chore(algebra/algebra/basic): add missing lemmas (#7412)

src/algebra/algebra/basic.lean

@@ -529,17 +529,25 @@ protected def id : A →ₐ[R] A :=
 { commutes' := λ _, rfl,
   ..ring_hom.id A  }
 
+@[simp] lemma coe_id : ⇑(alg_hom.id R A) = id := rfl
+
+@[simp] lemma id_to_ring_hom : (alg_hom.id R A : A →+* A) = ring_hom.id _ := rfl
+
 end
 
-@[simp] lemma id_apply (p : A) : alg_hom.id R A p = p := rfl
+lemma id_apply (p : A) : alg_hom.id R A p = p := rfl
 
 /-- Composition of algebra homeomorphisms. -/
 def comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=
 { commutes' := λ r : R, by rw [← φ₁.commutes, ← φ₂.commutes]; refl,
   .. φ₁.to_ring_hom.comp ↑φ₂ }
 
-@[simp] lemma comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) :
-  φ₁.comp φ₂ p = φ₁ (φ₂ p) := rfl
+@[simp] lemma coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ := rfl
+
+lemma comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) := rfl
+
+lemma comp_to_ring_hom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :
+  ⇑(φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ := rfl
 
 @[simp] theorem comp_id : φ.comp (alg_hom.id R A) = φ :=
 ext $ λ x, rfl
@@ -750,8 +758,9 @@ instance : inhabited (A₁ ≃ₐ[R] A₁) := ⟨1⟩
 @[refl]
 def refl : A₁ ≃ₐ[R] A₁ := 1
 
-@[simp] lemma coe_refl : (@refl R A₁ _ _ _ : A₁ →ₐ[R] A₁) = alg_hom.id R A₁ :=
-alg_hom.ext (λ x, rfl)
+@[simp] lemma refl_to_alg_hom : ↑(refl : A₁ ≃ₐ[R] A₁) = alg_hom.id R A₁ := rfl
+
+@[simp] lemma coe_refl : ⇑(refl : A₁ ≃ₐ[R] A₁) = id := rfl
 
 /-- Algebra equivalences are symmetric. -/
 @[symm]
@@ -794,7 +803,10 @@ def trans (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) : A₁ ≃
 @[simp] lemma symm_apply_apply (e : A₁ ≃ₐ[R] A₂) : ∀ x, e.symm (e x) = x :=
   e.to_equiv.symm_apply_apply
 
-@[simp] lemma trans_apply (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) (x : A₁) :
+@[simp] lemma coe_trans (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) :
+  ⇑(e₁.trans e₂) = e₂ ∘ e₁ := rfl
+
+lemma trans_apply (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) (x : A₁) :
   (e₁.trans e₂) x = e₂ (e₁ x) := rfl
 
 @[simp] lemma comp_symm (e : A₁ ≃ₐ[R] A₂) :
@@ -805,6 +817,10 @@ by { ext, simp }
   alg_hom.comp ↑e.symm (e : A₁ →ₐ[R] A₂) = alg_hom.id R A₁ :=
 by { ext, simp }
 
+theorem left_inverse_symm (e : A₁ ≃ₐ[R] A₂) : function.left_inverse e.symm e := e.left_inv
+
+theorem right_inverse_symm (e : A₁ ≃ₐ[R] A₂) : function.right_inverse e.symm e := e.right_inv
+
 /-- If `A₁` is equivalent to `A₁'` and `A₂` is equivalent to `A₂'`, then the type of maps
 `A₁ →ₐ[R] A₂` is equivalent to the type of maps `A₁' →ₐ[R] A₂'`. -/
 def arrow_congr {A₁' A₂' : Type*} [semiring A₁'] [semiring A₂'] [algebra R A₁'] [algebra R A₂']