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chore(RingTheory/TensorProduct): golf the mul definition #6195

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chore(RingTheory/TensorProduct): golf the proof and construction
eric-wieser Jul 27, 2023
a711a3a
Merge branch 'master' into tidy-RingTheory.tensor
eric-wieser Jul 28, 2023
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33 changes: 4 additions & 29 deletions Mathlib/RingTheory/TensorProduct.lean
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Expand Up @@ -158,41 +158,16 @@ variable {B : Type v₂} [Semiring B] [Algebra R B]
### The `R`-algebra structure on `A ⊗[R] B`
-/


/-- (Implementation detail)
The multiplication map on `A ⊗[R] B`,
for a fixed pure tensor in the first argument,
as an `R`-linear map.
-/
def mulAux (a₁ : A) (b₁ : B) : A ⊗[R] B →ₗ[R] A ⊗[R] B :=
TensorProduct.map (LinearMap.mulLeft R a₁) (LinearMap.mulLeft R b₁)
#align algebra.tensor_product.mul_aux Algebra.TensorProduct.mulAux

@[simp]
theorem mulAux_apply (a₁ a₂ : A) (b₁ b₂ : B) :
(mulAux a₁ b₁) (a₂ ⊗ₜ[R] b₂) = (a₁ * a₂) ⊗ₜ[R] (b₁ * b₂) :=
rfl
#align algebra.tensor_product.mul_aux_apply Algebra.TensorProduct.mulAux_apply
#noalign algebra.tensor_product.mul_aux
#noalign algebra.tensor_product.mul_aux_apply

/-- (Implementation detail)
The multiplication map on `A ⊗[R] B`,
as an `R`-bilinear map.
-/
def mul : A ⊗[R] B →ₗ[R] A ⊗[R] B →ₗ[R] A ⊗[R] B :=
TensorProduct.lift <|
LinearMap.mk₂ R mulAux
(fun x₁ x₂ y =>
TensorProduct.ext' fun x' y' => by
simp only [mulAux_apply, LinearMap.add_apply, add_mul, add_tmul])
(fun c x y =>
TensorProduct.ext' fun x' y' => by
simp only [mulAux_apply, LinearMap.smul_apply, smul_tmul', smul_mul_assoc])
(fun x y₁ y₂ =>
TensorProduct.ext' fun x' y' => by
simp only [mulAux_apply, LinearMap.add_apply, add_mul, tmul_add])
fun c x y =>
TensorProduct.ext' fun x' y' => by
simp only [mulAux_apply, LinearMap.smul_apply, smul_tmul, smul_tmul', smul_mul_assoc]
TensorProduct.homTensorHomMap R A B A B
∘ₗ TensorProduct.map (LinearMap.mul R A) (LinearMap.mul R B)
#align algebra.tensor_product.mul Algebra.TensorProduct.mul

@[simp]
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