-- Mathlib/Algebra/Algebra/NonUnitalHom.lean
theorem coe_injective : @Function.Injective (A →ₛₙₐ[φ] B) (A → B) (↑)

Proof state:
R : Type u
S : Type u₁
inst✝⁵ : Monoid R
inst✝⁴ : Monoid S
φ : R →* S
A : Type v
B : Type w
inst✝³ : NonUnitalNonAssocSemiring A
inst✝² : DistribMulAction R A
inst✝¹ : NonUnitalNonAssocSemiring B
inst✝ : DistribMulAction S B
⊢ Function.Injective DFunLike.coe

Tactic:
rintro ⟨⟨⟨f, _⟩, _⟩, _⟩ ⟨⟨⟨g, _⟩, _⟩, _⟩ h

Result:
case mk.mk.mk.mk.mk.mk
R : Type u
S : Type u₁
inst✝⁵ : Monoid R
inst✝⁴ : Monoid S
φ : R →* S
A : Type v
B : Type w
inst✝³ : NonUnitalNonAssocSemiring A
inst✝² : DistribMulAction R A
inst✝¹ : NonUnitalNonAssocSemiring B
inst✝ : DistribMulAction S B
f : A → B
map_smul'✝¹ : ∀ (m : R) (x : A), f (m • x) = φ m • f x
map_zero'✝¹ : { toFun := f, map_smul' := map_smul'✝¹ }.toFun 0 = 0
map_add'✝¹ :
  ∀ (x y : A),
    { toFun := f, map_smul' := map_smul'✝¹ }.toFun (x + y) =
      { toFun := f, map_smul' := map_smul'✝¹ }.toFun x + { toFun := f, map_smul' := map_smul'✝¹ }.toFun y
map_mul'✝¹ :
  ∀ (x y : A),
    { toFun := f, map_smul' := map_smul'✝¹, map_zero' := map_zero'✝¹, map_add' := map_add'✝¹ }.toFun (x * y) =
      { toFun := f, map_smul' := map_smul'✝¹, map_zero' := map_zero'✝¹, map_add' := map_add'✝¹ }.toFun x *
        { toFun := f, map_smul' := map_smul'✝¹, map_zero' := map_zero'✝¹, map_add' := map_add'✝¹ }.toFun y
g : A → B
map_smul'✝ : ∀ (m : R) (x : A), g (m • x) = φ m • g x
map_zero'✝ : { toFun := g, map_smul' := map_smul'✝ }.toFun 0 = 0
map_add'✝ :
  ∀ (x y : A),
    { toFun := g, map_smul' := map_smul'✝ }.toFun (x + y) =
      { toFun := g, map_smul' := map_smul'✝ }.toFun x + { toFun := g, map_smul' := map_smul'✝ }.toFun y
map_mul'✝ :
  ∀ (x y : A),
    { toFun := g, map_smul' := map_smul'✝, map_zero' := map_zero'✝, map_add' := map_add'✝ }.toFun (x * y) =
      { toFun := g, map_smul' := map_smul'✝, map_zero' := map_zero'✝, map_add' := map_add'✝ }.toFun x *
        { toFun := g, map_smul' := map_smul'✝, map_zero' := map_zero'✝, map_add' := map_add'✝ }.toFun y
h :
  ⇑{ toFun := f, map_smul' := map_smul'✝¹, map_zero' := map_zero'✝¹, map_add' := map_add'✝¹, map_mul' := map_mul'✝¹ } =
    ⇑{ toFun := g, map_smul' := map_smul'✝, map_zero' := map_zero'✝, map_add' := map_add'✝, map_mul' := map_mul'✝ }
⊢ { toFun := f, map_smul' := map_smul'✝¹, map_zero' := map_zero'✝¹, map_add' := map_add'✝¹, map_mul' := map_mul'✝¹ } =
    { toFun := g, map_smul' := map_smul'✝, map_zero' := map_zero'✝, map_add' := map_add'✝, map_mul' := map_mul'✝ }

Proof state:
case mk.mk.mk.mk.mk.mk
R : Type u
S : Type u₁
inst✝⁵ : Monoid R
inst✝⁴ : Monoid S
φ : R →* S
A : Type v
B : Type w
inst✝³ : NonUnitalNonAssocSemiring A
inst✝² : DistribMulAction R A
inst✝¹ : NonUnitalNonAssocSemiring B
inst✝ : DistribMulAction S B
f : A → B
map_smul'✝¹ : ∀ (m : R) (x : A), f (m • x) = φ m • f x
map_zero'✝¹ : { toFun := f, map_smul' := map_smul'✝¹ }.toFun 0 = 0
map_add'✝¹ :
  ∀ (x y : A),
    { toFun := f, map_smul' := map_smul'✝¹ }.toFun (x + y) =
      { toFun := f, map_smul' := map_smul'✝¹ }.toFun x + { toFun := f, map_smul' := map_smul'✝¹ }.toFun y
map_mul'✝¹ :
  ∀ (x y : A),
    { toFun := f, map_smul' := map_smul'✝¹, map_zero' := map_zero'✝¹, map_add' := map_add'✝¹ }.toFun (x * y) =
      { toFun := f, map_smul' := map_smul'✝¹, map_zero' := map_zero'✝¹, map_add' := map_add'✝¹ }.toFun x *
        { toFun := f, map_smul' := map_smul'✝¹, map_zero' := map_zero'✝¹, map_add' := map_add'✝¹ }.toFun y
g : A → B
map_smul'✝ : ∀ (m : R) (x : A), g (m • x) = φ m • g x
map_zero'✝ : { toFun := g, map_smul' := map_smul'✝ }.toFun 0 = 0
map_add'✝ :
  ∀ (x y : A),
    { toFun := g, map_smul' := map_smul'✝ }.toFun (x + y) =
      { toFun := g, map_smul' := map_smul'✝ }.toFun x + { toFun := g, map_smul' := map_smul'✝ }.toFun y
map_mul'✝ :
  ∀ (x y : A),
    { toFun := g, map_smul' := map_smul'✝, map_zero' := map_zero'✝, map_add' := map_add'✝ }.toFun (x * y) =
      { toFun := g, map_smul' := map_smul'✝, map_zero' := map_zero'✝, map_add' := map_add'✝ }.toFun x *
        { toFun := g, map_smul' := map_smul'✝, map_zero' := map_zero'✝, map_add' := map_add'✝ }.toFun y
h :
  ⇑{ toFun := f, map_smul' := map_smul'✝¹, map_zero' := map_zero'✝¹, map_add' := map_add'✝¹, map_mul' := map_mul'✝¹ } =
    ⇑{ toFun := g, map_smul' := map_smul'✝, map_zero' := map_zero'✝, map_add' := map_add'✝, map_mul' := map_mul'✝ }
⊢ { toFun := f, map_smul' := map_smul'✝¹, map_zero' := map_zero'✝¹, map_add' := map_add'✝¹, map_mul' := map_mul'✝¹ } =
    { toFun := g, map_smul' := map_smul'✝, map_zero' := map_zero'✝, map_add' := map_add'✝, map_mul' := map_mul'✝ }

Tactic:
congr

Result:
no goals