simp only [Subalgebra.algebraMap_eq, Algebra.id.map_eq_id, RingHomCompTriple.comp_eq, coeff_map,
  RingHom.coe_coe, Subalgebra.coe_val, one_div, map_sub, map_one, coeff_add, coeff_sub,
  PNat.pos, pow_eq_zero_iff, this, mul_add, IsPrimitiveRoot.val_unit'_coe,
  IsPrimitiveRoot.coe_unit'_coe]
case a.a
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
i : ℕ
this :
  (ζ - 1) ^ ↑p * ↑((poly hp hζ u hcong).coeff i) =
    (((C ((algebraMap (𝓞 K) K) ↑hζ.unit') - 1) * X - 1) ^ ↑p).coeff i + (C ((algebraMap (𝓞 K) K) ↑u)).coeff i
⊢ (fun x => (ζ - 1) ^ ↑p * x) ((map (algebraMap (𝓞 K) K) (poly hp hζ u hcong)).coeff i) =
    (fun x => (ζ - 1) ^ ↑p * x) (((X - C (1 / (ζ - 1))) ^ ↑p + C ((algebraMap (𝓞 K) K) ↑u / (ζ - 1) ^ ↑p)).coeff i)
case a.a
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
i : ℕ
this :
  (ζ - 1) ^ ↑p * ↑((poly hp hζ u hcong).coeff i) =
    (((C ((algebraMap (𝓞 K) K) ↑hζ.unit') - 1) * X - 1) ^ ↑p).coeff i + (C ((algebraMap (𝓞 K) K) ↑u)).coeff i
⊢ (((C ζ - 1) * X - 1) ^ ↑p).coeff i + (C ((algebraMap (𝓞 K) K) ↑u)).coeff i =
    (ζ - 1) ^ ↑p * ((X - C (ζ - 1)⁻¹) ^ ↑p).coeff i +
      (ζ - 1) ^ ↑p * (C ((algebraMap (𝓞 K) K) ↑u / (ζ - 1) ^ ↑p)).coeff i