| {"state":"R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra S A\ninst✝¹ : SMulCommClass R S A\ninst✝ : IsScalarTower R S A\nr : R\nx : A\n⊢ (RingHom.toOpposite (algebraMap R A)\n (_ : ∀ (x y : R), (algebraMap R A) x * (algebraMap R A) y = (algebraMap R A) y * (algebraMap R A) x))\n r *\n op x =\n op x *\n (RingHom.toOpposite (algebraMap R A)\n (_ : ∀ (x y : R), (algebraMap R A) x * (algebraMap R A) y = (algebraMap R A) y * (algebraMap R A) x))\n r","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Equiv\nimport Mathlib.Algebra.Module.Opposites\nimport Mathlib.Algebra.Ring.Opposite\n\n/-!\n# Algebra structures on the multiplicative opposite\n\n## Main definitions\n\n* `MulOpposite.instAlgebra`: the algebra on `Aᵐᵒᵖ`\n* `AlgHom.op`/`AlgHom.unop`: simultaneously convert the domain and codomain of a morphism to the\n opposite algebra.\n* `AlgHom.opComm`: swap which side of a morphism lies in the opposite algebra.\n* `AlgEquiv.op`/`AlgEquiv.unop`: simultaneously convert the source and target of an isomorphism to\n the opposite algebra.\n* `AlgEquiv.opOp`: any algebra is isomorphic to the opposite of its opposite.\n* `AlgEquiv.toOpposite`: in a commutative algebra, the opposite algebra is isomorphic to the\n original algebra.\n* `AlgEquiv.opComm`: swap which side of an isomorphism lies in the opposite algebra.\n-/\n\n\nvariable {R S A B : Type*}\n\nopen MulOpposite\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\nvariable [Algebra R S] [Algebra R A] [Algebra R B] [Algebra S A] [SMulCommClass R S A]\nvariable [IsScalarTower R S A]\n\nnamespace MulOpposite\n\ninstance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n toRingHom := (algebraMap R A).toOpposite fun x y => Algebra.commutes _ _\n smul_def' c x := unop_injective <| by\n simp only [unop_smul, RingHom.toOpposite_apply, Function.comp_apply, unop_mul, op_mul,\n Algebra.smul_def, Algebra.commutes, op_unop, unop_op]\n commutes' r := MulOpposite.rec' fun x => by\n ","nextTactic":"simp only [RingHom.toOpposite_apply, Function.comp_apply, ← op_mul, Algebra.commutes]","declUpToTactic":"instance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n toRingHom := (algebraMap R A).toOpposite fun x y => Algebra.commutes _ _\n smul_def' c x := unop_injective <| by\n simp only [unop_smul, RingHom.toOpposite_apply, Function.comp_apply, unop_mul, op_mul,\n Algebra.smul_def, Algebra.commutes, op_unop, unop_op]\n commutes' r := MulOpposite.rec' fun x => by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Opposite.40_0.Ep4LYRAHgoq5qI9","decl":"instance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n toRingHom "} | |
| {"state":"R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra S A\ninst✝¹ : SMulCommClass R S A\ninst✝ : IsScalarTower R S A\nc : R\nx : Aᵐᵒᵖ\n⊢ unop (c • x) =\n unop\n ((RingHom.toOpposite (algebraMap R A)\n (_ : ∀ (x y : R), (algebraMap R A) x * (algebraMap R A) y = (algebraMap R A) y * (algebraMap R A) x))\n c *\n x)","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Equiv\nimport Mathlib.Algebra.Module.Opposites\nimport Mathlib.Algebra.Ring.Opposite\n\n/-!\n# Algebra structures on the multiplicative opposite\n\n## Main definitions\n\n* `MulOpposite.instAlgebra`: the algebra on `Aᵐᵒᵖ`\n* `AlgHom.op`/`AlgHom.unop`: simultaneously convert the domain and codomain of a morphism to the\n opposite algebra.\n* `AlgHom.opComm`: swap which side of a morphism lies in the opposite algebra.\n* `AlgEquiv.op`/`AlgEquiv.unop`: simultaneously convert the source and target of an isomorphism to\n the opposite algebra.\n* `AlgEquiv.opOp`: any algebra is isomorphic to the opposite of its opposite.\n* `AlgEquiv.toOpposite`: in a commutative algebra, the opposite algebra is isomorphic to the\n original algebra.\n* `AlgEquiv.opComm`: swap which side of an isomorphism lies in the opposite algebra.\n-/\n\n\nvariable {R S A B : Type*}\n\nopen MulOpposite\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\nvariable [Algebra R S] [Algebra R A] [Algebra R B] [Algebra S A] [SMulCommClass R S A]\nvariable [IsScalarTower R S A]\n\nnamespace MulOpposite\n\ninstance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n toRingHom := (algebraMap R A).toOpposite fun x y => Algebra.commutes _ _\n smul_def' c x := unop_injective <| by\n ","nextTactic":"simp only [unop_smul, RingHom.toOpposite_apply, Function.comp_apply, unop_mul, op_mul,\n Algebra.smul_def, Algebra.commutes, op_unop, unop_op]","declUpToTactic":"instance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n toRingHom := (algebraMap R A).toOpposite fun x y => Algebra.commutes _ _\n smul_def' c x := unop_injective <| by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Opposite.40_0.Ep4LYRAHgoq5qI9","decl":"instance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n toRingHom "} | |