have := congr_arg (fun P : (𝓞 K)[X] ↦ (↑(coeff P i) : K)) (poly_spec hp hζ u hcong)
case 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
⊢ (map (algebraMap (𝓞 K) K) (poly hp hζ u hcong)).coeff i =
    ((X - C (1 / (ζ - 1))) ^ ↑p + C ((algebraMap (𝓞 K) K) ↑u / (ζ - 1) ^ ↑p)).coeff i
case 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 :
  (fun P => ↑(P.coeff i)) (C ((↑hζ.unit' - 1) ^ ↑p) * poly hp hζ u hcong) =
    (fun P => ↑(P.coeff i)) ((C (↑hζ.unit' - 1) * X - 1) ^ ↑p + C ↑u)
⊢ (map (algebraMap (𝓞 K) K) (poly hp hζ u hcong)).coeff i =
    ((X - C (1 / (ζ - 1))) ^ ↑p + C ((algebraMap (𝓞 K) K) ↑u / (ζ - 1) ^ ↑p)).coeff i