External

📁 Source: Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean

Statistics

Metric	Count
Definitions gradedComm, gradedCommAux, gradedMul, instModuleFstSnd	4
Theorems algebraMap_gradedMul, gradedCommAux_comp_gradedCommAux, gradedCommAux_lof_tmul, gradedComm_algebraMap, gradedComm_algebraMap_tmul, gradedComm_gradedMul, gradedComm_of_tmul_of, gradedComm_of_zero_tmul, gradedComm_one, gradedComm_one_tmul, gradedComm_symm, gradedComm_tmul_algebraMap, gradedComm_tmul_of_zero, gradedComm_tmul_one, gradedMul_algebraMap, gradedMul_assoc, gradedMul_def, gradedMul_one, one_gradedMul, tmul_of_gradedMul_of_tmul	20
Total	24

TensorProduct

Definitions

Name	Category	Theorems
gradedComm 📖	CompOp	13 math math: gradedComm_tmul_one, gradedMul_def, gradedComm_algebraMap_tmul, GradedTensorProduct.auxEquiv_comm, gradedComm_algebraMap, gradedComm_tmul_of_zero, gradedComm_one, gradedComm_symm, gradedComm_of_zero_tmul, gradedComm_tmul_algebraMap, gradedComm_one_tmul, gradedComm_of_tmul_of, gradedComm_gradedMul
gradedCommAux 📖	CompOp	2 math math: gradedCommAux_comp_gradedCommAux, gradedCommAux_lof_tmul
gradedMul 📖	CompOp	10 math math: gradedMul_assoc, gradedMul_def, algebraMap_gradedMul, GradedTensorProduct.auxEquiv_mul, gradedMul_one, GradedTensorProduct.mulHom_apply, gradedMul_algebraMap, tmul_of_gradedMul_of_tmul, one_gradedMul, gradedComm_gradedMul
instModuleFstSnd 📖	CompOp	2 math math: gradedCommAux_comp_gradedCommAux, gradedCommAux_lof_tmul

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
algebraMap_gradedMul 📖	mathematical	—	DFunLike.coe LinearMap CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid leftModule DirectSum.instSMulCommClass smulCommClass_self CommRing.toCommMonoid DistribMulAction.toMulAction MonoidWithZero.toMonoid Semiring.toMonoidWithZero SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup Module.toDistribMulAction LinearMap.instFunLike LinearMap.addCommMonoid LinearMap.module smulCommClass_left gradedMul tmul RingHom DirectSum.semiring AddMonoidWithOne.toAddMonoid AddCommMonoidWithOne.toAddMonoidWithOne NonAssocSemiring.toAddCommMonoidWithOne DirectSum.GRing.toGSemiring RingHom.instFunLike algebraMap DirectSum.instAlgebra DirectSum.instOne MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonAssocSemiring.toNonUnitalNonAssocSemiring GradedMonoid.GMonoid.toGOne AddCommMonoid.toAddMonoid NonUnitalNonAssocSemiring.toAddCommMonoid DirectSum.GSemiring.toGMonoid instSMul	—	DirectSum.instSMulCommClass smulCommClass_self smulCommClass_left AlgebraTensorModule.curry_injective DirectSum.instIsScalarTower IsScalarTower.left isScalarTower DirectSum.linearMap_ext IsScalarTower.to_smulCommClass LinearMap.ext RingHomCompTriple.ids tmul_of_gradedMul_of_tmul MulZeroClass.zero_mul uzpow_zero one_smul smul_tmul' one_mul Algebra.smul_def DFunLike.congr_fun
gradedCommAux_comp_gradedCommAux 📖	mathematical	—	LinearMap.comp DirectSum TensorProduct CommRing.toCommSemiring AddCommGroup.toAddCommMonoid addCommMonoid CommSemiring.toSemiring instAddCommMonoidDirectSum DirectSum.instModule instModuleFstSnd RingHom.id Semiring.toNonAssocSemiring RingHomCompTriple.ids gradedCommAux LinearMap.id	—	DirectSum.linearMap_ext RingHomCompTriple.ids AlgebraTensorModule.curry_injective IsScalarTower.left DirectSum.instIsScalarTower isScalarTower smulCommClass_self LinearMap.ext IsScalarTower.to_smulCommClass DirectSum.ext gradedCommAux_lof_tmul LinearMap.map_smul_of_tower LinearMap.CompatibleSMul.units LinearMap.CompatibleSMul.intModule smul_smul mul_comm Int.units_mul_self one_smul
gradedCommAux_lof_tmul 📖	mathematical	—	DFunLike.coe LinearMap CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring DirectSum TensorProduct AddCommGroup.toAddCommMonoid addCommMonoid instAddCommMonoidDirectSum DirectSum.instModule instModuleFstSnd LinearMap.instFunLike gradedCommAux DirectSum.lof tmul Units Int.instMonoid Units.instSMul SubNegMonoid.toZSMul AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DirectSum.instAddCommGroup addCommGroup Int.instUnitsPow Units.instNeg NonUnitalNonAssocRing.toHasDistribNeg NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing Int.instCommRing Units.instOne Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring	—	gradedCommAux.eq_1 DirectSum.toModule_lof LinearMap.comp.congr_simp mul_comm
gradedComm_algebraMap 📖	mathematical	—	DFunLike.coe LinearEquiv CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid instModule EquivLike.toFunLike DFinsupp.instEquivLikeLinearEquiv gradedComm RingHom Algebra.TensorProduct.instSemiring DirectSum.semiring AddMonoidWithOne.toAddMonoid AddCommMonoidWithOne.toAddMonoidWithOne NonAssocSemiring.toAddCommMonoidWithOne DirectSum.instAlgebra RingHom.instFunLike algebraMap Algebra.TensorProduct.instAlgebra	—	RingHomInvPair.ids gradedComm_algebraMap_tmul Algebra.TensorProduct.algebraMap_apply'
gradedComm_algebraMap_tmul 📖	mathematical	—	DFunLike.coe LinearEquiv CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid instModule EquivLike.toFunLike DFinsupp.instEquivLikeLinearEquiv gradedComm tmul RingHom DirectSum.semiring AddMonoidWithOne.toAddMonoid AddCommMonoidWithOne.toAddMonoidWithOne NonAssocSemiring.toAddCommMonoidWithOne RingHom.instFunLike algebraMap DirectSum.instAlgebra	—	gradedComm_of_zero_tmul
gradedComm_gradedMul 📖	mathematical	—	DFunLike.coe LinearEquiv CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid instModule EquivLike.toFunLike DFinsupp.instEquivLikeLinearEquiv gradedComm LinearMap leftModule DirectSum.instSMulCommClass smulCommClass_self CommRing.toCommMonoid DistribMulAction.toMulAction MonoidWithZero.toMonoid Semiring.toMonoidWithZero SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup Module.toDistribMulAction LinearMap.instFunLike LinearMap.addCommMonoid LinearMap.module smulCommClass_left gradedMul	—	DirectSum.instSMulCommClass smulCommClass_self smulCommClass_left isScalarTower DirectSum.instIsScalarTower IsScalarTower.left RingHomInvPair.ids RingHomCompTriple.ids AlgebraTensorModule.curry_injective LinearMap.instIsScalarTower DirectSum.linearMap_ext IsScalarTower.to_smulCommClass LinearMap.ext gradedComm_of_tmul_of tmul_of_gradedMul_of_tmul Int.cast_smul_eq_zsmul LinearEquiv.map_smul map_smul SemilinearMapClass.toMulActionSemiHomClass LinearMap.semilinearMapClass DirectSum.of_mul_of smul_smul mul_add Distrib.leftDistribClass add_mul Distrib.rightDistribClass mul_comm Mathlib.Tactic.Abel.subst_into_add Mathlib.Tactic.Abel.term_atom Mathlib.Tactic.Abel.term_add_const zero_add Mathlib.Tactic.Abel.const_add_term Mathlib.Tactic.Abel.term_add_term Mathlib.Meta.NormNum.IsNat.to_eq Mathlib.Meta.NormNum.isNat_add Mathlib.Meta.NormNum.isNat_ofNat Mathlib.Tactic.Abel.term_eq one_nsmul add_zero two_nsmul uzpow_add Int.units_mul_self one_mul LinearMap.congr_fun₂
gradedComm_of_tmul_of 📖	mathematical	—	DFunLike.coe LinearEquiv CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid instModule EquivLike.toFunLike DFinsupp.instEquivLikeLinearEquiv gradedComm tmul LinearMap LinearMap.instFunLike DirectSum.lof Units Int.instMonoid Units.instSMul SubNegMonoid.toZSMul AddGroup.toSubNegMonoid AddCommGroup.toAddGroup addCommGroup DirectSum.instAddCommGroup Int.instUnitsPow Units.instNeg NonUnitalNonAssocRing.toHasDistribNeg NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing Int.instCommRing Units.instOne Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring	—	RingHomInvPair.ids gradedCommAux_comp_gradedCommAux gradedComm.eq_1 IsScalarTower.to_smulCommClass DirectSum.instSMulCommClass directSum_lof_tmul_lof gradedCommAux_lof_tmul Units.smul_def Int.cast_smul_eq_zsmul LinearEquiv.map_smul directSum_symm_lof_tmul
gradedComm_of_zero_tmul 📖	mathematical	—	DFunLike.coe LinearEquiv CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid instModule EquivLike.toFunLike DFinsupp.instEquivLikeLinearEquiv gradedComm tmul LinearMap MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonAssocSemiring.toNonUnitalNonAssocSemiring LinearMap.instFunLike DirectSum.lof	—	RingHomCompTriple.ids RingHomInvPair.ids smulCommClass_left smulCommClass_self SMulCommClass.symm DirectSum.linearMap_ext LinearMap.ext gradedComm_of_tmul_of MulZeroClass.mul_zero uzpow_zero one_smul DFunLike.congr_fun
gradedComm_one 📖	mathematical	—	DFunLike.coe LinearEquiv CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid instModule EquivLike.toFunLike DFinsupp.instEquivLikeLinearEquiv gradedComm Algebra.TensorProduct.instOneTensorProduct NonAssocSemiring.toAddCommMonoidWithOne DirectSum.semiring AddMonoidWithOne.toAddMonoid AddCommMonoidWithOne.toAddMonoidWithOne	—	gradedComm_one_tmul
gradedComm_one_tmul 📖	mathematical	—	DFunLike.coe LinearEquiv CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid instModule EquivLike.toFunLike DFinsupp.instEquivLikeLinearEquiv gradedComm tmul DirectSum.instOne MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonAssocSemiring.toNonUnitalNonAssocSemiring	—	gradedComm_of_zero_tmul
gradedComm_symm 📖	mathematical	—	LinearEquiv.symm TensorProduct CommRing.toCommSemiring DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule CommSemiring.toSemiring addCommMonoid instModule RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids gradedComm	—	RingHomInvPair.ids
gradedComm_tmul_algebraMap 📖	mathematical	—	DFunLike.coe LinearEquiv CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid instModule EquivLike.toFunLike DFinsupp.instEquivLikeLinearEquiv gradedComm tmul RingHom DirectSum.semiring AddMonoidWithOne.toAddMonoid AddCommMonoidWithOne.toAddMonoidWithOne NonAssocSemiring.toAddCommMonoidWithOne RingHom.instFunLike algebraMap DirectSum.instAlgebra	—	gradedComm_tmul_of_zero
gradedComm_tmul_of_zero 📖	mathematical	—	DFunLike.coe LinearEquiv CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid instModule EquivLike.toFunLike DFinsupp.instEquivLikeLinearEquiv gradedComm tmul LinearMap MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonAssocSemiring.toNonUnitalNonAssocSemiring LinearMap.instFunLike DirectSum.lof	—	RingHomCompTriple.ids RingHomInvPair.ids SMulCommClass.symm smulCommClass_left smulCommClass_self DirectSum.linearMap_ext LinearMap.ext gradedComm_of_tmul_of MulZeroClass.zero_mul uzpow_zero one_smul DFunLike.congr_fun
gradedComm_tmul_one 📖	mathematical	—	DFunLike.coe LinearEquiv CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid instModule EquivLike.toFunLike DFinsupp.instEquivLikeLinearEquiv gradedComm tmul DirectSum.instOne MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonAssocSemiring.toNonUnitalNonAssocSemiring	—	gradedComm_tmul_of_zero
gradedMul_algebraMap 📖	mathematical	—	DFunLike.coe LinearMap CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid leftModule DirectSum.instSMulCommClass smulCommClass_self CommRing.toCommMonoid DistribMulAction.toMulAction MonoidWithZero.toMonoid Semiring.toMonoidWithZero SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup Module.toDistribMulAction LinearMap.instFunLike LinearMap.addCommMonoid LinearMap.module smulCommClass_left gradedMul tmul RingHom DirectSum.semiring AddMonoidWithOne.toAddMonoid AddCommMonoidWithOne.toAddMonoidWithOne NonAssocSemiring.toAddCommMonoidWithOne DirectSum.GRing.toGSemiring RingHom.instFunLike algebraMap DirectSum.instAlgebra DirectSum.instOne MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonAssocSemiring.toNonUnitalNonAssocSemiring GradedMonoid.GMonoid.toGOne AddCommMonoid.toAddMonoid NonUnitalNonAssocSemiring.toAddCommMonoid DirectSum.GSemiring.toGMonoid instSMul	—	DirectSum.instSMulCommClass smulCommClass_self SMulCommClass.symm smulCommClass_left AlgebraTensorModule.curry_injective DirectSum.instIsScalarTower IsScalarTower.left isScalarTower DirectSum.linearMap_ext IsScalarTower.to_smulCommClass LinearMap.ext RingHomCompTriple.ids tmul_of_gradedMul_of_tmul MulZeroClass.mul_zero uzpow_zero one_smul smul_tmul' mul_one Algebra.smul_def Algebra.commutes DFunLike.congr_fun
gradedMul_assoc 📖	mathematical	—	DFunLike.coe LinearMap CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid leftModule DirectSum.instSMulCommClass smulCommClass_self CommRing.toCommMonoid DistribMulAction.toMulAction MonoidWithZero.toMonoid Semiring.toMonoidWithZero SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup Module.toDistribMulAction LinearMap.instFunLike LinearMap.addCommMonoid LinearMap.module smulCommClass_left gradedMul	—	DirectSum.instSMulCommClass smulCommClass_self smulCommClass_left RingHomCompTriple.ids LinearMap.instSMulCommClass SMulCommClass.symm RingHomInvPair.ids AlgebraTensorModule.curry_injective DirectSum.instIsScalarTower IsScalarTower.left LinearMap.instIsScalarTower isScalarTower DirectSum.linearMap_ext IsScalarTower.to_smulCommClass LinearMap.ext tmul_of_gradedMul_of_tmul Int.cast_smul_eq_zsmul LinearMap.map_smul₂ map_smul SemilinearMapClass.toMulActionSemiHomClass LinearMap.semilinearMapClass DirectSum.of_mul_of mul_assoc smul_smul add_mul Distrib.rightDistribClass mul_add Distrib.leftDistribClass Mathlib.Tactic.Abel.subst_into_add Mathlib.Tactic.Abel.term_atom Mathlib.Tactic.Abel.term_add_const zero_add Mathlib.Tactic.Abel.const_add_term add_zero DFunLike.congr_fun
gradedMul_def 📖	mathematical	—	gradedMul curry CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring CommSemiring.toSemiring TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid leftModule LinearMap.comp NonUnitalNonAssocSemiring.toAddCommMonoid DirectSum.instNonUnitalNonAssocSemiring Distrib.toAdd NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring DirectSum.GSemiring.toGNonUnitalNonAssocSemiring AddMonoidWithOne.toAddMonoid AddCommMonoidWithOne.toAddMonoidWithOne NonAssocSemiring.toAddCommMonoidWithOne DirectSum.GRing.toGSemiring instModule map LinearMap.mul' LinearEquiv.toLinearMap LinearEquiv.symm assoc LinearMap.lTensor LinearMap.rTensor gradedComm	—	DirectSum.instSMulCommClass smulCommClass_self smulCommClass_left
gradedMul_one 📖	mathematical	—	DFunLike.coe LinearMap CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid leftModule DirectSum.instSMulCommClass smulCommClass_self CommRing.toCommMonoid DistribMulAction.toMulAction MonoidWithZero.toMonoid Semiring.toMonoidWithZero SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup Module.toDistribMulAction LinearMap.instFunLike LinearMap.addCommMonoid LinearMap.module smulCommClass_left gradedMul Algebra.TensorProduct.instOneTensorProduct NonAssocSemiring.toAddCommMonoidWithOne DirectSum.semiring AddMonoidWithOne.toAddMonoid AddCommMonoidWithOne.toAddMonoidWithOne DirectSum.GRing.toGSemiring	—	DirectSum.instSMulCommClass smulCommClass_self smulCommClass_left RingHom.map_one one_smul gradedMul_algebraMap
one_gradedMul 📖	mathematical	—	DFunLike.coe LinearMap CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid leftModule DirectSum.instSMulCommClass smulCommClass_self CommRing.toCommMonoid DistribMulAction.toMulAction MonoidWithZero.toMonoid Semiring.toMonoidWithZero SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup Module.toDistribMulAction LinearMap.instFunLike LinearMap.addCommMonoid LinearMap.module smulCommClass_left gradedMul Algebra.TensorProduct.instOneTensorProduct NonAssocSemiring.toAddCommMonoidWithOne DirectSum.semiring AddMonoidWithOne.toAddMonoid AddCommMonoidWithOne.toAddMonoidWithOne DirectSum.GRing.toGSemiring	—	DirectSum.instSMulCommClass smulCommClass_self smulCommClass_left RingHom.map_one one_smul algebraMap_gradedMul
tmul_of_gradedMul_of_tmul 📖	mathematical	—	DFunLike.coe LinearMap CommSemiring.toSemiring CommRing.toCommSemiring RingHom.id Semiring.toNonAssocSemiring TensorProduct DirectSum AddCommGroup.toAddCommMonoid instAddCommMonoidDirectSum DirectSum.instModule addCommMonoid leftModule DirectSum.instSMulCommClass smulCommClass_self CommRing.toCommMonoid DistribMulAction.toMulAction MonoidWithZero.toMonoid Semiring.toMonoidWithZero SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup Module.toDistribMulAction LinearMap.instFunLike LinearMap.addCommMonoid LinearMap.module smulCommClass_left gradedMul tmul DirectSum.lof Units Int.instMonoid Units.instSMul SubNegMonoid.toZSMul AddGroupWithOne.toAddGroup Ring.toAddGroupWithOne Algebra.TensorProduct.instRing DirectSum.ring AddMonoidWithOne.toAddMonoid AddCommMonoidWithOne.toAddMonoidWithOne NonAssocSemiring.toAddCommMonoidWithOne DirectSum.instAlgebra DirectSum.GRing.toGSemiring DirectSum.semiring Int.instUnitsPow Units.instNeg NonUnitalNonAssocRing.toHasDistribNeg NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing Int.instCommRing Units.instOne Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring DirectSum.instMul Distrib.toAdd DirectSum.GSemiring.toGNonUnitalNonAssocSemiring	—	DirectSum.instSMulCommClass smulCommClass_self smulCommClass_left gradedMul_def mul_comm RingHomInvPair.ids gradedComm_of_tmul_of Units.smul_def Int.cast_smul_eq_zsmul smul_tmul' LinearEquiv.map_smul tmul_smul CompatibleSMul.isScalarTower DirectSum.instIsScalarTower IsScalarTower.left isScalarTower map_smul SemilinearMapClass.toMulActionSemiHomClass LinearMap.semilinearMapClass

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