-- Mathlib/Algebra/Star/StarAlgHom.lean
theorem mk_coe (f : A →⋆ₐ[R] B) (h₁ h₂ h₃ h₄ h₅ h₆) :
    (⟨⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩, h₆⟩ : A →⋆ₐ[R] B) = f

Proof state:
case h
R : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
inst✝³ : Star A
inst✝² : Semiring B
inst✝¹ : Algebra R B
inst✝ : Star B
f : A →⋆ₐ[R] B
h₁ : f 1 = 1
h₂ :
  ∀ (x y : A),
    { toFun := ⇑f, map_one' := h₁ }.toFun (x * y) =
      { toFun := ⇑f, map_one' := h₁ }.toFun x * { toFun := ⇑f, map_one' := h₁ }.toFun y
h₃ : (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun 0 = 0
h₄ :
  ∀ (x y : A),
    (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun (x + y) =
      (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun x +
        (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun y
h₅ :
  ∀ (r : R),
    (↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄ }).toFun ((algebraMap R A) r) =
      (algebraMap R B) r
h₆ :
  ∀ (x : A),
    (↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄,
                  commutes' := h₅ }.toRingHom).toFun
        (star x) =
      star
        ((↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄,
                    commutes' := h₅ }.toRingHom).toFun
          x)
x✝ : A
⊢ { toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅, map_star' := h₆ }
      x✝ =
    f x✝

Tactic:
rfl

Result:
no goals

Proof state:
R : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
inst✝³ : Star A
inst✝² : Semiring B
inst✝¹ : Algebra R B
inst✝ : Star B
f : A →⋆ₐ[R] B
h₁ : f 1 = 1
h₂ :
  ∀ (x y : A),
    { toFun := ⇑f, map_one' := h₁ }.toFun (x * y) =
      { toFun := ⇑f, map_one' := h₁ }.toFun x * { toFun := ⇑f, map_one' := h₁ }.toFun y
h₃ : (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun 0 = 0
h₄ :
  ∀ (x y : A),
    (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun (x + y) =
      (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun x +
        (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun y
h₅ :
  ∀ (r : R),
    (↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄ }).toFun ((algebraMap R A) r) =
      (algebraMap R B) r
h₆ :
  ∀ (x : A),
    (↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄,
                  commutes' := h₅ }.toRingHom).toFun
        (star x) =
      star
        ((↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄,
                    commutes' := h₅ }.toRingHom).toFun
          x)
⊢ { toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅, map_star' := h₆ } = f

Tactic:
ext

Result:
case h
R : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
inst✝³ : Star A
inst✝² : Semiring B
inst✝¹ : Algebra R B
inst✝ : Star B
f : A →⋆ₐ[R] B
h₁ : f 1 = 1
h₂ :
  ∀ (x y : A),
    { toFun := ⇑f, map_one' := h₁ }.toFun (x * y) =
      { toFun := ⇑f, map_one' := h₁ }.toFun x * { toFun := ⇑f, map_one' := h₁ }.toFun y
h₃ : (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun 0 = 0
h₄ :
  ∀ (x y : A),
    (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun (x + y) =
      (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun x +
        (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun y
h₅ :
  ∀ (r : R),
    (↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄ }).toFun ((algebraMap R A) r) =
      (algebraMap R B) r
h₆ :
  ∀ (x : A),
    (↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄,
                  commutes' := h₅ }.toRingHom).toFun
        (star x) =
      star
        ((↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄,
                    commutes' := h₅ }.toRingHom).toFun
          x)
x✝ : A
⊢ { toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅, map_star' := h₆ }
      x✝ =
    f x✝