simp_rw [← smul_eq_mul K, ← coeff_smul]
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
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