rw [← (div_zeta_sub_one_Injective hp hζ e hy).eq_iff, zeta_sub_one_dvd_root_spec,
  Ideal.Quotient.eq_zero_iff_dvd, ← Ideal.mem_span_singleton (α := 𝓞 K),
  ← Ideal.dvd_span_singleton, ← div_zeta_sub_one_dvd_gcd_spec (hy := hy),
  ← dvd_gcd_mul_iff_dvd_mul, gcd_comm, gcd_zeta_sub_one_eq_one hζ hy, one_mul]
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
inst✝ : Fintype (ClassGroup (𝓞 K))
hreg : (↑p).Coprime (Fintype.card (ClassGroup (𝓞 K)))
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
x y z : 𝓞 K
ε : (𝓞 K)ˣ
m : ℕ
e : x ^ ↑p + y ^ ↑p = ↑ε * (π ^ (m + 1) * z) ^ ↑p
hy : ¬π ∣ y
hz : ¬π ∣ z
η : { x // x ∈ nthRootsFinset (↑p) (𝓞 K) }
⊢ 𝔭 ∣ 𝔠 η ↔ η = η₀
no goals