feat(LinearAlgebra/TensorProduct): add TensorProduct.map₂ (#10986)

Author: jjaassoonn

Mathlib/LinearAlgebra/TensorProduct.lean

@@ -945,6 +945,19 @@ def homTensorHomMap : (M →ₗ[R] P) ⊗[R] (N →ₗ[R] Q) →ₗ[R] M ⊗[R]
 
 variable {R M N P Q}
 
+/--
+This is a binary version of `TensorProduct.map`: Given a bilinear map `f : M ⟶ P ⟶ Q` and a
+bilinear map `g : N ⟶ S ⟶ T`, if we think `f` and `g` as linear maps with two inputs, then
+`map₂ f g` is a bilinear map taking two inputs `M ⊗ N → P ⊗ S → Q ⊗ S` defined by
+`map₂ f g (m ⊗ n) (p ⊗ s) = f m p ⊗ g n s`.
+
+Mathematically, `TensorProduct.map₂` is defined as the composition
+`M ⊗ N -map→ Hom(P, Q) ⊗ Hom(S, T) -homTensorHomMap→ Hom(P ⊗ S, Q ⊗ T)`.
+-/
+def map₂ (f : M →ₗ[R] P →ₗ[R] Q) (g : N →ₗ[R] S →ₗ[R] T) :
+    M ⊗[R] N →ₗ[R] P ⊗[R] S →ₗ[R] Q ⊗[R] T :=
+  homTensorHomMap R _ _ _ _ ∘ₗ map f g
+
 @[simp]
 theorem mapBilinear_apply (f : M →ₗ[R] P) (g : N →ₗ[R] Q) : mapBilinear R M N P Q f g = map f g :=
   rfl
@@ -968,6 +981,10 @@ theorem homTensorHomMap_apply (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
   rfl
 #align tensor_product.hom_tensor_hom_map_apply TensorProduct.homTensorHomMap_apply
 
+@[simp]
+theorem map₂_apply_tmul (f : M →ₗ[R] P →ₗ[R] Q) (g : N →ₗ[R] S →ₗ[R] T) (m : M) (n : N) :
+    map₂ f g (m ⊗ₜ n) = map (f m) (g n) := rfl
+
 @[simp]
 theorem map_zero_left (g : N →ₗ[R] Q) : map (0 : M →ₗ[R] P) g = 0 :=
   (mapBilinear R M N P Q).map_zero₂ _

Mathlib/RingTheory/TensorProduct.lean

@@ -213,40 +213,15 @@ variable [CommSemiring R]
 variable [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A]
 variable [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B 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.map₂ (LinearMap.mul R A) (LinearMap.mul R B)
 #align algebra.tensor_product.mul Algebra.TensorProduct.mul
 
 @[simp]