theorem coe_linearEquiv_injective : Injective ((↑) : (L₁ ≃ₗ⁅R⁆ L₂) → L₁ ≃ₗ[R] L₂) := by
  rintro ⟨⟨⟨⟨f, -⟩, -⟩, -⟩, f_inv⟩ ⟨⟨⟨⟨g, -⟩, -⟩, -⟩, g_inv⟩
  intro h
  simp only [to_linearEquiv_mk, LinearEquiv.mk.injEq, LinearMap.mk.injEq, AddHom.mk.injEq] at h
  congr
  

Proof state:
case mk.mk.mk.mk.mk.mk.mk.mk.e_toLieHom.e_toLinearMap.e_toAddHom.e_toFun
R : Type u
L₁ : Type v
L₂ : Type w
L₃ : Type w₁
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L₁
inst✝⁴ : LieRing L₂
inst✝³ : LieRing L₃
inst✝² : LieAlgebra R L₁
inst✝¹ : LieAlgebra R L₂
inst✝ : LieAlgebra R L₃
f_inv : L₂ → L₁
f : L₁ → L₂
map_add'✝¹ : ∀ (x y : L₁), f (x + y) = f x + f y
map_smul'✝¹ :
  ∀ (r : R) (x : L₁),
    AddHom.toFun { toFun := f, map_add' := map_add'✝¹ } (r • x) =
      (RingHom.id R) r • AddHom.toFun { toFun := f, map_add' := map_add'✝¹ } x
map_lie'✝¹ :
  ∀ {x y : L₁},
    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'✝¹ }, map_smul' := map_smul'✝¹ }.toAddHom ⁅x, y⁆ =
      ⁅AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'✝¹ }, map_smul' := map_smul'✝¹ }.toAddHom x,
        AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'✝¹ }, map_smul' := map_smul'✝¹ }.toAddHom y⁆
left_inv✝¹ :
  LeftInverse f_inv
    (↑{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'✝¹ }, map_smul' := map_smul'✝¹ },
            map_lie' := map_lie'✝¹ }).toAddHom.toFun
right_inv✝¹ :
  Function.RightInverse f_inv
    (↑{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'✝¹ }, map_smul' := map_smul'✝¹ },
            map_lie' := map_lie'✝¹ }).toAddHom.toFun
g_inv : L₂ → L₁
g : L₁ → L₂
map_add'✝ : ∀ (x y : L₁), g (x + y) = g x + g y
map_smul'✝ :
  ∀ (r : R) (x : L₁),
    AddHom.toFun { toFun := g, map_add' := map_add'✝ } (r • x) =
      (RingHom.id R) r • AddHom.toFun { toFun := g, map_add' := map_add'✝ } x
map_lie'✝ :
  ∀ {x y : L₁},
    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'✝ }, map_smul' := map_smul'✝ }.toAddHom ⁅x, y⁆ =
      ⁅AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'✝ }, map_smul' := map_smul'✝ }.toAddHom x,
        AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'✝ }, map_smul' := map_smul'✝ }.toAddHom y⁆
left_inv✝ :
  LeftInverse g_inv
    (↑{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'✝ }, map_smul' := map_smul'✝ },
            map_lie' := map_lie'✝ }).toAddHom.toFun
right_inv✝ :
  Function.RightInverse g_inv
    (↑{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'✝ }, map_smul' := map_smul'✝ },
            map_lie' := map_lie'✝ }).toAddHom.toFun
h : f = g ∧ f_inv = g_inv
⊢ f = g
case mk.mk.mk.mk.mk.mk.mk.mk.e_invFun
R : Type u
L₁ : Type v
L₂ : Type w
L₃ : Type w₁
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L₁
inst✝⁴ : LieRing L₂
inst✝³ : LieRing L₃
inst✝² : LieAlgebra R L₁
inst✝¹ : LieAlgebra R L₂
inst✝ : LieAlgebra R L₃
f_inv : L₂ → L₁
f : L₁ → L₂
map_add'✝¹ : ∀ (x y : L₁), f (x + y) = f x + f y
map_smul'✝¹ :
  ∀ (r : R) (x : L₁),
    AddHom.toFun { toFun := f, map_add' := map_add'✝¹ } (r • x) =
      (RingHom.id R) r • AddHom.toFun { toFun := f, map_add' := map_add'✝¹ } x
map_lie'✝¹ :
  ∀ {x y : L₁},
    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'✝¹ }, map_smul' := map_smul'✝¹ }.toAddHom ⁅x, y⁆ =
      ⁅AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'✝¹ }, map_smul' := map_smul'✝¹ }.toAddHom x,
        AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'✝¹ }, map_smul' := map_smul'✝¹ }.toAddHom y⁆
left_inv✝¹ :
  LeftInverse f_inv
    (↑{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'✝¹ }, map_smul' := map_smul'✝¹ },
            map_lie' := map_lie'✝¹ }).toAddHom.toFun
right_inv✝¹ :
  Function.RightInverse f_inv
    (↑{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'✝¹ }, map_smul' := map_smul'✝¹ },
            map_lie' := map_lie'✝¹ }).toAddHom.toFun
g_inv : L₂ → L₁
g : L₁ → L₂
map_add'✝ : ∀ (x y : L₁), g (x + y) = g x + g y
map_smul'✝ :
  ∀ (r : R) (x : L₁),
    AddHom.toFun { toFun := g, map_add' := map_add'✝ } (r • x) =
      (RingHom.id R) r • AddHom.toFun { toFun := g, map_add' := map_add'✝ } x
map_lie'✝ :
  ∀ {x y : L₁},
    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'✝ }, map_smul' := map_smul'✝ }.toAddHom ⁅x, y⁆ =
      ⁅AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'✝ }, map_smul' := map_smul'✝ }.toAddHom x,
        AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'✝ }, map_smul' := map_smul'✝ }.toAddHom y⁆
left_inv✝ :
  LeftInverse g_inv
    (↑{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'✝ }, map_smul' := map_smul'✝ },
            map_lie' := map_lie'✝ }).toAddHom.toFun
right_inv✝ :
  Function.RightInverse g_inv
    (↑{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'✝ }, map_smul' := map_smul'✝ },
            map_lie' := map_lie'✝ }).toAddHom.toFun
h : f = g ∧ f_inv = g_inv
⊢ f_inv = g_inv

Next tactic:
exacts [h.1, h.2]