTheory.MeasurePreserving.ergodic_conjugate_iff, image_eq_preimage_symm, LinearIsometryEquiv.toMeasurableEquiv_symm, finTwoArrow_symm_apply, MeasurableEmbedding.equivRange_symm_apply_mk, Homeomorph.toMeasurableEquiv_symm_coe, symm_smul₀, symm_comp_self, ProbabilityTheory.Kernel.compProd_assoc, ProbabilityTheory.Kernel.compProd_def, coe_toEquiv_symm, symm_smul, withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul', Ergodic.symm, setOf_symm_apply, symm_refl, self_trans_symm, MeasureTheory.measurePreserving_sumPiEquivProdPi_symm, coe_piCurry_symm, gaussianReal_map_symm_apply, MeasureTheory.Measure.pi_map_piOptionEquivProd, Ergodic.symm_iff, eq_image_iff_symm_image_eq, Complex.measurableEquivRealProd_symm_apply, MeasureTheory.volume_measurePreserving_sumPiEquivProdPi_symm, MeasureTheory.Measure.infinitePi_map_piCurry_symm, symm_vadd, EuclideanSpace.coe_measurableEquiv_symm, symm_neg, map_symm, ProbabilityTheory.Kernel.map_apply_eq_iff_map_symm_apply_eq, unitInterval.symm_symmMeasurableEquiv, MeasureTheory.MeasurableEquiv.measurePreserving_symm, coe_toLp_symm, piMeasurableEquivTProd_symm_apply, symm_mk, piFinSuccAbove_symm_apply, LinearIsometryEquiv.coe_symm_toMeasurableEquiv

MeasureTheory.AEDisjoint

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	MeasureTheory.AEDisjoint	—	—	eq_1 Set.inter_comm

MeasureTheory.Measure.MutuallySingular

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	MeasureTheory.Measure.MutuallySingular	—	—	MeasurableSet.compl compl_compl

MeasureTheory.MeasurePreserving

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	MeasureTheory.MeasurePreserving DFunLike.coe MeasurableEquiv EquivLike.toFunLike MeasurableEquiv.instEquivLike	MeasurableEquiv.symm	—	MeasurableEquiv.measurable map_eq MeasureTheory.Measure.map_map MeasurableEquiv.symm_comp_self MeasureTheory.Measure.map_id

MeasureTheory.VectorMeasure.MutuallySingular

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	MeasureTheory.VectorMeasure.MutuallySingular	—	—	MeasurableSet.compl compl_compl

Metric.AreSeparated

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Metric.AreSeparated	—	—	PseudoEMetricSpace.edist_comm

ModelWithCorners

Definitions

Name	Category	Theorems
symm 📖	CompOp	32 math math: symm_continuousWithinAt_comp_right_iff, mdifferentiableOn_symm, image_eq, modelWithCornersSelf_coe_symm, mk_symm, uniqueDiffOn_preimage, symm_comp_self, prod_symm_apply, OpenPartialHomeomorph.extend_target, modelWithCorners_prod_coe_symm, OpenPartialHomeomorph.extend_coe_symm, symm_map_nhdsWithin_range, leftInverse, contDiffWithinAtProp_self_target, right_inv, continuousOn_symm, hasMFDerivWithinAt_symm, continuousWithinAt_symm, continuous_symm, symm_map_nhdsWithin_image, extChartAt_target, continuousAt_symm, extChartAt_coe_symm, contMDiffOn_model_symm, coe_transContinuousLinearEquiv_symm, uniqueDiffOn_preimage_source, rightInvOn, left_inv, mdifferentiableWithinAt_symm, differentiableWithinAtProp_self_target, toHomeomorph_symm_apply, toPartialEquiv_coe_symm

Monovary

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Monovary PartialOrder.toPreorder SemilatticeInf.toPartialOrder Lattice.toSemilatticeInf DistribLattice.toLattice instDistribLatticeOfLinearOrder	—	—	le_of_not_gt LT.lt.not_ge

MonovaryOn

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	MonovaryOn PartialOrder.toPreorder SemilatticeInf.toPartialOrder Lattice.toSemilatticeInf DistribLattice.toLattice instDistribLatticeOfLinearOrder	—	—	le_of_not_gt LT.lt.not_ge

MoritaEquivalence

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

MulEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	332 math math: CommGrpCat.uliftFunctor_map, strictMono_symm, Submonoid.mem_map_equiv, MulAut.symm_inv, galRestrict_symm_algebraMap_apply, Subgroup.centerCongr_symm_apply_coe, zpowersMulHom_symm_apply, SemidirectProduct.congr_symm_apply_left, monoidHomCongrLeft_apply, AddEquiv.toMultiplicative_apply_symm_apply, coe_monoidHom_symm_comp_coe_monoidHom, CoxeterSystem.reindex_mulEquiv, Subgroup.comap_equiv_eq_map_symm, WithZero.withZeroUnitsEquiv_symm_apply_coe, mulEquivOfOrderOfEq_symm_apply_gen, AddAutAdditive_apply_symm_apply, MulAut.inv_apply, toContinuousMulEquiv_symm_apply, CategoryTheory.InducedCategory.endEquiv_symm_apply_hom, MulAut.conjNormal_symm_apply, val_inv_unitarySubgroupUnitsEquiv_symm_apply_coe, Submonoid.leftInvEquiv_symm_mul, CategoryTheory.Iso.commGroupIsoToMulEquiv_symm_apply, OrderMonoidIso.withZero_symm_apply_symm_apply, IsCyclic.val_mulAutMulEquiv_apply, symm_mk, symm_comp_self, Submonoid.fromLeftInv_leftInvEquiv_symm, MulAction.stabilizerEquivStabilizer_symm_apply, associatesEquivOfUniqueUnits_symm_apply, Subsemigroup.centerCongr_symm_apply_coe, OrderMonoidIso.val_unitsCongr_symm_apply, eq_symm_apply, Set.unitEquivUnitsInteger_symm_apply_coe, Localization.mulEquivOfQuotient_symm_monoidOf, val_unitarySubgroupUnitsEquiv_symm_apply_coe, comapSubgroup_symm_apply, Subgroup.comap_equiv_eq_map_symm', autEquivZmod_symm_apply_natCast, map_dvd_iff_dvd_symm, toSingleObjEquiv_inverse_map, SkewMonoidAlgebra.domCongrAlg_apply, Matrix.SpecialLinearGroup.center_equiv_rootsOfUnity'_symm_apply_coe_coe, NonUnitalSubsemiring.centerCongr_symm_apply_coe, CategoryTheory.Functor.FullyFaithful.autMulEquivOfFullyFaithful_symm_apply_hom, autAdjoinRootXPowSubCEquiv_symm_smul, MonoidHom.ofLeftInverse_symm_apply, AddOpposite.opMulEquiv_symm_apply, uliftZPowersHom_symm_apply, AddAutAdditive_symm_apply_symm_apply, Subsemiring.centerCongr_symm_apply_coe, prodMultiplicative_symm_apply, val_piUnits_symm_apply, MulAutMultiplicative_symm_apply_apply, opOp_symm_apply, monoidHomCongrLeftEquiv_apply, comp_symm_eq, op_apply_symm_apply, ZMod.AddAutEquivUnits_symm_apply, HNNExtension.inv_t_mul_of, HNNExtension.toSubgroupEquiv_neg_one, eq_symm_comp, StarMulEquiv.toMulEquiv_symm, Submonoid.LocalizationMap.ofMulEquivOfDom_comp_symm, Submonoid.topEquiv_symm_apply_coe, AddEquiv.toMultiplicativeLeft_apply_symm_apply, mk_coe', coprodPUnit_symm_apply, invFun_eq_symm, CategoryTheory.Aut.toEnd_apply, AffineEquiv.linear_equivUnitsAffineMap_symm_apply, uliftPowersHom_symm_apply, coprodAssoc_symm_apply_inr_inr, Subgroup.mem_map_equiv, AddEquiv.toMultiplicativeRight_apply_symm_apply, restrictRootsOfUnity_symm, Subsemigroup.topEquiv_symm_apply_coe, symm_trans_self, MulAut.coe_inv, unitary.linearIsometryEquiv_coe_symm_apply, withZero_apply_symm_apply, SemidirectProduct.congr'_apply_right, OrderMonoidIso.val_inv_unitsCongr_symm_apply, symm_monoidHomCongrRight, ofLeftInverse_symm_apply, Con.comapQuotientEquivOfSurj_symm_mk', QuotientGroup.congr_symm, Subgroup.map_symm_eq_iff_map_eq, val_inv_unitsNonZeroDivisorsEquiv_symm_apply_coe, Unitary.coe_symm_linearIsometryEquiv_apply, MulAutMultiplicative_symm_apply_symm_apply, toSingleObjEquiv_unitIso_hom, SpecialLinearGroup.congr_linearEquiv_symm, Submonoid.val_inv_unitsTypeEquivIsUnitSubmonoid_symm_apply, Nonneg.val_unitsEquivPos_symm_apply_coe, MonoidHom.restrictHomKerEquiv_symm_coe_apply, eq_comp_symm, GroupExtension.Equiv.coe_symm, IsFreeGroup.mulEquiv_def, AlgEquiv.symm_toMulEquiv, InfiniteGalois.mulEquivToLimit_symm_continuous, IsCyclic.val_inv_mulAutMulEquiv_apply, WithZero.withZeroUnitsEquiv_symm_strictMono, Monoid.PushoutI.NormalWord.cons_head, prodUnique_symm_apply, OrderMonoidIso.withZero_apply_symm_apply, coe_prodComm_symm, WithZero.unitsWithZeroEquiv_symm_apply, Subgroup.subgroupOfEquivOfLe_symm_apply_coe_coe, autEquivZmod_symm_apply_intCast, MonoidHom.toMulEquiv_symm_apply, MulOpposite.coe_symm_opMulEquiv, associatesNonZeroDivisorsEquiv_symm_mk_mk, UniqueFactorizationMonoid.normalizedFactorsEquiv_symm_apply, apply_eq_iff_symm_apply, refl_symm, IsFreeGroup.toFreeGroup_symm_apply, QuotientGroup.quotientKerEquivOfRightInverse_symm_apply, piCongrRight_symm, Units.mapEquiv_symm, withOneCongr_symm, self_comp_symm, SemidirectProduct.congr'_apply_left, symm_apply_eq, toSingleObjEquiv_counitIso_hom, SpecialLinearGroup.centerEquivRootsOfUnity_symm_apply, ConjAct.of_mul_symm_eq, Unitization.val_unitsFstOne_mulEquiv_quasiregular_symm_apply_coe, SemidirectProduct.congr'_symm_apply_right, toMonCatIso_inv, Unitization.val_inv_unitsFstOne_mulEquiv_quasiregular_symm_apply_coe, val_unitsNonZeroDivisorsEquiv_symm_apply_coe, NonUnitalSubring.centerCongr_symm_apply_coe, Subgroup.val_centerUnitsEquivUnitsCenter_symm_apply_coe, Monoid.IsTorsion.torsionMulEquiv_symm_apply_coe, MulHom.toMulEquiv_symm_apply, AddEquiv.toMultiplicativeLeft_symm_apply_symm_apply, Submonoid.centerToMulOpposite_symm_apply_coe, symm_monoidHomCongrRightEquiv, CategoryTheory.Functor.FullyFaithful.mulEquivEnd_symm_apply, toMultiplicative_toAdditive_symm_apply, CategoryTheory.PreGaloisCategory.autMulEquivAutGalois_symm_app, toCommGrpIso_inv, HNNExtension.of_mul_t, Subgroup.topEquiv_symm_apply_coe, Unitary.linearIsometryEquiv_coe_symm_apply, toSingleObjEquiv_unitIso_inv, WithZero.toMulBot_coe_ofAdd, cast_symm_apply, Con.quotientKerEquivOfRightInverse_symm_apply, mulAutArrow_apply_symm_apply, mapSubgroup_symm_apply, AlgEquiv.autCongr_symm, inv'_symm_apply, equivLike_inv_eq_symm, ofBijective_apply_symm_apply, CategoryTheory.Hom.mulEquivCongrRight_symm_apply, uniqueProd_symm_apply, Submonoid.centerCongr_symm_apply_coe, AffineEquiv.equivUnitsAffineMap_symm_apply_apply, symm_monoidHomCongrLeftEquiv, toSingleObjEquiv_counitIso_inv, galRestrictHom_symm_algebraMap_apply, freeGroupEquivCoprodI_symm_apply, Con.comapQuotientEquivOfSurj_symm_mk, Matrix.SpecialLinearGroup.toLin_equiv.symm_toLinearMap_eq, Subsemigroup.centerToMulOpposite_symm_apply_coe, PowerBasis.quotientEquivQuotientMinpolyMap_symm_apply, MulDistribMulAction.toMulEquiv_symm_apply, NumberField.RingOfIntegers.withValEquiv_apply, Localization.mulEquivOfQuotient_symm_mk, val_unitsCentralizerEquiv_symm_apply_coe, Submodule.unitsQuotEquivRelPic_symm_apply, symm_toContinuousMulEquiv, Equiv.permCongrHom_symm, symmEquiv_apply_apply, funUnique_symm_apply, AlgEquiv.val_algHomUnitsEquiv_symm_apply, CategoryTheory.Functor.FullyFaithful.homMulEquiv_symm_apply, Submonoid.map_equiv_eq_comap_symm, submonoidMap_symm_apply, mulEquivOfOrderOfEq_symm, Submonoid.LocalizationMap.mulEquivOfLocalizations_symm_apply, monoidEndToAdditive_symm_apply_apply, SemidirectProduct.mulEquivProd_symm_apply_left, AddEquiv.toMultiplicativeRight_symm_apply_symm_apply, symm_trans_apply, punitCoprod_symm_apply, coprodComm_symm_apply, AffineEquiv.equivUnitsAffineMap_symm_apply_symm_apply, abelianizationCongr_symm, multiplicativeAdditive_symm_apply, IsFreeGroup.toFreeGroup_apply, AddConstEquiv.equivUnits_symm_apply_apply, AlgEquiv.opComm_apply_symm_apply, coe_prodAssoc_symm, toSemigrpIso_inv, Subsemigroup.mem_map_equiv, MonCat.uliftFunctor_map, CategoryTheory.Functor.FullyFaithful.autMulEquivOfFullyFaithful_symm_apply_inv, MonoidAlgebra.domCongr_symm, ClassGroup.equivPic_symm_apply, CategoryTheory.Groupoid.vertexGroupIsomOfMap_symm_apply, symm_ofLexMulEquiv, SemidirectProduct.congr_symm_apply_right, RingEquiv.piCongrLeft_apply, AffineEquiv.equivUnitsAffineMap_symm_apply_toFun, MulAction.zpowersQuotientStabilizerEquiv_symm_apply, SemidirectProduct.congr'_symm_apply_left, symmEquiv_symm_apply_apply, Subsemigroup.map_equiv_eq_comap_symm, Subring.centerCongr_symm_apply_coe, CoxeterSystem.map_mulEquiv, MulAut.conj_symm_apply, prodProdProdComm_symm, MonoidHom.ker_comp_mulEquiv, AddEquiv.toMultiplicative_symm_apply_symm_apply, toMagmaCatIso_inv, Subgroup.map_equiv_eq_comap_symm', FreeGroupBasis.lift_symm_apply, symmEquiv_symm_apply_symm_apply, toEquiv_symm, MulEquivClass.coe_symm_apply_apply, piUnique_symm_apply, LinearMap.GeneralLinearGroup.congrLinearEquiv_symm, ValuationSubring.coe_unitGroupMulEquiv_symm_apply, AlgEquiv.val_inv_algHomUnitsEquiv_symm_apply, MulAction.stabilizerEquivStabilizer_symm, MulEquivClass.apply_coe_symm_apply, MulAut.congr_apply, Submonoid.leftInvEquiv_symm_eq_inv, ofLeftInverse'_symm_apply, IsCyclic.mulAutMulEquiv_symm_apply_symm_apply, toAdditive_symm_apply_symm_apply, galRestrictHom_symm_apply, inv_symm, Submonoid.LocalizationMap.ofMulEquivOfLocalizations_eq_iff_eq, MonoidAlgebra.mapDomainRingEquiv_apply, toGrpIso_inv, toAdditive_apply_symm_apply, Submonoid.LocalizationMap.mulEquivOfLocalizations_symm_eq_mulEquivOfLocalizations, Equiv.Perm.equivUnitsEnd_symm_apply_symm_apply, Algebra.IsAlgebraic.algEquivEquivAlgHom_symm_apply, toCommMonCatIso_inv, toUnits_symm_apply, Units.coe_opEquiv_symm, MulAutMultiplicative_apply_symm_apply, CommMonCat.uliftFunctor_map, MulAut.inv_symm, nonZeroDivisorsEquivUnits_symm_apply_coe, WithZero.withZeroUnitsEquiv_symm_apply, coprodCongr_symm_apply, PresentedGroup.equivPresentedGroup_symm_apply_of, IsGaloisGroup.mulEquivAlgEquiv_symm_apply, self_trans_symm, Equiv.Perm.equivUnitsEnd_symm_apply_apply, Action.resEquiv_inverse, coprodAssoc_symm_apply_inl, FDRep.of_ρ, Subgroup.IsComplement'.QuotientMulEquiv_symm_apply, symm_comp_eq, MulAut.inv_def, Submonoid.comap_equiv_eq_map_symm, IsCyclic.mulAutMulEquiv_symm_apply_apply, op_symm_apply_symm_apply, CoxeterMatrix.reindexGroupEquiv_symm_apply_simple, Submonoid.leftInvEquiv_symm_fromLeftInv, AddConstEquiv.equivUnits_symm_apply_symm_apply, SemidirectProduct.mulEquivSubgroup_symm_apply, Unitary.conjStarAlgAut_symm_unitaryLinearIsometryEquiv, symm_monoidHomCongrLeft, MonoidAlgebra.symm_mapDomainRingEquiv, coe_toEquiv_symm, symm_bijective, symm_toLexMulEquiv, SemidirectProduct.mulEquivProd_symm_apply_right, Submonoid.LocalizationMap.symm_comp_ofMulEquivOfLocalizations_apply, MulAction.stabilizerEquivStabilizer_inv, ConjAct.to_mul_symm_eq, Subgroup.equivSMul_symm_apply_coe, coprodAssoc_symm_apply_inr_inl, symm_symm, RingEquiv.coe_toMulEquiv_symm, GrpCat.uliftFunctor_map, HNNExtension.equiv_symm_eq_conj, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_coe, QuotientGroup.equivQuotientZPowOfEquiv_symm, subsemigroupMap_symm_apply_coe, monoidHomCongrRightEquiv_symm_apply, subgroupMap_symm_apply, Nonneg.val_inv_unitsEquivPos_symm_apply_coe, Submonoid.val_unitsTypeEquivIsUnitSubmonoid_symm_apply, MonoidHom.toHomUnitsMulEquiv_symm_apply, coe_monoidHom_comp_coe_monoidHom_symm, Equiv.mulEquiv_symm_apply, symm_comapSubgroup, Subgroup.normalizerMonoidHom_apply_symm_apply_coe, symm_mapSubgroup, Subgroup.centerToMulOpposite_symm_apply_coe, Localization.mulEquivOfQuotient_symm_mk', Submonoid.powLogEquiv_symm_apply, FreeGroup.freeGroupCongr_symm, MonoidHom.apply_ofInjective_symm, powersMulHom_symm_apply, MulOpposite.opMulEquiv_symm_apply, QuotientGroup.quotientBot_symm_apply, Submonoid.LocalizationMap.symm_comp_ofMulEquivOfLocalizations_apply', symm_apply_apply, WithZero.toMulBot_symm_bot, rootsOfUnityEquivOfPrimitiveRoots_symm_apply, MonoidAlgebra.domCongr_apply, AddAutAdditive_symm_apply_apply, SkewMonoidAlgebra.domCongr_symm, Subgroup.map_equiv_eq_comap_symm, subgroupCongr_symm_apply, CategoryTheory.ActionCategory.stabilizerIsoEnd_symm_apply, AffineEquiv.equivUnitsAffineMap_symm_apply_invFun, IsCyclotomicExtension.autEquivPow_symm_apply, Rat.ringOfIntegersWithValEquiv_apply, symmEquiv_apply_symm_apply, addMonoidEndToMultiplicative_symm_apply_apply, Subsemigroup.comap_equiv_eq_map_symm, ValuationSubring.principalUnitGroup_symm_apply, Abelianization.equivOfComm_symm_apply, val_inv_piUnits_symm_apply, apply_symm_apply, AddAut.congr_symm_apply, withZero_symm_apply_symm_apply, MulAut.congr_symm_apply, GrpWithZero.Iso.mk_inv, piMultiplicative_symm_apply, Submonoid.mul_leftInvEquiv_symm

MvQPF.wEquiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	MvQPF.WEquiv	—	—	—

NNReal.HolderConjugate

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	NNReal.HolderConjugate	—	—	NNReal.HolderTriple.symm

NNReal.HolderTriple

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	NNReal.HolderTriple	—	—	inv_add_inv_eq_inv add_comm right_pos left_pos

Nat.ModEq

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Nat.ModEq	—	—	—

NumberField.ComplexEmbedding.IsConj

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	NumberField.ComplexEmbedding.IsConj	AlgEquiv.symm Semifield.toCommSemiring Field.toSemifield DivisionSemiring.toSemiring Semifield.toDivisionSemiring	—	RingHom.ext AlgEquiv.apply_symm_apply RingHomCompTriple.comp_apply RingHomInvPair.triples₂ RingHomInvPair.instStarRingEnd RingHomCompTriple.right_ids eq

OneOneEquiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	OneOneEquiv	—	—	—

OpenPartialHomeomorph

Definitions

Name	Category	Theorems
symm 📖	CompOp	252 math math: NumberField.mixedEmbedding.integral_comp_polarSpaceCoord_symm, Trivialization.proj_symm_apply', symm_map_nhds_eq, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart', NumberField.mixedEmbedding.lintegral_comp_polarCoord_symm, NumberField.mixedEmbedding.measurable_polarCoordReal_symm, Trivialization.preimageHomeomorph_symm_apply, UniqueMDiffOn.uniqueMDiffOn_preimage, IsImage.symm_apply_mem_iff, subtypeRestr_symm_trans_subtypeRestr, Complex.norm_polarCoord_symm, eventually_left_inverse', restr_target, contDiff_unitBallBall_symm, symm_toPartialEquiv, Complex.integral_comp_pi_polarCoord_symm, symm_piecewise, symm_image_target_eq_source, image_eq_target_inter_inv_preimage, TopologicalSpace.Opens.chartAt_subtype_val_symm_eventuallyEq, StructureGroupoid.LocalInvariantProp.liftPropAt_chart_symm, UniqueMDiffWithinAt.preimage_PartialHomeomorph, NumberField.mixedEmbedding.normAtPlace_polarCoord_symm_of_isComplex, lift_openEmbedding_trans_apply, continuousOn_univBall_symm, transHomeomorph_symm_apply, NumberField.mixedEmbedding.fundamentalCone.completeBasis_apply_of_ne, ext_iff, ChartedSpace.LiftPropWithinAt.prop, isLittleO_congr, trans'_symm_apply, coe_ofContinuousOpenRestrict_symm, contMDiffOn_chart_symm, Trivialization.contMDiffOn_symm_trans, MDifferentiable.mdifferentiableAt_symm, symm_mapsTo, univUnitBall_symm_apply_zero, self_trans_symm, contMDiffOn_isOpenEmbedding_symm, source_inter_preimage_inv_preimage, toHomeomorphOfSourceEqUnivTargetEqUniv_symm_apply, Complex.integral_comp_polarCoord_symm, HasStrictFDerivAt.localInverse_def, homeomorphOfImageSubsetSource_symm_apply, prod_symm_apply, NumberField.mixedEmbedding.normAtComplexPlaces_polarSpaceCoord_symm, VectorBundleCore.localTriv_symm_fst, Unitary.openPartialHomeomorph_symm_apply, TopologicalSpace.Opens.chartAt_inclusion_symm_eventuallyEq, range_stereographic_symm, eventually_left_inverse, isOpenEmbedding_stereographic_symm, lintegral_comp_pi_polarCoord_symm, eventually_nhdsWithin, trans_source'', Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm, mk_coe_symm, refl_symm, continuous_polarCoord_symm, target_inter_inv_preimage_preimage, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source_aux, Trivialization.domExtend_symm_apply, inv_image_trans_target, leftInvOn, symm_trans_mem_contDiffGroupoid, isBigO_congr, NumberField.mixedEmbedding.polarCoord_symm_apply, lift_openEmbedding_trans, Trivialization.symm_apply_apply, eventually_right_inverse, integral_comp_polarCoord_symm, StructureGroupoid.compatible_of_mem_maximalAtlas, left_inv, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_aux, symm_image_target_inter_eq, ChartedSpace.liftPropAt_iff, Trivialization.map_target, prod_symm, symm_trans_self, Topology.IsOpenEmbedding.toPartialHomeomorph_left_inv, continuousWithinAt_iff_continuousWithinAt_comp_right, extend_coe_symm, restrContDiff_target, UniqueMDiffWithinAt.preimage_openPartialHomeomorph, continuousAt_iff_continuousAt_comp_right, contDiffOn_univUnitBall_symm, subtypeRestr_symm_apply, ChartedSpace.liftProp_iff, image_source_inter_eq, StructureGroupoid.compatible, MDifferentiable.comp_symm_deriv, ext_coord_change_source, restrContDiff_symm_apply, polarCoord_symm_apply, ImplicitFunctionData.implicitFunction_apply, StructureGroupoid.liftPropWithinAt_self_target, isLocalStructomorphOn_contDiffGroupoid_iff, tangentBundle_model_space_coe_chartAt_symm, symm_target, analyticAt_symm', map_target, Trivialization.symm_apply, restrContDiff_source, Complex.lintegral_comp_polarCoord_symm, IsManifold.compatible_of_mem_maximalAtlas, NumberField.mixedEmbedding.fundamentalCone.expMap_symm_apply, Trivialization.apply_symm_apply, NumberField.mixedEmbedding.polarSpaceCoord_symm_apply, Complex.measurableEquivRealProd_symm_polarCoord_symm_apply, IsImage.symm_eqOn_of_inter_eq_of_eqOn, hasFDerivAt_polarCoord_symm, IsImage.restr_symm_apply, StructureGroupoid.LocalInvariantProp.right_invariance', prod_symm_trans_prod, symm_bijective, coe_restrOpen_symm, isOpen_symm_image_iff_of_subset_target, NumberField.mixedEmbedding.normAtPlace_polarCoord_symm_of_isReal, ofSet_symm, IsImage.symm_mapsTo, StructureGroupoid.LocalInvariantProp.liftPropAt_symm_of_mem_maximalAtlas, hasFPowerSeriesAt_symm, NumberField.mixedEmbedding.polarCoord_symm_eq, image_source_inter_eq', Trivialization.pullback_symm_apply_snd, lift_openEmbedding_symm, Trivialization.prod_symm_apply_proj, Bundle.Trivial.toOpenPartialHomeomorph_trivialization_symm_apply, Trivialization.prod_symm_apply_snd, IsImage.leftInvOn_piecewise, trans_target', StructureGroupoid.trans_restricted, symm_trans_restr, invFun_eq_coe, tangentMap_chart_symm, Trivialization.symm_coe_proj, IsImage.iff_symm_preimage_eq, symm_image_eq_source_inter_preimage, EqOnSource.symm_eqOn_target, NumberField.mixedEmbedding.fundamentalCone.expMap_single_symm_apply, eventually_nhdsWithin', eq_symm_apply, eventually_nhds', contMDiffAt_symm_of_mem_maximalAtlas, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_iff, eventually_right_inverse', StructureGroupoid.LocalInvariantProp.right_invariance, trans_symm_eq_symm_trans_symm, contDiffOn_restrContDiff_target, extend_symm_continuousWithinAt_comp_right_iff, FiberBundle.writtenInExtChartAt_trivializationAt_symm, pi_symm_apply, Trivialization.apply_symm_apply', StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source, mdifferentiableAt_atlas_symm, restr_symm_apply, Trivialization.mk_coordChange, Trivialization.preimageSingletonHomeomorph_symm_apply, subtypeRestr_symm_eqOn, NumberField.mixedEmbedding.lintegral_comp_polarCoordReal_symm, Trivialization.mk_symm, NumberField.mixedEmbedding.measurable_polarCoord_symm, Trivialization.proj_symm_apply, injOn_pi_polarCoord_symm, MDifferentiable.symm_comp_deriv, StructureGroupoid.symm', NumberField.mixedEmbedding.hasFDerivAt_polarCoordReal_symm, rightInvOn, hasFDerivAt_pi_polarCoord_symm, IsImage.symm_image_eq, Homeomorph.toOpenPartialHomeomorphOfImageEq_symm_apply, Trivialization.pullback_symm_apply_proj, lintegral_comp_polarCoord_symm, subtypeRestr_symm_eqOn_of_le, Trivialization.sourceHomeomorphBaseSetProd_symm_apply, Homeomorph.symm_toPartialHomeomorph, Homeomorph.transOpenPartialHomeomorph_symm_apply, NumberField.mixedEmbedding.polarCoordReal_symm_target_ae_eq_univ, AddCircle.openPartialHomeomorphCoe_symm_apply, lift_openEmbedding_symm_target, Homeomorph.toOpenPartialHomeomorph_symm_apply, contDiffOn_univBall_symm, writtenInExtChartAt_chartAt_symm_comp, symm_source, invOn, IsImage.iff_symm_preimage_eq', writtenInExtChartAt_chartAt_symm, Real.expPartialHomeomorph_symm_apply, isBigOWith_congr, Trivialization.contMDiffOn_symm, StructureGroupoid.LocalInvariantProp.liftPropOn_symm_of_mem_maximalAtlas, coe_ofContinuousOpen_symm, continuousOn_symm, Structomorph.mem_groupoid, Real.coe_tanPartialHomeomorph_symm, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target_aux, Complex.lintegral_comp_pi_polarCoord_symm, Complex.polarCoord_symm_apply, stereographic'_symm_apply, Trivialization.apply_symm_apply_eq_coordChangeL, TangentBundle.coe_chartAt_symm_fst, pi_polarCoord_symm_target_ae_eq_univ, preimage_eventuallyEq_target_inter_preimage_inter, NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace_expMap_symm, extChartAt_coe_symm, StructureGroupoid.symm, symm_symm, EqOnSource.symm', Trivialization.symm_apply_mk_proj, Topology.IsOpenEmbedding.toOpenPartialHomeomorph_right_inv, NumberField.mixedEmbedding.fundamentalCone.sum_expMap_symm_apply, tendsto_symm, lift_openEmbedding_symm_source, coe_trans_symm, continuousAt_symm, isOpen_inter_preimage_symm, Bundle.Trivial.trivialization_symm_apply_proj, IsImage.symm, IsImage.symm_preimage_eq', StructureGroupoid.LocalInvariantProp.liftPropOn_chart_symm, coe_coe_symm, extend_coord_change_source, contMDiffOn_symm_of_mem_maximalAtlas, unitBallBall_symm_apply, NumberField.mixedEmbedding.integral_comp_polarCoord_symm, univUnitBall_symm_apply, FiberBundle.chartedSpace_chartAt_symm_fst, Topology.IsOpenEmbedding.toPartialHomeomorph_right_inv, IsImage.symm_preimage_eq, trans_target, Topology.IsOpenEmbedding.toOpenPartialHomeomorph_left_inv, IsImage.symm_iff, mdifferentiableOn_atlas_symm, Homeomorph.symm_toOpenPartialHomeomorph, NumberField.mixedEmbedding.integral_comp_polarCoordReal_symm, integral_comp_pi_polarCoord_symm, NumberField.mixedEmbedding.measurable_polarSpaceCoord_symm, isOpen_image_symm_of_subset_target, univBall_symm_apply_center, Trivialization.coordChangeL_apply', HasGroupoid.compatible, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart, Bundle.Trivial.trivialization_symm_apply_snd, restr_symm_trans, toHomeomorphSourceTarget_symm_apply_coe, FiberBundleCore.localTriv_symm_apply, mem_maximalAtlas_iff, mem_groupoid_of_pregroupoid, right_inv, MDifferentiable.symm, NumberField.mixedEmbedding.lintegral_comp_polarSpaceCoord_symm, ChartedSpace.liftPropWithinAt_iff'

OpenPartialHomeomorph.IsImage

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	OpenPartialHomeomorph.IsImage	OpenPartialHomeomorph.symm	—	PartialEquiv.IsImage.symm toPartialEquiv

OpenPartialHomeomorph.MDifferentiable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	OpenPartialHomeomorph.MDifferentiable	OpenPartialHomeomorph.symm	—	—

OrderAddMonoidIso

Definitions

Name	Category	Theorems
symm 📖	CompOp	19 math math: symm_comp_self, toEquiv_symm, symm_comp_eq, strictMono_symm, eq_symm_comp, equivLike_neg_eq_symm, coe_toEquiv_symm, symm_apply_eq, refl_symm, eq_comp_symm, self_comp_symm, symm_bijective, apply_symm_apply, symm_symm, comp_symm_eq, apply_eq_iff_symm_apply, invFun_eq_symm, symm_apply_apply, eq_symm_apply

OrderIso

Definitions

Name	Category	Theorems
symm 📖	CompOp	213 math math: IsLocalization.AtPrime.coe_primeSpectrumOrderIso_symm_apply_asIdeal, isLUB_image, CategoryTheory.ComposableArrows.opEquivalence_counitIso_inv_app_app, pnatIsoNat_symm_apply, symm_trans, toRelIsoLT_symm, arrowCongr_apply, WithBot.orderIsoPUnitSumLex_symm_inl, isGLB_preimage', symm_trans_apply, coe_dualDual_symm, finSuccAboveOrderIso_symm_apply_last, StrictMono.orderIsoOfSurjective_symm_apply_self, WithBot.subtypeOrderIso_symm_apply, mulRight₀_symm, Sublattice.map_equiv_eq_comap_symm, sumLexIicIoi_symm_apply_of_lt, symm_mk, List.SortedLT.coe_getIso_symm_apply, preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd_inv_app_hom_coe, invENNReal_symm_apply, Ordinal.toNimber_symm_eq, preimage_Ioc, Tropical.tropOrderIso_symm_coe_fn, WithBot.orderIsoPUnitSumLex_symm_inr, image_symm_image, ENNReal.orderIsoRpow_symm_apply, Flag.symm_map, UpperSet.mem_map, sumLexIioIci_symm_apply_Ici, dualAntisymmetrization_symm_apply, sumCongr_symm, apply_symm_apply, isGLB_image, EReal.expOrderIso_symm, sumLexAssoc_symm_apply_inr_inr, StrictMono.orderIsoOfSurjective_self_symm_apply, Finset.orderIsoOfFin_symm_apply, addLeft_symm, SemilatSupCat.Iso.mk_inv_toFun, IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe, HeytAlg.Iso.mk_inv, infIrredUpperSet_symm_apply, image_setOf_maximal, strictConvexOn_symm, toRelIsoGT_symm, symm_symm, IsOrderRightAdjoint.comp_orderIso, Fin.symm_castOrderIso, sumLexIicIoi_symm_apply_of_le, map_sInf_eq_sInf_symm_preimage, supIrredLowerSet_symm_apply, CircleDeg1Lift.coe_toOrderIso_symm, PrimeSpectrum.isIdempotentElemEquivClopens_symm_bot, DistLat.Iso.mk_inv, to_galoisConnection, CategoryTheory.ComposableArrows.opEquivalence_unitIso_inv_app, coe_symm_toEquiv, preimage_Ioo, sumComm_symm, WithTop.orderIsoSumLexPUnit_symm_inl, toEquiv_symm, UpperSet.symm_map, LowerSet.symm_map, AddSubgroup.toIntSubmodule_symm, Lat.Iso.mk_inv, concaveOn_symm, sumLexIioIci_symm_apply_Iio, mem_subalgebraEquivIntermediateField_symm, WithTop.toDualBotEquiv_symm_top, AddEquiv.symm_mapAddSubgroup, PrimeSpectrum.isIdempotentElemEquivClopens_symm_inf, map_sSup_eq_sSup_symm_preimage, List.Sorted.coe_getIso_symm_apply, sumDualDistrib_symm_inl, Real.log_of_pos, RingEquiv.mapTwoSidedIdeal_symm, withTopCongr_symm, Frm.Iso.mk_inv, ENNReal.orderIsoIicCoe_symm_apply_coe, NonemptyFinLinOrd.Iso.mk_inv, ENNReal.logOrderIso_symm, symm_bijective, conj_apply, symm_apply_eq, FiniteMulArchimedeanClass.withTopOrderIso_symm_apply, sumLexIicIoi_symm_apply_Iic, arrowCongr_symm_apply, WithTop.subtypeOrderIso_symm_apply, CategoryTheory.ComposableArrows.opEquivalence_inverse_obj, preimage_Iio, image_setOf_minimal, symm_preimage_preimage, IsGalois.intermediateFieldEquivSubgroup_symm_apply_toDual, sumLexIicIoi_symm_apply_Ioi, CategoryTheory.ComposableArrows.opEquivalence_unitIso_hom_app, lt_symm_apply, AddSubgroup.toZModSubmodule_symm, Ideal.idealProdEquiv_symm_apply, Submodule.orderIsoMapComap_symm_apply', RingEquiv.idealComapOrderIso_symm_apply, PrimeSpectrum.isIdempotentElemEquivClopens_symm_sup, AddSubmonoid.toNatSubmodule_symm, equivalence_counitIso, Fintype.coe_finsetOrderIsoSet_symm, sumAssoc_symm_apply_inr_inl, preimage_Icc, LowerSet.mem_map, Homeomorph.opensCongr_symm, Submodule.orderIsoMapComap_symm_apply, idealFactorsEquivOfQuotEquiv_symm, WithBot.toDualTopEquiv_symm_bot, coe_symm_toRelIsoGT, IsGalois.intermediateFieldEquivSubgroup_symm_apply, sumLexAssoc_symm_apply_inr_inl, Finsupp.coe_orderIsoMultiset_symm, WithBot.toDualTopEquiv_symm_coe, Module.AEval.mem_mapSubmodule_symm_apply, Sublattice.comap_equiv_eq_map_symm, FiniteMulArchimedeanClass.congrOrderIso_symm, Antisymmetrization.prodEquiv_symm_apply_mk, apply_eq_iff_eq_symm_apply, SemilatInfCat.Iso.mk_inv_toFun, PrimeSpectrum.basicOpen_isIdempotentElemEquivClopens_symm, LinOrd.Iso.mk_inv, BddOrd.Iso.mk_inv, sumDualDistrib_symm_inr, NNReal.orderIsoIccZeroCoe_symm_apply_coe, preimage_Ici, PrimeSpectrum.comapEquiv_symm, Fin.revOrderIso_symm_apply, symm_trans_self, IsGaloisGroup.intermediateFieldEquivSubgroup_symm_apply, FiniteArchimedeanClass.congrOrderIso_symm, mulLeft_symm, AddSubgroup.MapSubtype.orderIso_symm_apply, coe_symm_toRelIsoLT, symm_apply_le, FiniteArchimedeanClass.withTopOrderIso_symm_apply, WithTop.toDualBotEquiv_symm_coe, NNReal.orderIsoRpow_symm_eq, equivalence_inverse, equivClosureOperator_symm_apply, sumLexDualAntidistrib_symm_inl, FinPartOrd.Iso.mk_inv, preimage_Ioi, PrimeSpectrum.isIdempotentElemEquivClopens_symm_top, preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd_inv_app_hom_coe', le_symm_apply, ofHomInv_symm_apply, symm_apply_lt, sumLexIioIci_symm_apply_of_ge, prodComm_symm, sumLexCongr_symm, CategoryTheory.ComposableArrows.opEquivalence_inverse_map, IsDedekindDomain.primesOverEquivPrimesOver_symm_apply, NatOrdinal.toOrdinal_symm_eq, sumAssoc_symm_apply_inr_inr, orderIsoShrink_symm_apply, nucleusIsoSublocale.symm_eq_toNucleus, preimage_symm_preimage, AddEquiv.symm_comapAddSubgroup, DivisibleHull.archimedeanClassOrderIso_symm_apply, WithTop.orderIsoSumLexPUnit_symm_inr, Preord.Iso.mk_inv, symm_apply_apply, PartOrd.Iso.mk_inv, convexOn_symm, funUnique_symm_apply, mulLeft₀_symm, image_eq_preimage_symm, BooleanSubalgebra.mem_map_equiv, Sublattice.map_symm_eq_iff_eq_map, symm_refl, mulRight_symm, Nimber.toOrdinal_symm_eq, Ideal.comap_fiberIsoOfBijectiveResidueField_symm, CategoryTheory.Equivalence.toOrderIso_symm_apply, alexDiscEquivPreord_counitIso, addRight_symm, strictConcaveOn_symm, symm_image_image, sumAssoc_symm_apply_inl, isLUB_preimage', MulEquiv.symm_comapSubgroup, MulEquiv.symm_mapSubgroup, CategoryTheory.ComposableArrows.opEquivalence_counitIso_hom_app_app, sumLexDualAntidistrib_symm_inr, ofRelIsoLT_symm, Real.log_of_ne_zero, dual_symm_apply, coe_toHomeomorph_symm, sumLexAssoc_symm_apply_inl, preimage_Ico, equivalence_unitIso, PrimeSpectrum.isIdempotentElemEquivClopens_symm_compl, Finset.sumEquiv_symm_apply, Sublattice.mem_map_equiv, CompleteLat.Iso.mk_inv, sumLexIioIci_symm_apply_of_lt, ENNReal.orderIsoIicOneBirational_symm_apply, self_trans_symm, IsGaloisGroup.intermediateFieldEquivSubgroup_symm_apply_toDual, PartOrdEmb.Iso.mk_inv, Ordinal.toNatOrdinal_symm_eq, Module.mapEvalEquiv_symm_apply, symm_injective, IsOrderRightAdjoint.orderIso_comp, dualDual_symm_apply, symm_dual, preimage_Iic, Subgroup.MapSubtype.orderIso_symm_apply, withBotCongr_symm

OrderMonoidIso

Definitions

Name	Category	Theorems
symm 📖	CompOp	28 math math: apply_symm_apply, val_inv_unitsWithZero_symm_apply, withZero_symm_apply_symm_apply, val_unitsCongr_symm_apply, Submonoid.topOrderMonoidIso_symm_apply_coe, invFun_eq_symm, symm_symm, symm_apply_apply, symm_comp_self, val_inv_unitsCongr_symm_apply, withZeroUnits_symm_apply, equivLike_inv_eq_symm, val_unitsWithZero_symm_apply, eq_symm_apply, withZero_apply_symm_apply, symm_comp_eq, PNat.equivNonZeroDivisorsNat_symm_apply_coe, strictMono_symm, self_comp_symm, apply_eq_iff_symm_apply, comp_symm_eq, symm_apply_eq, refl_symm, coe_toEquiv_symm, symm_bijective, eq_symm_comp, eq_comp_symm, toEquiv_symm

OrderRingIso

Definitions

Name	Category	Theorems
symm 📖	CompOp	11 math math: symm_bijective, apply_symm_apply, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_symm_apply, symm_symm, symm_apply_apply, lt_symm_apply, self_trans_symm, LinearOrderedField.inducedOrderRingIso_symm, symm_trans_self, symm_apply_lt, Valuation.IsEquiv.orderRingIso_symm_apply

Ordnode.BalancedSz

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Ordnode.BalancedSz	—	—	add_comm

PEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	17 math math: toMatrix_symm, mem_iff_mem, trans_symm_eq_iff_forall_isSome, symm_symm, ofSet_symm, mul_toMatrix_apply, symm_single, Equiv.toPEquiv_symm, isSome_symm_get, symm_trans_self, symm_refl, symm_bijective, eq_some_iff, symm_trans_rev, symm_injective, symm_bot, self_trans_symm

PNat.Coprime

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	PNat.Coprime	—	—	PNat.gcd_comm

PSet.Equiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	PSet.Equiv	—	—	euc refl

PartialDiffeomorph

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

PartialEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	248 math math: restr_target, contMDiffAt_iff_of_mem_source, OpenPartialHomeomorph.extend_preimage_inter_eq, SmoothBumpFunction.support_eq_symm_image, extChartAt_to_inv, ApproximatesLinearOn.inverse_continuousOn, OpenPartialHomeomorph.symm_toPartialEquiv, Pretrivialization.preimage_symm_proj_inter, MDifferentiableWithinAt.differentiableWithinAt_mpullbackWithin_vectorField, mdifferentiableOn_iff, mdifferentiableWithinAt_iff_of_mem_maximalAtlas, ChartedSpaceCore.continuousOn_toFun, contMDiffAt_iff_source, eventually_enorm_mfderivWithin_symm_extChartAt_lt, Equidecomp.restr_invFun, TangentBundle.symmL_trivializationAt, Topology.RelCWComplex.continuousOn_symm, ContinuousWithinAt.extChartAt_symm_preimage_inter_range_eventuallyEq, map_extChartAt_symm_nhdsWithin, symm_symm, map_target, eq_symm_apply, mdifferentiableWithinAt_iff_target_inter, contMDiffWithinAt_iff_image, tangentCoordChange_def, map_extChartAt_symm_nhdsWithin', FiberPrebundle.continuous_trivChange, map_extChartAt_symm_nhdsWithin_range', IsImage.iff_symm_preimage_eq, extChartAt_preimage_inter_eq, FiberPrebundle.continuous_symm_of_mem_pretrivializationAtlas, map_extChartAt_symm_nhdsWithin_range, map_extChartAt_nhdsWithin', OpenPartialHomeomorph.coe_ofContinuousOpenRestrict_symm, inTangentCoordinates_eq_mfderiv_comp, Equiv.transPartialEquiv_symm_apply, symm_image_target_eq_source, MDifferentiableWithinAt.differentiableWithinAt_writtenInExtChartAt, ModelWithCorners.mk_symm, OpenPartialHomeomorph.continuousAt_extend_symm', transEquiv_symm_apply, IsImage.restr_symm_apply, extChartAt_preimage_mem_nhdsWithin', Equiv.toPartialEquiv_symm_apply, trans_target', symm_trans_self, eventually_norm_mfderivWithin_symm_extChartAt_lt, image_eq_target_inter_inv_preimage, UniqueMDiffOn.uniqueDiffOn_target_inter, mapsTo_extChartAt, preimage_extChartAt_eventuallyEq_compl_singleton, trans_source'', Pretrivialization.mk_symm, Pretrivialization.preimage_symm_proj_baseSet, UniqueMDiffOn.uniqueMDiffOn_target_inter, contMDiffAt_iff_source_of_mem_source, symm_piecewise, Manifold.IsImmersionAt.writtenInCharts, OpenPartialHomeomorph.mk_coe_symm, symm_image_target_inter_eq, SmoothBumpFunction.tsupport_subset_symm_image_closedBall, Trivialization.domExtend_symm_apply, TangentBundle.trivializationAt_apply, prod_coe_symm, IsImage.symm_preimage_eq, OpenPartialHomeomorph.contDiffOn_extend_coord_change, OpenPartialHomeomorph.continuousWithinAt_writtenInExtend_iff, Trivialization.prod_symm_apply, Equiv.toPartialEquivOfImageEq_symm_apply, mem_symm_trans_source, OpenPartialHomeomorph.map_extend_symm_nhdsWithin, IsImage.leftInvOn_piecewise, extChartAt_mem_closure_interior, mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm, restr_coe_symm, extChartAt_preimage_mem_nhds, EqOnSource.symm_eqOn, IsImage.symm_mapsTo, invFun_as_coe, OpenPartialHomeomorph.contDiffWithinAt_extend_coord_change', contMDiffWithinAt_iff_of_mem_source', mdifferentiableWithinAt_iff_image, OpenPartialHomeomorph.extend_coord_change_source_mem_nhdsWithin', injective_symm_of_target_eq_univ, mdifferentiableOn_extChartAt_symm, OpenPartialHomeomorph.continuousOn_extend_symm, VectorField.mlieBracketWithin_def, OpenPartialHomeomorph.extend_left_inv, single_symm_apply, Pretrivialization.apply_symm_apply, target_subset_preimage_source, mdifferentiable_iff, contMDiffAt_iff, OpenPartialHomeomorph.extend_coe_symm, contMDiffOn_iff_of_mem_maximalAtlas', extChartAt_preimage_mem_nhdsWithin, contMDiffOn_extChartAt_symm, extChartAt_preimage_mem_nhds', contMDiffWithinAt_extChartAt_symm_range_self, Pretrivialization.symm_apply, OpenPartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq_aux, OpenPartialHomeomorph.continuousOn_writtenInExtend_iff, mdifferentiableOn_iff_of_subset_source, contMDiffWithinAt_iff_source_of_mem_maximalAtlas, ext_coord_change_source, contMDiffOn_iff_of_subset_source', Pretrivialization.symm_apply_mk_proj, FiberPrebundle.isOpen_target_of_mem_pretrivializationAtlas_inter, Equidecomp.left_inv, source_inter_preimage_inv_preimage, right_inv, Pretrivialization.proj_symm_apply', hasFDerivWithinAt_tangentCoordChange, ContinuousWithinAt.nhdsWithin_extChartAt_symm_preimage_inter_range, MDifferentiableWithinAt.mfderivWithin, contMDiffWithinAt_iff, symm_bijective, OpenPartialHomeomorph.extend_image_source_inter, ofSet_symm, piecewise_symm_apply, Pretrivialization.symm_coe_proj, image_symm_image_of_subset_target, OpenPartialHomeomorph.extend_preimage_mem_nhds, symm_image_target_inter_eq', tangentBundleCore_coordChange, Pretrivialization.symm_trans_target_eq, contDiffOn_fderiv_coord_change, SmoothBumpFunction.isClosed_symm_image_closedBall, refl_symm, symm_source, mdifferentiableWithinAt_iff', Equidecomp.right_inv, symm_mapsTo, continuousAt_extChartAt_symm, contMDiffWithinAt_iff', Trivialization.symm_trans_target_eq, ModelWithCorners.coe_extChartAt_transContinuousLinearEquiv_symm, image_source_inter_eq', uniqueMDiffWithinAt_iff, VectorField.eventuallyEq_mpullback_mpullbackWithin_extChartAt, IsImage.symm_image_eq, map_extChartAt_nhdsWithin, contMDiffOn_iff, EqOnSource.symm', UniqueMDiffOn.uniqueDiffOn_inter_preimage, mdifferentiableWithinAt_iff_target_inter', contMDiffOn_iff_source_of_mem_maximalAtlas, contMDiffWithinAt_extChartAt_symm_target_self, contMDiff_iff, mdifferentiableOn_iff_of_mem_maximalAtlas', uniqueMDiffWithinAt_iff_inter_range, continuousOn_extChartAt_symm, symm_target, contDiffOn_ext_coord_change, writtenInExtChartAt_extChartAt_symm, symm_image_eq_source_inter_preimage, mdifferentiableWithinAt_iff_of_mem_source', VectorField.contMDiffWithinAt_mpullbackWithin_extChartAt_symm, mdifferentiableWithinAt_iff_source_of_mem_maximalAtlas, mdifferentiableWithinAt_iff, mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm', Pretrivialization.trans_source, contMDiffWithinAt_extChartAt_symm_target, prod_symm, FiberBundle.extChartAt_target, mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt', OpenPartialHomeomorph.extend_symm_continuousWithinAt_comp_right_iff, contMDiffOn_iff_of_mem_maximalAtlas, OpenPartialHomeomorph.map_extend_nhdsWithin, mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt, ext_iff, IsImage.symm_iff, Real.cosPartialEquiv_symm_apply, image_source_inter_eq, contMDiffWithinAt_iff_source, Pretrivialization.symm_apply_apply, contMDiffWithinAt_iff_source_of_mem_source, contMDiffOn_iff_of_subset_source, contMDiffWithinAt_iff_of_mem_maximalAtlas, trans_symm_eq_symm_trans_symm, disjointUnion_symm_apply, mdifferentiableAt_iff_source_of_mem_source, mdifferentiableAt_iff_of_mem_source, Manifold.IsImmersionAtOfComplement.writtenInCharts, OpenPartialHomeomorph.continuousAt_extend_symm, mdifferentiableWithinAt_extChartAt_symm, contMDiffWithinAt_extChartAt_symm_range, coe_symm_mk, ChartedSpaceCore.open_source, eventually_norm_mfderivWithin_symm_extChartAt_comp_lt, OpenPartialHomeomorph.extend_preimage_mem_nhdsWithin, Trivialization.symm_trans_source_eq, Pretrivialization.continuousLinearMap_symm_apply, contMDiffOn_extend_symm, symm_image_image_of_subset_source, surjective_symm_of_source_eq_univ, IsImage.symm, mdifferentiableWithinAt_iff_source_of_mem_source, leftInvOn, Set.BijOn.toPartialEquiv_symm_apply, OpenPartialHomeomorph.coe_ofContinuousOpen_symm, inv_image_trans_target, IsImage.symm_eq_on_of_inter_eq_of_eqOn, trans'_symm_apply, left_inv, OpenPartialHomeomorph.extend_left_inv', extChartAt_coe_symm, continuousAt_extChartAt_symm'', mdifferentiableWithinAt_iff_of_mem_source, OpenPartialHomeomorph.map_extend_symm_nhdsWithin_range, isInvertible_mfderivWithin_extChartAt_symm, VectorField.mlieBracketWithin_apply, target_inter_inv_preimage_preimage, Pretrivialization.symm_trans_symm, trans_target, Pretrivialization.proj_symm_apply, VectorField.eventually_contMDiffWithinAt_mpullbackWithin_extChartAt_symm, self_trans_symm, OpenPartialHomeomorph.coe_coe_symm, OpenPartialHomeomorph.extend_coord_change_source, Topology.CWComplex.continuousOn_symm, Pretrivialization.apply_symm_apply', Equiv.symm_toPartialEquiv, Equidecomp.symm_toPartialEquiv, OpenPartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq, Circle.argPartialEquiv_symm_apply, continuousAt_extChartAt_symm', invOn, IsImage.symm_apply_mem_iff, OpenPartialHomeomorph.mapsTo_extend, Equidecomp.restr_target, Pretrivialization.symm_trans_source_eq, mdifferentiableOn_iff_of_mem_maximalAtlas, writtenInExtChartAt_comp, Pretrivialization.target_inter_preimage_symm_source_eq, mdifferentiableOn_iff_of_subset_source', rightInvOn, FiberBundleCore.localTrivAsPartialEquiv_symm, coe_trans_symm, FiberBundleCore.localTrivAsPartialEquiv_trans, ApproximatesLinearOn.to_inv, contMDiffWithinAt_iff_of_mem_source, pi_symm, ModelWithCorners.toPartialEquiv_coe_symm, SmoothBumpFunction.image_eq_inter_preimage_of_subset_support, tangentBundleCore_coordChange_achart, pi_symm_apply, SmoothBumpFunction.isCompact_symm_image_closedBall

PartialEquiv.IsImage

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	PartialEquiv.IsImage	PartialEquiv.symm	—	symm_apply_mem_iff

Path

Definitions

Name	Category	Theorems
symm 📖	CompOp	27 math math: symm_bijective, CurveIntegrable.symm, uniformContinuous_symm, extend_symm_apply, continuous_symm, curveIntegralFun_symm_apply, Homotopic.trans_symm, Homotopic.symm₂, symm_continuous_family, GenLoop.fromLoop_symm_toLoop, symm_apply, symm_symm, symm_cast, curveIntegral_symm, segment_symm, curveIntegrable_symm, Homotopic.symm_trans, trans_symm, refl_symm, symm_subpath, extend_symm, map_symm, curveIntegralFun_symm, cast_symm, Homotopic.Quotient.mk_symm, Homotopy.symm₂_apply, symm_range

Path.Homotopic

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	Path.Homotopic	—	Nonempty.map

Path.Homotopic.Quotient

Definitions

Name	Category	Theorems
symm 📖	CompOp	3 math math: trans_symm, symm_trans, mk_symm

Path.Homotopy

Definitions

Name	Category	Theorems
symm 📖	CompOp	4 math math: symm_trans, symm_bijective, symm_apply, symm_symm

Pretrivialization

Definitions

Name	Category	Theorems
symm 📖	CompOp	17 math math: linearEquivAt_symm_apply, continuousLinearMap_symm_apply', VectorPrebundle.mk_contMDiffCoordChange, symm_apply_apply_mk, continuousLinearMapCoordChange_apply, mk_symm, apply_mk_symm, symm_apply, symm_proj_apply, symmₗ_apply, VectorPrebundle.exists_coordChange, VectorPrebundle.mk_coordChange, VectorPrebundle.IsContMDiff.exists_contMDiffCoordChange, symm_apply_of_notMem, VectorPrebundle.coordChange_apply, coe_symm_of_notMem, VectorPrebundle.contMDiffCoordChange_apply

ProbabilityTheory.CondIndep

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	MeasurableSpace MeasurableSpace.instLE ProbabilityTheory.CondIndep	—	—	ProbabilityTheory.CondIndepSets.symm

ProbabilityTheory.CondIndepFun

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	MeasurableSpace MeasurableSpace.instLE ProbabilityTheory.CondIndepFun	—	—	ProbabilityTheory.Kernel.IndepFun.symm

ProbabilityTheory.CondIndepSets

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	MeasurableSpace MeasurableSpace.instLE ProbabilityTheory.CondIndepSets	—	—	ProbabilityTheory.Kernel.IndepSets.symm

ProbabilityTheory.IdentDistrib

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	ProbabilityTheory.IdentDistrib	—	—	aemeasurable_snd aemeasurable_fst map_eq

ProbabilityTheory.Indep

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	ProbabilityTheory.Indep	—	—	ProbabilityTheory.IndepSets.symm

ProbabilityTheory.IndepFun

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	ProbabilityTheory.IndepFun	—	—	ProbabilityTheory.Kernel.IndepFun.symm

ProbabilityTheory.IndepSets

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	ProbabilityTheory.IndepSets	—	—	ProbabilityTheory.Kernel.IndepSets.symm

ProbabilityTheory.Kernel.Indep

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	ProbabilityTheory.Kernel.Indep	—	—	ProbabilityTheory.Kernel.IndepSets.symm

ProbabilityTheory.Kernel.IndepFun

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	ProbabilityTheory.Kernel.IndepFun	—	—	ProbabilityTheory.Kernel.Indep.symm

ProbabilityTheory.Kernel.IndepSets

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	ProbabilityTheory.Kernel.IndepSets	—	—	Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass Filter.univ_mem' Set.inter_comm mul_comm

PseudoMetric

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	DFunLike.coe PseudoMetric instFunLikeForall	—	symm'

PythagoreanTriple

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	PythagoreanTriple	—	—	pythagoreanTriple_comm

QPF.Wequiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	QPF.Wequiv	—	—	—

QuadraticMap.Equivalent

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	QuadraticMap.Equivalent	—	—

QuadraticMap.IsOrtho

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	QuadraticMap.IsOrtho	—	—	QuadraticMap.isOrtho_comm

QuadraticMap.IsometryEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	11 math math: QuadraticForm.tensorRId_symm_apply, QuadraticForm.tensorLId_symm_apply, QuadraticForm.tensorAssoc_symm_apply, QuadraticModuleCat.toIsometry_inv_rightUnitor, QuadraticModuleCat.toIsometry_inv_leftUnitor, CliffordAlgebra.equivOfIsometry_symm, QuadraticForm.tensorComm_symm, QuadraticModuleCat.ofIso_inv, QuadraticModuleCat.ofIso_symm, CategoryTheory.Iso.toIsometryEquiv_symm, QuadraticModuleCat.hom_inv_associator

Rack.PreEnvelGroupRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Rack.PreEnvelGroupRel	—	—	Rack.PreEnvelGroupRel'.rel

Real.HolderConjugate

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Real.HolderConjugate	—	—	Real.HolderTriple.symm

Real.HolderTriple

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Real.HolderTriple	—	—	inv_add_inv_eq_inv add_comm right_pos left_pos

RelEmbedding

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	—	—	—	map_rel_iff symm

RelIso

Definitions

Name	Category	Theorems
symm 📖	CompOp	149 math math: IsLocalization.AtPrime.coe_primeSpectrumOrderIso_symm_apply_asIdeal, TwoSidedIdeal.orderIsoRingCon_symm_apply, ULift.orderIso_symm_apply_down, YoungDiagram.transposeOrderIso_symm_apply, OrderIso.toRelIsoLT_symm, OrderIso.mulRight₀_symm_apply, TopologicalSpace.Closeds.complOrderIso_symm_apply, AddSubsemigroup.coe_toSubsemigroup_symm_apply, PrimeSpectrum.equivSubtype_symm_apply_asIdeal, relHomCongr_symm_apply, OrderIso.divRight_symm_apply, ENNReal.mulRightOrderIso_symm_apply, coe_fn_symm_mk, OrderHom.piIso_symm_apply, Ordinal.enum_symm_apply_coe, MulEquiv.comapSubgroup_symm_apply, Submonoid.coe_invOrderIso_symm_apply, Ordinal.isInitialIso_symm_apply_coe, Subsemigroup.opEquiv_symm_apply, preimage_eq_image_symm, IsLocalization.coe_primeSpectrumOrderIso_symm_apply_asIdeal, OrderIso.inv_symm_apply, IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe, TopologicalSpace.Opens.complOrderIso_symm_apply, OrderIso.toRelIsoGT_symm, Equiv.toOrderIsoSet_symm_apply, symm_symm, Prod.Lex.prodLexAssoc_symm_apply, WithZero.expOrderIso_symm_apply, PrimeSpectrum.coe_preimageOrderIsoFiber_symm_apply_coe_asIdeal, Fin.insertNthOrderIso_symm_apply, symm_bijective, cast_symm, OrderIso.divRight₀_symm_apply, OrderIso.finsetSetFinite_symm_apply, TwoSidedIdeal.orderIsoIsTwoSided_symm_apply, TwoSidedIdeal.opOrderIso_symm_apply, Prod.Lex.sumLexProdLexDistrib_symm_apply, OrderIso.subRight_symm_apply, OrderIso.withBotCongr_symm_apply, upperSetIsoLowerSet_symm_apply, symm_apply_eq, OrderIso.withZeroUnits_symm_apply, relHomCongr_apply, LieSubmodule.orderIsoMapComap_symm_apply, AddSubsemigroup.opEquiv_symm_apply, sumLexEmpty_symm_apply, InitialSeg.antisymm_symm, subrelUnivIso_symm_apply, StrictMono.orderIsoOfRightInverse_symm_apply, SimpleGraph.boxProdComm_symm_apply, OrderIso.setCongr_symm_apply, OrderIso.smulRight_symm_apply, infIooOrderIsoIooSup_symm_apply_coe, AddSubmonoid.coe_negOrderIso_symm_apply, self_trans_symm, Con.orderIsoOp_symm_apply, symm_apply_apply, rel_symm_apply, AddEquiv.mapAddSubgroup_symm_apply, Subgroup.coe_toAddSubgroup_symm_apply, Sum.Lex.toLexRelIsoLE_symm_coe, OrderIso.arrowCongr_symm_apply, OrderIso.compl_symm_apply, OrderIso.prodAssoc_symm_apply, OrderIso.ofUnique_symm_apply, apply_eq_iff_eq_symm_apply, ENNReal.mulLeftOrderIso_symm_apply, symm_apply_rel, symm_comp_self, AddSubgroup.orderIsoAddCon_symm_apply_coe, refl_symm, MulEquiv.mapSubgroup_symm_apply, symm_symm_apply, Submodule.orderIsoMapComapOfBijective_symm_apply, Fin.snocOrderIso_symm_apply, Subsemiring.opEquiv_symm_apply, AddSubgroup.opEquiv_symm_apply, PrincipalSeg.subrelIso_symm_apply, Ordinal.ToType.mk_symm_apply_coe, apply_symm_apply, OrderIso.coe_symm_toRelIsoGT, compl_symm_apply, symm_trans_apply, OrderIso.neg_symm_apply, symm_trans_self, RingCon.opOrderIso_symm_apply, Fin.castLEOrderIso_symm_apply, Fin.consOrderIso_symm_apply, SimpleGraph.Iso.boxProdSumDistrib_symm_apply, IntermediateField.extendScalars.orderIso_symm_apply_coe, OrderIso.mulLeft₀_symm_apply, IsLocalization.orderIsoOfPrime_symm_apply_coe, Fin.castOrderIso_symm_apply, TwoSidedIdeal.orderIsoMatrix_symm_apply_ringCon_r, Subgroup.opEquiv_symm_apply, OrderIso.coe_symm_toRelIsoLT, mkFactorOrderIsoOfFactorDvdEquiv_symm_apply_coe, CategoryTheory.Subobject.mapIsoToOrderIso_symm_apply, Subsemigroup.coe_toAddSubsemigroup_symm_apply, AddSubmonoid.coe_toSubmonoid_symm_apply, SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning_symm_apply, emptySumLex_symm_apply, Subrepresentation.subrepresentationSubmoduleOrderIso_symm_apply, CategoryTheory.Subfunctor.orderIsoSubobject_symm_apply, SimpleGraph.boxProdAssoc_symm_apply, image_eq_preimage_symm, OrderHom.curry_symm_apply, Fin.orderIsoSubtype_symm_apply, Ideal.coe_piOrderIso_symm_apply, OrderIso.subLeft_symm_apply, Subgroup.orderIsoCon_symm_apply_coe, SimpleGraph.Iso.sumBoxProdDistrib_symm_apply, Submonoid.coe_toAddSubmonoid_symm_apply, Subalgebra.opEquiv_symm_apply, self_comp_symm, Subring.opEquiv_symm_apply, SimpleGraph.induceUnivIso_symm_apply_coe, OrderIso.smulRightDual_symm_apply, PrimeSpectrum.coe_primesOverOrderIsoFiber_symm_apply_coe, OrderHom.prodIso_symm_apply, ValuationSubring.coe_primeSpectrumOrderEquiv_symm_apply_asIdeal, OrderIso.divLeft_symm_apply, OrderIso.setIsotypicComponents_symm_apply, Subfield.extendScalars.orderIso_symm_apply_coe, Sum.Lex.toLexRelIsoLT_symm_coe, infIccOrderIsoIccSup_symm_apply_coe, AddCon.orderIsoOp_symm_apply, OrderIso.withTopCongr_symm_apply, Concept.swapEquiv_symm_apply, AddSubmonoid.opEquiv_symm_apply, Finset.orderIsoColex_symm_apply, preimage_symm_apply, Ordinal.toZFSetIso_symm_apply, OrderIso.ofRelIsoLT_symm, OrderIso.asBoolAlgAsBoolRing_symm_apply, CategoryTheory.Abelian.subobjectIsoSubobjectOp_symm_apply, Sublattice.prodEquiv_symm_apply, AddSubgroup.coe_toSubgroup_symm_apply, WithZero.val_logOrderIso_symm_apply, Submonoid.opEquiv_symm_apply, Real.sinhOrderIso_symm_apply, SimpleGraph.Iso.sumAssoc_symm_apply, WithZero.val_inv_logOrderIso_symm_apply, eq_symm_apply, AddEquiv.comapAddSubgroup_symm_apply, SimpleGraph.Iso.sumComm_symm_apply, Subrepresentation.submoduleSubrepresentationOrderIso_symm_apply, AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv_symm_apply_coe

Relation.SymmGen

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Relation.SymmGen	—	—	—

Relator.BiTotal

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Relator.BiTotal	—	—	Relator.RightTotal.symm Relator.LeftTotal.symm

Relator.LeftTotal

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	Relator.LeftTotal	Relator.RightTotal	—	—

Relator.RightTotal

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	Relator.RightTotal	Relator.LeftTotal	—	Relator.LeftTotal.symm

RingCon

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	DFunLike.coe RingCon instFunLikeForallProp	—	—	Setoid.symm'

RingEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	311 math math: RCLike.sqrt_eq_ite, apply_symm_apply, restrict_symm_apply_coe, toRingCatIso_inv, Polynomial.roots_expand, toOpposite_symm_apply, symm_mk, AlgebraicGeometry.StructureSheaf.globalSectionsIso_inv, iterateFrobeniusEquiv_symm_add, BitVec.equivFin_symm_apply_toFin, MvPolynomial.mapEquiv_symm, iterateFrobeniusEquiv_symm_add_apply, coe_frobeniusEquiv_symm_comp_coe_frobenius, coe_prodComm_symm, Ideal.quotientEquiv_symm_apply, AdjoinRoot.quotMapOfEquivQuotMapCMapSpanMk_symm_mk, DoubleQuot.coe_quotQuotEquivQuotOfLEₐ_symm, FDRep.endRingEquiv_symm_comp_ρ, RingHom.quotientKerEquivOfSurjective_symm_apply, WittVector.IsocrystalHom.frob_equivariant, NumberField.CMExtension.algebraMap_equivMaximalRealSubfield_symm_apply, WittVector.frobeniusEquiv_symm_apply, Ideal.comap_symm, NonUnitalSubsemiring.mem_map_equiv, MvPolynomial.mvPolynomialEquivMvPolynomial_symm_apply, PowerBasis.quotientEquivQuotientMinpolyMap_apply, NonUnitalSubring.map_equiv_eq_comap_symm, Submodule.LinearDisjoint.op, AlgebraicGeometry.localRingHom_comp_stalkIso_apply, toCommSemiRingCatIso_inv, mopMatrix_symm_apply, DoubleQuot.quotQuotEquivQuotOfLEₐ_symm_toRingEquiv, NonUnitalSubsemiring.centerCongr_symm_apply_coe, ofLeftInverse_symm_apply, op_apply_symm_apply, RingCon.comapQuotientEquivRange_symm_mk, piCongrLeft_symm_apply, RCLike.realRingEquiv_symm_apply, Ring.DirectLimit.congr_symm_apply_of, BoolRing.Iso.mk_inv_hom', RingCon.comapQuotientEquivOfSurj_symm_mk, piCongrLeft'_symm, ofHomInv_symm_apply, Subsemiring.centerCongr_symm_apply_coe, MonoidAlgebra.opRingEquiv_symm_apply, NumberField.RingOfIntegers.withValEquiv_symm_apply, RingHom.quotientKerEquivOfSurjective_symm_comp, ModuleCat.endRingEquiv_symm_apply_hom, frobeniusEquiv_symm_comp_frobenius, IsLocalization.isLocalization_iff_of_base_ringEquiv, MonoidHom.map_frobeniusEquiv_symm, AddMonoidAlgebra.opRingEquiv_symm_single, AdjoinRoot.symm_mapRingEquiv, RingCon.comapQuotientEquivOfSurj_symm_mk', ofLeftInverseS_symm_apply, AddMonoidAlgebra.opRingEquiv_symm_apply, NonUnitalSubsemiring.centerToMulOpposite_symm_apply_coe, quotientBot_symm_mk, Ideal.comap_of_equiv, Polynomial.mapEquiv_symm_apply, FractionalIdeal.mapEquiv_symm, symm_toRingHom_comp_toRingHom, asBoolRingAsBoolAlg_symm_apply, HahnSeries.toMvPowerSeries_symm_apply_coeff, PreTilt.coeff_frobeniusEquiv_symm, DividedPowers.ofRingEquiv_dpow, DividedPowers.equiv_apply, RingCon.comapQuotientEquivRangeS_symm_mk, moduleEndSelf_symm_apply, Matrix.transposeRingEquiv_symm_apply, NonUnitalSubring.topEquiv_symm_apply_coe, FractionalIdeal.canonicalEquiv_symm, Polynomial.roots_X_pow_char_pow_sub_C_pow, WithVal.apply_symm_equiv, IsLocalization.ringEquivOfRingEquiv_symm_apply, symm_apply_apply, IsLocalization.AtPrime.equivQuotientMapOfIsMaximal_symm_apply_mk, normalizedFactorsEquivOfQuotEquiv_symm, Matrix.piRingEquiv_symm_apply, WithAbs.tendsto_one_div_one_add_pow_nhds_one, symm_apply_eq, piEquivPiSubtypeProd_symm_apply, MonoidAlgebra.symm_mapRangeRingEquiv, Algebra.norm_eq_of_equiv_equiv, prodZeroRing_symm_apply, Subring.comap_equiv_eq_map_symm, zeroRingProd_symm_apply, AdjoinRoot.quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk_symm_quotQuotMk, IsFractionRing.isFractionRing_iff_of_base_ringEquiv, StarAlgEquiv.toRingEquiv_symm, WittVector.IsocrystalEquiv.frob_equivariant, Ideal.quotientEquiv_symm_mk, toCommRingCatIso_inv, WithVal.equivWithVal_symm, mapTwoSidedIdeal_symm, symm_toRingHom_apply_toRingHom_apply, Subring.centerToMulOpposite_symm_apply_coe, Subsemiring.mem_map_equiv, MonoidAlgebra.symm_commRingEquiv, MvPolynomial.isEmptyRingEquiv_symm_apply, Subring.mopRingEquivOp_symm_apply, NonUnitalSubsemiring.map_equiv_eq_comap_symm, Subsemiring.centerToMulOpposite_symm_apply_coe, TruncatedWittVector.commutes_symm', invFun_eq_symm, IsPerfectClosure.equiv_symm, MonoidAlgebra.opRingEquiv_symm_single, PreTilt.coeff_iterate_frobeniusEquiv_symm, Subring.mem_map_equiv, Ideal.symm_apply_mem_of_equiv_iff, Matrix.compRingEquiv_symm_apply, ZMod.ringEquivCongr_symm, StarAlgEquiv.symm_to_ringEquiv, NonUnitalSubring.centerCongr_symm_apply_coe, MonoidAlgebra.curryRingEquiv_symm_single, ofRingHom_symm_apply, Int.quotientSpanEquivZMod_comp_castRingHom, LocallyConstant.congrLeftRingEquiv_symm_apply_apply, frobeniusEquiv_symm_pow, subsemiringMap_symm_apply_coe, mapTwoSidedIdeal_apply, WittVector.dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff, frobenius_comp_frobeniusEquiv_symm, iterateFrobeniusEquiv_symm, Subring.topEquiv_symm_apply_coe, IsAlgClosure.equivOfEquiv_symm_algebraMap, symm_trans_self, DoubleQuot.coe_quotQuotEquivQuotSupₐ_symm, comp_ofBijective_symm, Polynomial.roots_X_pow_char_sub_C_pow, WittVector.inv_pair₂, MulSemiringAction.toRingEquiv_symm_apply, coe_frobenius_comp_coe_frobeniusEquiv_symm, sumArrowEquivProdArrow_symm_apply, symm_bijective, RingHom.quotientKerEquivOfRightInverse.Symm.apply, PowerBasis.quotientEquivQuotientMinpolyMap_symm_apply, symm_toNonUnitalRingHom_comp_toNonUnitalRingHom, WittVector.inv_pair₁, Matrix.conjTransposeRingEquiv_symm_apply, PrimeSpectrum.comapEquiv_apply, cast_symm_apply, idealFactorsEquivOfQuotEquiv_symm, IsLocalization.AtPrime.algebraMap_equivQuotMaximalIdeal_symm_apply, piUnique_symm_apply, Ideal.pointwise_smul_eq_comap, RingHom.map_iterate_frobeniusEquiv_symm, Polynomial.opRingEquiv_symm_C_mul_X_pow, Polynomial.opRingEquiv_symm_monomial, IsLocalization.algEquivOfAlgEquiv_symm_apply, Submodule.linearDisjoint_op, NonUnitalSubring.comap_equiv_eq_map_symm, PerfectRing.lift_apply, PerfectionMap.comp_symm_equiv, RingHom.rangeRestrictFieldEquiv_apply_symm_apply, RCLike.complexRingEquiv_symm_apply, ofLeftInverse'_symm_apply, WithAbs.equivWithAbs_equiv_symm_apply, IsLocalization.isLocalization_of_base_ringEquiv, Subring.map_equiv_eq_comap_symm, NonUnitalSubsemiring.comap_equiv_eq_map_symm, HahnSeries.coeff_toPowerSeries_symm, MvPolynomial.isEmptyRingEquiv_symm_toRingHom, WithAbs.equiv_equivWithAbs_symm_apply, IsPerfectClosure.equiv_apply, Matrix.uniqueRingEquiv_symm_apply, IsAlgClosure.equivOfEquiv_symm_comp_algebraMap, Ideal.polynomialQuotientEquivQuotientPolynomial_symm_mk, iterate_frobeniusEquiv_symm_pow_p_pow, symm_ofNonUnitalRingHom, toSemilinearEquiv_symm_apply, NonUnitalSubring.centerToMulOpposite_symm_apply_coe, Ideal.map_of_equiv, prodProdProdComm_symm, RingAut.inv_apply, op_symm_apply_symm_apply, nonUnitalSubsemiringMap_symm_apply_coe, AddMonoidAlgebra.curryRingEquiv_symm_single, image_eq_preimage, WittVector.StandardOneDimIsocrystal.frobenius_apply, PrimeSpectrum.comapEquiv_symm, Subring.centerCongr_symm_apply_coe, polynomial_expand_eq, coe_toEquiv_symm, NonUnitalSubring.mem_map_equiv, Subring.ringEquivOpMop_symm_apply_coe, HahnSeries.coeff_toMvPowerSeries_symm, MonoidAlgebra.uniqueRingEquiv_symm_apply, Ideal.map_comap_of_equiv, symm_trans_apply, Ideal.map_symm, mk_coe', LocallyConstant.congrRightRingEquiv_symm_apply_apply, AddMonoidAlgebra.uniqueRingEquiv_symm_apply, Subsemiring.ringEquivOpMop_symm_apply_coe, ofNonUnitalRingHom_symm_apply, AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot_symm_mk_mk, ofRingHom_symm, Ring.DirectLimit.ringEquiv_symm_mk, symm_trans, frobeniusEquiv_symm_pow_p, AddMonoidAlgebra.symm_mapDomainRingEquiv, frobenius_apply_frobeniusEquiv_symm, IsFractionRing.ringEquivOfRingEquiv_symm, trace_quotient_eq_trace_localization_quotient, Perfection.coeff_frobeniusEquiv_symm, Rep.Action_ρ_eq_ρ, AdjoinRoot.Polynomial.quotQuotEquivComm_symm_mk_mk, LinearEquiv.conjRingEquiv_symm_apply_apply, piCongrLeft'_symm_apply, toSemilinearEquiv_apply, Perfection.lift_apply_apply_coe, WithAbs.equivWithAbs_symm, RingHomInvPair.of_ringEquiv, PreTilt.untilt_iterate_frobeniusEquiv_symm_pow, PerfectionMap.surjective, Ideal.quotEquivOfEq_symm, Subsemiring.topEquiv_symm_apply_coe, symm_symm, image_eq_preimage_symm, WithVal.equivWithVal_apply, AlgEquiv.ofRingEquiv_toEquiv, IsLocalization.ringEquivOfRingEquiv_symm, Padic.coe_withValRingEquiv_symm, sofLeftInverse'_symm_apply, Polynomial.roots_X_pow_char_sub_C, toNonUnitalRingHom_apply_symm_toNonUnitalRingHom_apply, AdjoinRoot.quotEquivQuotMap_symm_apply, toRingHom_apply_symm_toRingHom_apply, IsPerfectClosure.equiv_symm_apply, StarAlgEquiv.to_ringEquiv_symm, FDRep.of_ρ, Rat.ringOfIntegersEquiv_symm_apply_coe, Polynomial.opRingEquiv_symm_C, piFinTwo_symm_apply, Module.Basis.mapCoeffs_repr, symm_comp, HahnSeries.ofPowerSeries_apply, Polynomial.roots_expand_pow, Subsemiring.map_equiv_eq_comap_symm, coe_ringEquiv_lpBCF_symm, Polynomial.toFinsuppIso_symm_apply, symm_toNonUnitalRingHom_apply_toNonUnitalRingHom_apply, PerfectionMap.comp_symm_equiv', Language.reverseIso_symm_apply, MonoidAlgebra.symm_mapDomainRingEquiv, Submodule.equivOpposite_symm_apply, DoubleQuot.quotQuotEquivQuotOfLE_symm_comp_mk, coe_toAddEquiv_symm, Perfection.coeff_iterate_frobeniusEquiv_symm, Subsemiring.comap_equiv_eq_map_symm, DoubleQuot.quotQuotEquivQuotSup_symm_quotQuotMk, Submodule.mulMap_op, eq_symm_apply, addMonoidEndRingEquivInt_symm_apply, addMonoidAlgebraRingEquivDirectSum_symm_apply, image_symm_eq_preimage, coe_toMulEquiv_symm, AddMonoidAlgebra.symm_mapRangeRingEquiv, RingCon.quotientQuotientEquivQuotient_symm_mk, toRingHom_comp_symm_toRingHom, IsLocalization.AtPrime.equivQuotMaximalIdeal_symm_apply_mk, AlgEquiv.symm_toRingEquiv, RatFunc.toFractionRingRingEquiv_symm_eq, DoubleQuot.quotQuotEquivQuotOfLE_symm_mk, opOp_symm_apply, WithVal.valuation_equiv_symm, Subsemiring.mopRingEquivOp_symm_apply, piCongrRight_symm, Module.compHom.toLinearEquiv_symm_apply, PerfectRing.liftAux_apply, AddMonoidAlgebra.symm_commRingEquiv, Rep.ihom_obj_ρ, Algebra.trace_eq_of_equiv_equiv, MonoidHom.map_iterate_frobeniusEquiv_symm, ofHomInv'_symm_apply, Module.End.ringEquivEndFinsupp_symm_apply_apply, piOptionEquivProd_symm_apply, Equiv.ringEquiv_symm_apply, PerfectionMap.lift_apply, DoubleQuot.quotQuotEquivQuotSupₐ_symm_toRingEquiv, HahnSeries.toPowerSeries_symm_apply_coeff, AlgebraicGeometry.Proj.stalkIso'_symm_mk, WithVal.equivWithVal_symm_apply, toNonUnitalRingHomm_comp_symm_toNonUnitalRingHom, AlgEquiv.toRingEquiv_symm, ofBijective_symm_comp, NumberField.RingOfIntegers.mapRingEquiv_symm_apply, IsPerfectClosure.equiv_symm_toRingHom, AlgEquiv.ofRingEquiv_symm_apply, Real.ringEquivCauchy_symm_apply_cauchy, symm_refl, NumberField.Ideal.primesOverSpanEquivMonicFactorsMod_symm_apply, Int.quotientSpanNatEquivZMod_comp_castRingHom, self_trans_symm, Polynomial.opRingEquiv_symm_X, RingHom.map_frobeniusEquiv_symm, WithAbs.equivWithAbs_symm_equiv_symm_apply, toSemiRingCatIso_inv, TruncatedWittVector.commutes_symm, mapMatrix_symm, frobeniusEquiv_symm_apply_frobenius, Polynomial.roots_X_pow_char_pow_sub_C, prodCongr_symm_apply, DoubleQuot.quotQuotEquivComm_symm, NonUnitalSubsemiring.topEquiv_symm_apply_coe, moduleEndSelfOp_symm_apply, IsLocalRing.ResidueField.mapEquiv.symm, Valuation.IsEquiv.orderRingIso_symm_apply, comp_symm, KummerDedekind.quotMapEquivQuotQuotMap_symm_apply

RingHomInvPair

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	RingHomInvPair	—	comp_eq₂ comp_eq

RootPairing.Equiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	4 math math: toEndUnit_inv, inv_indexEquiv, inv_coweightMap, inv_weightMap

RootPairing.InvariantForm

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	LinearMap.IsSymm CommRing.toCommSemiring AddCommGroup.toAddCommMonoid RingHom.id Semiring.toNonAssocSemiring CommSemiring.toSemiring form	—	—

RootPairing.IsOrthogonal

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	RootPairing.IsOrthogonal	—	—	—

RootPairing.RootPositiveForm

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	LinearMap.IsSymm CommRing.toCommSemiring AddCommGroup.toAddCommMonoid RingHom.id Semiring.toNonAssocSemiring CommSemiring.toSemiring form	—	—

SModEq

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SModEq	—	—	—

SMulCommClass

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	SMulCommClass	—	smul_comm

SSet.Truncated.HomotopicL

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	SSet.Truncated.HomotopicL	—	SSet.Truncated.Quasicategory₂.fill31

SSet.Truncated.HomotopicR

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	SSet.Truncated.HomotopicR	—	SSet.Truncated.Quasicategory₂.fill32

SameRay

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SameRay	—	—	—

Sbtw

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Sbtw	—	—	sbtw_comm

SeparatedNhds

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SeparatedNhds	—	—	Disjoint.symm

Set.BijOn

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Set.InvOn Set.BijOn	—	—	Set.RightInvOn.mapsTo surjOn Set.LeftInvOn.injOn Set.RightInvOn.surjOn mapsTo

Set.EqOn

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Set.EqOn	—	—	—

Set.InvOn

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Set.InvOn	—	—	—

SetRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SetRel Set.instMembership	—	—	symm_of

SetTheory.PGame.Equiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SetTheory.PGame SetTheory.PGame.setoid	—	—	symm IsEquiv.toSymm Quotient.instIsEquivEquiv

SetTheory.PGame.Identical

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SetTheory.PGame.Identical	—	—	Relator.BiTotal.symm

SetTheory.PGame.Relabelling

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

Similar

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Similar	—	—	inv_ne_zero ENNReal.coe_inv ENNReal.div_eq_inv_mul ENNReal.eq_div_iff Nat.cast_zero ENNReal.coe_ne_top

SimpleGraph

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	Symmetric Adj	—	—

SimpleGraph.Adj

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SimpleGraph.Adj	—	—	SimpleGraph.symm

SimpleGraph.Dart

Definitions

Name	Category	Theorems
symm 📖	CompOp	10 math math: edge_fiber, SimpleGraph.Walk.mem_darts_reverse, symm_involutive, SimpleGraph.dart_edge_eq_iff, edge_symm, SimpleGraph.Walk.darts_reverse, edge_comp_symm, symm_toProd, symm_symm, symm_mk

SimpleGraph.IsBipartiteWith

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SimpleGraph.IsBipartiteWith	—	—	Disjoint.symm disjoint mem_of_adj SimpleGraph.Adj.symm

SimpleGraph.IsCompleteBetween

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SimpleGraph.IsCompleteBetween	—	—	SimpleGraph.isCompleteBetween_comm

SimpleGraph.IsEdgeReachable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SimpleGraph.IsEdgeReachable	—	—	SimpleGraph.Reachable.symm

SimpleGraph.IsFiveWheelLike

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SimpleGraph.IsFiveWheelLike	—	—	SimpleGraph.IsPathGraph3Compl.symm Finset.inter_comm

SimpleGraph.IsPathGraph3Compl

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SimpleGraph.IsPathGraph3Compl	—	—	SimpleGraph.Adj.symm

SimpleGraph.IsUniform

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	Symmetric Finset SimpleGraph.IsUniform	—	SimpleGraph.edgeDensity_comm

SimpleGraph.Iso

Definitions

Name	Category	Theorems
symm 📖	CompOp	12 math math: SimpleGraph.UnitDistEmbedding.iso_p_apply, connectedComponentEquiv_symm_apply, symm_apply_reachable, map_symm_apply, SimpleGraph.ConnectedComponent.iso_inv_image_comp_eq_iff_eq_map, symm_toHom_comp_toHom, mapNeighborSet_symm_apply_coe, toHom_comp_symm_toHom, SimpleGraph.hasseDualIso_symm_apply, connectedComponentEquiv_symm, mapEdgeSet_symm_apply, comap_symm_apply

SimpleGraph.Reachable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	SimpleGraph.Reachable	—	elim

SimpleGraph.Subgraph

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	Symmetric Adj	—	—

SimpleGraph.Subgraph.Adj

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	SimpleGraph.Subgraph.Adj	—	—	SimpleGraph.Subgraph.symm

Specializes

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Specializes	—	—	R0Space.specializes_symmetric

StarAlgEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	33 math math: ofInjective_symm_apply, Matrix.kroneckerTMulStarAlgEquiv_symm_apply, ofStarAlgHom_symm_apply, Unitary.conjStarAlgAut_symm, toStarRingEquiv_symm, restrictScalars_symm_apply, toAlgEquiv_symm, gelfandStarTransform_symm_apply, rightInverse_symm, toRingEquiv_symm, aut_inv, leftInverse_symm, LinearMap.toMatrixOrthonormal_symm_apply, symm_to_ringEquiv, apply_symm_apply, symm_bijective, Unitary.conjStarAlgAut_symm_apply, coe_symm_toAlgEquiv, symm_apply_apply, CStarMatrix.reindexₐ_symm, LinearIsometryEquiv.symm_conjStarAlgEquiv_apply_apply, symm_trans_apply, symm_symm, LinearIsometryEquiv.symm_conjStarAlgEquiv, to_ringEquiv_symm, symm_mk, refl_symm, Homeomorph.compStarAlgEquiv'_symm_apply, ofLeftInverse'_symm_apply, ofAlgEquiv_symm, Unitization.inrRangeEquiv_symm_apply, invFun_eq_symm, Matrix.kroneckerStarAlgEquiv_symm_apply

StarMulEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	15 math math: symm_apply_apply, Unitary.mapEquiv_symm_apply, ofStarMonoidHom_symm_apply, toMulEquiv_symm, symm_symm, apply_symm_apply, leftInverse_symm, symm_bijective, unitary.mapEquiv_symm, symm_trans_apply, Unitary.mapEquiv_symm, rightInverse_symm, refl_symm, ofClass_symm_apply, invFun_eq_symm

StarRingEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	13 math math: symm_symm, CentroidHom.starCenterIsoCentroid_symm_apply_coe, StarAlgEquiv.toStarRingEquiv_symm, leftInverse_symm, rightInverse_symm, symm_apply_apply, apply_symm_apply, symm_mk, invFun_eq_symm, symm_trans_apply, symm_bijective, refl_symm, ofStarRingHom_symm_apply

Stream'.WSeq.Equiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Stream'.WSeq.Equiv	—	—	Stream'.WSeq.LiftRel.symm

Stream'.WSeq.LiftRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	Symmetric	Stream'.WSeq Stream'.WSeq.LiftRel	—	Symmetric.swap_eq swap

StrongFEPair

Definitions

Name	Category	Theorems
symm 📖	CompOp	2 math math: functional_equation, symm_Λ_eq

Structomorph

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

StructureGroupoid

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	OpenPartialHomeomorph StructureGroupoid instMembershipOpenPartialHomeomorphStructureGroupoid	OpenPartialHomeomorph.symm	—	symm'

Subalgebra.LinearDisjoint

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Subalgebra.LinearDisjoint CommSemiring.toSemiring	—	—	symm_of_commute mul_comm

Subgroup.Commensurable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Subgroup.Commensurable	—	—	—

Subgroup.IsComplement'

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Subgroup.IsComplement'	—	—	inv_inv mul_inv_rev Subgroup.isComplement'_def Subgroup.isComplement_iff_bijective Equiv.bijective_comp Equiv.comp_bijective

Submodule.IsOrtho

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Submodule.IsOrtho	—	—	LE.le.trans Submodule.le_orthogonal_orthogonal Submodule.orthogonal_le

Submodule.LinearDisjoint

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Submodule.LinearDisjoint CommSemiring.toSemiring	—	—	symm_of_commute mul_comm

Subtype

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	instHasEquiv_mathlib	—	—	—

Sym2.Rel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Sym2.Rel	—	—	—

TotalComplexShape

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

TotalComplexShapeSymmetry

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	ComplexShape.π	—	—

Trivialization

Definitions

Name	Category	Theorems
symm 📖	CompOp	19 math math: inCoordinates_apply_eq₂, VectorBundleCore.localTriv_symm_apply, prod_symm_apply, Bundle.Trivial.trivialization_symm_apply, symm_apply, pullback_symm_apply_snd, prod_symm_apply_snd, coe_symmₗ, mk_coordChangeL, symm_apply_of_notMem, symm_apply_apply_mk, continuousLinearEquivAt_symm_apply, coordChangeL_apply, mk_symm, apply_mk_symm, symmL_apply, symm_proj_apply, continuousOn_symm, linearEquivAt_symm_apply

Turing.BlankRel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Turing.BlankRel	—	—	—

TuringEquivalent

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	TuringEquivalent	—	—	equivalence

ULiftable

Definitions

Name	Category	Theorems
symm 📖	CompOp	—

UniformEquiv

Definitions

Name	Category	Theorems
symm 📖	CompOp	25 math math: self_comp_symm, continuous_symm, uniformContinuous_symm, prodCongr_symm, piCongrLeft_symm_apply, apply_symm_apply, coe_symm_toEquiv, symm_comp_self, image_symm, finTwoArrow_symm_apply, refl_symm, AbstractCompletion.mapEquiv_symm, funUnique_symm_apply, preimage_symm, Metric.Snowflaking.uniformEquiv_symm_apply, symm_apply_apply, UniformSpace.Completion.mapEquiv_symm, CompareReals.compare_uc_symm, uniformEquiv_mk_coe_symm, toHomeomorph_symm_apply, piCongrRight_symm, subtype_symm_apply_coe, piFinTwo_symm_apply, prodComm_symm, AbstractCompletion.uniformContinuous_compareEquiv_symm

UniformSpace

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	Filter.Tendsto uniformity	—	—

UniformSpace.Core

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	Filter.Tendsto uniformity	—	—

VAddCommClass

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	—	VAddCommClass	—	vadd_comm

Valuation.IsEquiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Valuation.IsEquiv	—	—	—

ValuativeRel.veq

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	ValuativeRel.veq	—	—	ValuativeRel.veq_comm

Wbtw

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	Wbtw	—	—	wbtw_comm

WeakFEPair

Definitions

Name	Category	Theorems
symm 📖	CompOp	4 math math: symm_Λ₀_eq, HurwitzZeta.hurwitzEvenFEPair_zero_symm, functional_equation₀, functional_equation

(root)

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	—	—	—	—	—

qrSign

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
symm 📖	mathematical	Odd Nat.instSemiring	qrSign	—	neg_one_pow mul_comm

unitInterval

Definitions

Name	Category	Theorems
symm 📖	CompOp	42 math math: symmHomeomorph_apply, strictAnti_symm, coe_symmMeasurableEquiv, symm_le_symm, symmMeasurableEquiv_apply, ProbabilityTheory.setBernoulli_singleton, measurePreserving_symm, symm_projIcc, ProbabilityTheory.setBernoulli_apply', lt_symm_comm, Path.Homotopy.symm_apply, measurable_symm, ContinuousMap.Homotopy.symm_apply, symm_one, coe_symm_eq, symmMeasurableEquiv_symm_apply, sigmoid_neg, symm_inj, symmHomeomorph_symm_apply, Path.symm_apply, symm_le_comm, ContinuousMap.HomotopyRel.symm_apply, symm_eq_zero, Set.Icc.convexCombo_symm, image_coe_preimage_symm, MeasureTheory.instIsProbabilityMeasureHAddMeasureHSMulNNRealToNNRealSymm, symm_zero, symm_lt_comm, symm_symm, ContinuousMap.HomotopyWith.symm_apply, toNNReal_symm_add_toNNReal, symm_lt_symm, half_le_symm_iff, symm_involutive, Path.Homotopy.symm₂_apply, symm_eq_one, ProbabilityTheory.setBernoulli_apply, symm_bijective, le_symm_comm, toNNReal_add_toNNReal_symm, continuous_symm, ProbabilityTheory.setBernoulli_eq_map

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