instance [Semiring R] : RingInvoClass (RingInvo R) R where
  coe f := f.toFun
  inv f := f.invFun
  coe_injective' e f h₁ h₂ := by
    rcases e with ⟨⟨tE, _⟩, _⟩; rcases f with ⟨⟨tF, _⟩, _⟩
    cases tE
    cases tF
    

Proof state:
case mk.mk.mk.mk.mk.mk
R : Type u_1
inst✝¹ inst✝ : Semiring R
toFun✝¹ : R → Rᵐᵒᵖ
invFun✝¹ : Rᵐᵒᵖ → R
left_inv✝¹ : Function.LeftInverse invFun✝¹ toFun✝¹
right_inv✝¹ : Function.RightInverse invFun✝¹ toFun✝¹
map_mul'✝¹ :
  ∀ (x y : R),
    Equiv.toFun { toFun := toFun✝¹, invFun := invFun✝¹, left_inv := left_inv✝¹, right_inv := right_inv✝¹ } (x * y) =
      Equiv.toFun { toFun := toFun✝¹, invFun := invFun✝¹, left_inv := left_inv✝¹, right_inv := right_inv✝¹ } x *
        Equiv.toFun { toFun := toFun✝¹, invFun := invFun✝¹, left_inv := left_inv✝¹, right_inv := right_inv✝¹ } y
map_add'✝¹ :
  ∀ (x y : R),
    Equiv.toFun { toFun := toFun✝¹, invFun := invFun✝¹, left_inv := left_inv✝¹, right_inv := right_inv✝¹ } (x + y) =
      Equiv.toFun { toFun := toFun✝¹, invFun := invFun✝¹, left_inv := left_inv✝¹, right_inv := right_inv✝¹ } x +
        Equiv.toFun { toFun := toFun✝¹, invFun := invFun✝¹, left_inv := left_inv✝¹, right_inv := right_inv✝¹ } y
involution'✝¹ :
  ∀ (x : R),
    MulOpposite.unop
        (Equiv.toFun
          { toEquiv := { toFun := toFun✝¹, invFun := invFun✝¹, left_inv := left_inv✝¹, right_inv := right_inv✝¹ },
              map_mul' := map_mul'✝¹, map_add' := map_add'✝¹ }.toEquiv
          (MulOpposite.unop
            (Equiv.toFun
              { toEquiv := { toFun := toFun✝¹, invFun := invFun✝¹, left_inv := left_inv✝¹, right_inv := right_inv✝¹ },
                  map_mul' := map_mul'✝¹, map_add' := map_add'✝¹ }.toEquiv
              x))) =
      x
toFun✝ : R → Rᵐᵒᵖ
invFun✝ : Rᵐᵒᵖ → R
left_inv✝ : Function.LeftInverse invFun✝ toFun✝
right_inv✝ : Function.RightInverse invFun✝ toFun✝
map_mul'✝ :
  ∀ (x y : R),
    Equiv.toFun { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ } (x * y) =
      Equiv.toFun { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ } x *
        Equiv.toFun { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ } y
map_add'✝ :
  ∀ (x y : R),
    Equiv.toFun { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ } (x + y) =
      Equiv.toFun { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ } x +
        Equiv.toFun { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ } y
involution'✝ :
  ∀ (x : R),
    MulOpposite.unop
        (Equiv.toFun
          { toEquiv := { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ },
              map_mul' := map_mul'✝, map_add' := map_add'✝ }.toEquiv
          (MulOpposite.unop
            (Equiv.toFun
              { toEquiv := { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ },
                  map_mul' := map_mul'✝, map_add' := map_add'✝ }.toEquiv
              x))) =
      x
h₁ :
  (fun f => f.toFun)
      {
        toRingEquiv :=
          { toEquiv := { toFun := toFun✝¹, invFun := invFun✝¹, left_inv := left_inv✝¹, right_inv := right_inv✝¹ },
            map_mul' := map_mul'✝¹, map_add' := map_add'✝¹ },
        involution' := involution'✝¹ } =
    (fun f => f.toFun)
      {
        toRingEquiv :=
          { toEquiv := { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ },
            map_mul' := map_mul'✝, map_add' := map_add'✝ },
        involution' := involution'✝ }
h₂ :
  (fun f => f.invFun)
      {
        toRingEquiv :=
          { toEquiv := { toFun := toFun✝¹, invFun := invFun✝¹, left_inv := left_inv✝¹, right_inv := right_inv✝¹ },
            map_mul' := map_mul'✝¹, map_add' := map_add'✝¹ },
        involution' := involution'✝¹ } =
    (fun f => f.invFun)
      {
        toRingEquiv :=
          { toEquiv := { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ },
            map_mul' := map_mul'✝, map_add' := map_add'✝ },
        involution' := involution'✝ }
⊢ {
      toRingEquiv :=
        { toEquiv := { toFun := toFun✝¹, invFun := invFun✝¹, left_inv := left_inv✝¹, right_inv := right_inv✝¹ },
          map_mul' := map_mul'✝¹, map_add' := map_add'✝¹ },
      involution' := involution'✝¹ } =
    {
      toRingEquiv :=
        { toEquiv := { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ },
          map_mul' := map_mul'✝, map_add' := map_add'✝ },
      involution' := involution'✝ }

Next tactic:
congr