chore(linear_algebra/multilinear_map): Add boring coercion lemmas cop…

…ied from ring_hom (#5099)

src/linear_algebra/multilinear.lean

@@ -80,8 +80,27 @@ variables [semiring R]
 
 instance : has_coe_to_fun (multilinear_map R M₁ M₂) := ⟨_, to_fun⟩
 
+initialize_simps_projections multilinear_map (to_fun → apply)
+
+@[simp] lemma to_fun_eq_coe : f.to_fun = f := rfl
+
+@[simp] lemma coe_mk (f : (Π i, M₁ i) → M₂) (h₁ h₂ ) :
+  ⇑(⟨f, h₁, h₂⟩ : multilinear_map R M₁ M₂) = f := rfl
+
+theorem congr_fun {f g : multilinear_map R M₁ M₂} (h : f = g) (x : Π i, M₁ i) : f x = g x :=
+congr_arg (λ h : multilinear_map R M₁ M₂, h x) h
+
+theorem congr_arg (f : multilinear_map R M₁ M₂) {x y : Π i, M₁ i} (h : x = y) : f x = f y :=
+congr_arg (λ x : Π i, M₁ i, f x) h
+
+theorem coe_inj ⦃f g : multilinear_map R M₁ M₂⦄ (h : ⇑f = g) : f = g :=
+by cases f; cases g; cases h; refl
+
 @[ext] theorem ext {f f' : multilinear_map R M₁ M₂} (H : ∀ x, f x = f' x) : f = f' :=
-by cases f; cases f'; congr'; exact funext H
+coe_inj (funext H)
+
+theorem ext_iff {f g : multilinear_map R M₁ M₂} : f = g ↔ ∀ x, f x = g x :=
+⟨λ h x, h ▸ rfl, λ h, ext h⟩
 
 @[simp] lemma map_add (m : Πi, M₁ i) (i : ι) (x y : M₁ i) :
   f (update m i (x + y)) = f (update m i x) + f (update m i y) :=