Domain implies char ne 2

lemma domain_implies_char_ne_2 [NeZero n] (ω : D) :  ¬CharP F 2 := fun hchar ↦ by  have hn : n ≠ 0 := NeZero.ne _  set k : Fin (2 ^ n) := ⟨2 ^ (n - 1), pow_lt_pow_right₀ (by decide) (by omega)⟩  have hk_ne_zero : k ≠ 0 := by simp [Fin.ext_iff, k]  have h_ne_val : ω k ≠ ω 0 := fun h => hk_ne_zero (CosetFftDomainClass.injective ω h)  have h_ne_one : ω k ≠ 1 := by rwa [apply_zero_eq_one] at h_ne_val  have h_kk : k + k = 0 := by    ext    simp only [Fin.val_add, Fin.coe_ofNat_eq_mod, Nat.zero_mod, k]    rcases n with _ | n <;> simp_all    ring_nf    simp  have h_sq : (ω k) ^ 2 = 1 := by    rw [sq, ←apply_add_eq_mul, h_kk, apply_zero_eq_one]  have h_eq : ω k = 1 ∨ ω k = -1 := sq_eq_one_iff.mp h_sq  have h_neg_eq_pos : (-1 : F) = 1 := by    have : (2 : F) = 0 := CharP.cast_eq_zero F 2    conv_rhs =>      rw [show (1 : F) = 2 - 1 by norm_num, this]    simp  rcases h_eq with h | h  · exact h_ne_one h  · exact h_ne_one (by rwa [h_neg_eq_pos] at h)