Theorem: NonUnitalStarAlgHom.mk_coe
File: Mathlib/Algebra/Star/StarAlgHom.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
D : Type u_5
inst✝¹² : Monoid R
inst✝¹¹ : NonUnitalNonAssocSemiring A
inst✝¹⁰ : DistribMulAction R A
inst✝⁹ : Star A
inst✝⁸ : NonUnitalNonAssocSemiring B
inst✝⁷ : DistribMulAction R B
inst✝⁶ : Star B
inst✝⁵ : NonUnitalNonAssocSemiring C
inst✝⁴ : DistribMulAction R C
inst✝³ : Star C
inst✝² : NonUnitalNonAssocSemiring D
inst✝¹ : DistribMulAction R D
inst✝ : Star D
f : A →⋆ₙₐ[R] B
h₁ : ∀ (m : R) (x : A), f (m • x) = (MonoidHom.id R) m • f x
h₂ : { toFun := ⇑f, map_smul' := h₁ }.toFun 0 = 0
h₃ :
  ∀ (x y : A),
    { toFun := ⇑f, map_smul' := h₁ }.toFun (x + y) =
      { toFun := ⇑f, map_smul' := h₁ }.toFun x + { toFun := ⇑f, map_smul' := h₁ }.toFun y
h₄ :
  ∀ (x y : A),
    { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun (x * y) =
      { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun x *
        { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun y
h₅ :
  ∀ (a : A),
    { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun (star a) =
      star ({ toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun a)
⊢ { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄, map_star' := h₅ } = f
state_after: case h
R : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
D : Type u_5
inst✝¹² : Monoid R
inst✝¹¹ : NonUnitalNonAssocSemiring A
inst✝¹⁰ : DistribMulAction R A
inst✝⁹ : Star A
inst✝⁸ : NonUnitalNonAssocSemiring B
inst✝⁷ : DistribMulAction R B
inst✝⁶ : Star B
inst✝⁵ : NonUnitalNonAssocSemiring C
inst✝⁴ : DistribMulAction R C
inst✝³ : Star C
inst✝² : NonUnitalNonAssocSemiring D
inst✝¹ : DistribMulAction R D
inst✝ : Star D
f : A →⋆ₙₐ[R] B
h₁ : ∀ (m : R) (x : A), f (m • x) = (MonoidHom.id R) m • f x
h₂ : { toFun := ⇑f, map_smul' := h₁ }.toFun 0 = 0
h₃ :
  ∀ (x y : A),
    { toFun := ⇑f, map_smul' := h₁ }.toFun (x + y) =
      { toFun := ⇑f, map_smul' := h₁ }.toFun x + { toFun := ⇑f, map_smul' := h₁ }.toFun y
h₄ :
  ∀ (x y : A),
    { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun (x * y) =
      { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun x *
        { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun y
h₅ :
  ∀ (a : A),
    { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun (star a) =
      star ({ toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun a)
x✝ : A
⊢ { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄, map_star' := h₅ } x✝ = f x✝
tactic: ext
state_before: case h
R : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
D : Type u_5
inst✝¹² : Monoid R
inst✝¹¹ : NonUnitalNonAssocSemiring A
inst✝¹⁰ : DistribMulAction R A
inst✝⁹ : Star A
inst✝⁸ : NonUnitalNonAssocSemiring B
inst✝⁷ : DistribMulAction R B
inst✝⁶ : Star B
inst✝⁵ : NonUnitalNonAssocSemiring C
inst✝⁴ : DistribMulAction R C
inst✝³ : Star C
inst✝² : NonUnitalNonAssocSemiring D
inst✝¹ : DistribMulAction R D
inst✝ : Star D
f : A →⋆ₙₐ[R] B
h₁ : ∀ (m : R) (x : A), f (m • x) = (MonoidHom.id R) m • f x
h₂ : { toFun := ⇑f, map_smul' := h₁ }.toFun 0 = 0
h₃ :
  ∀ (x y : A),
    { toFun := ⇑f, map_smul' := h₁ }.toFun (x + y) =
      { toFun := ⇑f, map_smul' := h₁ }.toFun x + { toFun := ⇑f, map_smul' := h₁ }.toFun y
h₄ :
  ∀ (x y : A),
    { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun (x * y) =
      { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun x *
        { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun y
h₅ :
  ∀ (a : A),
    { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun (star a) =
      star ({ toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun a)
x✝ : A
⊢ { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄, map_star' := h₅ } x✝ = f x✝
state_after: no goals
tactic: rfl