simp only [← prod_span_singleton, ← span_singleton_pow] at H₁
p : ℕ
hpri : Fact p.Prime
a b c : ℤ
h5p : 5 ≤ p
H : a ^ p + b ^ p = c ^ p
hgcd : {a, b, c}.gcd id = 1
caseI : ¬↑p ∣ a * b * c
ζ : 𝓞 (CyclotomicField ⟨p, ⋯⟩ ℚ)
hζ : ζ ∈ nthRootsFinset p (𝓞 (CyclotomicField ⟨p, ⋯⟩ ℚ))
hζ' : IsPrimitiveRoot ↑⋯.unit' ↑⟨p, ⋯⟩
H₁ :
  (fun x => span {x}) (∏ ζ ∈ nthRootsFinset p (𝓞 (CyclotomicField ⟨p, ⋯⟩ ℚ)), (↑a + ζ * ↑b)) =
    (fun x => span {x}) (↑c ^ p)
⊢ ∃ I, span {↑a + ζ * ↑b} = I ^ p
p : ℕ
hpri : Fact p.Prime
a b c : ℤ
h5p : 5 ≤ p
H : a ^ p + b ^ p = c ^ p
hgcd : {a, b, c}.gcd id = 1
caseI : ¬↑p ∣ a * b * c
ζ : 𝓞 (CyclotomicField ⟨p, ⋯⟩ ℚ)
hζ : ζ ∈ nthRootsFinset p (𝓞 (CyclotomicField ⟨p, ⋯⟩ ℚ))
hζ' : IsPrimitiveRoot ↑⋯.unit' ↑⟨p, ⋯⟩
H₁ : ∏ i ∈ nthRootsFinset p (𝓞 (CyclotomicField ⟨p, ⋯⟩ ℚ)), span {↑a + i * ↑b} = span {↑c} ^ p
⊢ ∃ I, span {↑a + ζ * ↑b} = I ^ p