Theorem: Hilbert92ish
Proof state before:
case neg.intro.intro.intro.intro.intro.intro.intro.right.refine_2
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H✝ : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H✝
inst✝⁷ : Fintype H✝
hCard : Fintype.card H✝ = ↑p
σ✝ : H✝
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H✝ G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
NE_p_pow : (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) NE = E ^ ↑p
H : Fin (r + 1) → Additive (𝓞 K)ˣ := unitlifts p hp hKL σ hσ S
N : Fin (r + 1) → Additive (𝓞 k)ˣ :=
  fun e => Additive.ofMul ((Units.map (RingOfIntegers.norm k)) (Additive.toMul (H e)))
η : Fin (r + 2) → Additive (𝓞 k)ˣ := Fin.snoc N (Additive.ofMul NE)
a : ℤ
ι : Fin (r + 2) → ℤ
i : Fin (r + 2)
ha : ∑ i : Fin (r + 2), ι i • η i = (a * ↑↑p) • Additive.ofMul ζ
ha' : ¬↑↑p ∣ ι i
ha'' : ζ = 1 → i ≠ Fin.last (r + 1)
H2 : Fin (r + 2) → Additive (𝓞 K)ˣ := Fin.snoc H (Additive.ofMul E)
J : (𝓞 K)ˣ := Additive.toMul (∑ i : Fin (r + 2), ι i • H2 i) * (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ζ ^ (-a)
ε : (𝓞 K)ˣ
hε : (algebraMap (𝓞 K) K) ↑J = (algebraMap (𝓞 K) K) ↑ε / σ ((algebraMap (𝓞 K) K) ↑ε)
⊢ (1 - zeta ↑p) • mkG ε = ∑ i : Fin (r + 1), (ι ∘ Fin.castSucc) i • S.units i
Tactic:
rw [← u_lemma2 p hp hKL σ hσ _ _ hε, unit_to_U_mul, toMul_sum, unit_to_U_prod,
  Fin.sum_univ_castSucc]
Proof state after:
case neg.intro.intro.intro.intro.intro.intro.intro.right.refine_2
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H✝ : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H✝
inst✝⁷ : Fintype H✝
hCard : Fintype.card H✝ = ↑p
σ✝ : H✝
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H✝ G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
NE_p_pow : (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) NE = E ^ ↑p
H : Fin (r + 1) → Additive (𝓞 K)ˣ := unitlifts p hp hKL σ hσ S
N : Fin (r + 1) → Additive (𝓞 k)ˣ :=
  fun e => Additive.ofMul ((Units.map (RingOfIntegers.norm k)) (Additive.toMul (H e)))
η : Fin (r + 2) → Additive (𝓞 k)ˣ := Fin.snoc N (Additive.ofMul NE)
a : ℤ
ι : Fin (r + 2) → ℤ
i : Fin (r + 2)
ha : ∑ i : Fin (r + 2), ι i • η i = (a * ↑↑p) • Additive.ofMul ζ
ha' : ¬↑↑p ∣ ι i
ha'' : ζ = 1 → i ≠ Fin.last (r + 1)
H2 : Fin (r + 2) → Additive (𝓞 K)ˣ := Fin.snoc H (Additive.ofMul E)
J : (𝓞 K)ˣ := Additive.toMul (∑ i : Fin (r + 2), ι i • H2 i) * (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ζ ^ (-a)
ε : (𝓞 K)ˣ
hε : (algebraMap (𝓞 K) K) ↑J = (algebraMap (𝓞 K) K) ↑ε / σ ((algebraMap (𝓞 K) K) ↑ε)
⊢ ∑ i : Fin (r + 1), mkG (Additive.toMul (ι i.castSucc • H2 i.castSucc)) +
        mkG (Additive.toMul (ι (Fin.last (r + 1)) • H2 (Fin.last (r + 1)))) +
      mkG ((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ζ ^ (-a)) =
    ∑ i : Fin (r + 1), (ι ∘ Fin.castSucc) i • S.units i