have h_n : n ≠ 0 := by
  by_contra h
  subst h
  simp only [_root_.pow_zero] at h_card
  rcases H with ⟨x, y, H⟩
  absurd H
  exact Fintype.card_le_one_iff.mp (Nat.le_of_eq h_card) x y
case intro
R : Type u_1
inst✝¹ : DecidableEq R
inst✝ : DivisionRing R
h : Fintype R
Z : Subring R := center R
n : ℕ
h_card : Fintype.card R = Fintype.card ↥Z ^ n
H : ∃ x y, x ≠ y
⊢ IsField R
case intro
R : Type u_1
inst✝¹ : DecidableEq R
inst✝ : DivisionRing R
h : Fintype R
Z : Subring R := center R
n : ℕ
h_card : Fintype.card R = Fintype.card ↥Z ^ n
H : ∃ x y, x ≠ y
h_n : n ≠ 0
⊢ IsField R