Theorem: Hilbert92ish (FltRegular/NumberTheory/Hilbert92.lean)

State before:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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 (Fin.last (r + 1))
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)
hε' : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
⊢ ∃ ε', ε' ^ ↑p = NE
Tactic:
rw [← (Nat.prime_iff_prime_int.mp hp).coprime_iff_not_dvd] at ha'
State after:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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' : IsCoprime (↑↑p) (ι (Fin.last (r + 1)))
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)
hε' : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
⊢ ∃ ε', ε' ^ ↑p = NE

State before:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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' : IsCoprime (↑↑p) (ι (Fin.last (r + 1)))
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)
hε' : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
⊢ ∃ ε', ε' ^ ↑p = NE
Tactic:
obtain ⟨α, β, hαβ⟩ := ha'
State after:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
hε' : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
⊢ ∃ ε', ε' ^ ↑p = NE

State before:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
hε' : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
⊢ ∃ ε', ε' ^ ↑p = NE
Tactic:
choose ι' hι' using hε'
State after:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
⊢ ∃ ε', ε' ^ ↑p = NE

State before:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
⊢ ∃ ε', ε' ^ ↑p = NE
Tactic:
rw [Fin.sum_univ_castSucc] at ha
State after:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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 + 1), ι i.castSucc • η i.castSucc + ι (Fin.last (r + 1)) • η (Fin.last (r + 1)) =
    (a * ↑↑p) • Additive.ofMul ζ
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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
⊢ ∃ ε', ε' ^ ↑p = NE

State before:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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 + 1), ι i.castSucc • η i.castSucc + ι (Fin.last (r + 1)) • η (Fin.last (r + 1)) =
    (a * ↑↑p) • Additive.ofMul ζ
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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
⊢ ∃ ε', ε' ^ ↑p = NE
Tactic:
simp (config := { zeta := false, proj := false }) only
  [hι', Fin.snoc_castSucc, Fin.snoc_last, mul_smul, η] at ha
State after:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
ha : ∑ x : Fin (r + 1), ↑↑p • ι' x • N x + ι (Fin.last (r + 1)) • Additive.ofMul NE = a • ↑↑p • Additive.ofMul ζ
⊢ ∃ ε', ε' ^ ↑p = NE

State before:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
ha : ∑ x : Fin (r + 1), ↑↑p • ι' x • N x + ι (Fin.last (r + 1)) • Additive.ofMul NE = a • ↑↑p • Additive.ofMul ζ
⊢ ∃ ε', ε' ^ ↑p = NE
Tactic:
rw [← smul_sum, add_comm, ← eq_sub_iff_add_eq, smul_comm, ← smul_sub] at ha
State after:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
ha : ι (Fin.last (r + 1)) • Additive.ofMul NE = ↑↑p • (a • Additive.ofMul ζ - ∑ x : Fin (r + 1), ι' x • N x)
⊢ ∃ ε', ε' ^ ↑p = NE

State before:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
ha :
  ↑↑p • α • Additive.ofMul NE + β • ι (Fin.last (r + 1)) • Additive.ofMul NE =
    ↑↑p • α • Additive.ofMul NE + β • ↑↑p • (a • Additive.ofMul ζ - ∑ x : Fin (r + 1), ι' x • N x)
⊢ ∃ ε', ε' ^ ↑p = NE
Tactic:
conv_rhs at ha => rw [smul_comm β, ← smul_add]
State after:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
ha :
  ↑↑p • α • Additive.ofMul NE + β • ι (Fin.last (r + 1)) • Additive.ofMul NE =
    ↑↑p • (α • Additive.ofMul NE + β • (a • Additive.ofMul ζ - ∑ x : Fin (r + 1), ι' x • N x))
⊢ ∃ ε', ε' ^ ↑p = NE

State before:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
ha :
  ↑↑p • α • Additive.ofMul NE + β • ι (Fin.last (r + 1)) • Additive.ofMul NE =
    ↑↑p • (α • Additive.ofMul NE + β • (a • Additive.ofMul ζ - ∑ x : Fin (r + 1), ι' x • N x))
⊢ ∃ ε', ε' ^ ↑p = NE
Tactic:
rw [smul_smul, smul_smul, ← add_smul, mul_comm _ α, hαβ, one_smul] at ha
State after:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
ha : Additive.ofMul NE = ↑↑p • (α • Additive.ofMul NE + β • (a • Additive.ofMul ζ - ∑ x : Fin (r + 1), ι' x • N x))
⊢ ∃ ε', ε' ^ ↑p = NE

State before:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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'' : 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)
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
α β : ℤ
hαβ : α * ↑↑p + β * ι (Fin.last (r + 1)) = 1
ι' : Fin (r + 1) → ℤ
hι' : ∀ (x : Fin (r + 1)), ι x.castSucc = ↑↑p * ι' x
ha : Additive.ofMul NE = ↑↑p • (α • Additive.ofMul NE + β • (a • Additive.ofMul ζ - ∑ x : Fin (r + 1), ι' x • N x))
⊢ ∃ ε', ε' ^ ↑p = NE
Tactic:
exact ⟨_, ha.symm⟩
State after:
no goals

State before:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
⊢ ∃ ε'', E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''
Tactic:
rw [← hε', map_pow, eq_comm, ← mul_inv_eq_one, ← inv_pow, ← mul_pow] at NE_p_pow
State after:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
NE_p_pow : (E * ((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε')⁻¹) ^ ↑p = 1
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
⊢ ∃ ε'', E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''

State before:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
NE_p_pow : (E * ((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε')⁻¹) ^ ↑p = 1
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
⊢ ∃ ε'', E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''
Tactic:
apply_fun ((↑) : (𝓞 K)ˣ → K) at NE_p_pow
State after:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ↑((E * ((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε')⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) ↑1
⊢ ∃ ε'', E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''

State before:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ↑((E * ((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε')⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) ↑1
⊢ ∃ ε'', E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''
Tactic:
simp only [RingHom.toMonoidHom_eq_coe, Units.val_pow_eq_pow_val, Units.val_mul,
  Units.coe_map_inv, MonoidHom.coe_coe, SubmonoidClass.coe_pow, Submonoid.coe_mul,
  Subsemiring.coe_toSubmonoid, Subalgebra.coe_toSubsemiring, Units.val_one,
  OneMemClass.coe_one, RingOfInteger.coe_algebraMap_apply] at NE_p_pow
State after:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
⊢ ∃ ε'', E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''

State before:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
⊢ ∃ ε'', E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''
Tactic:
obtain ⟨i, -, e⟩ := hζ''.eq_pow_of_pow_eq_one NE_p_pow p.pos
State after:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i✝ = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
i : ℕ
e : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h) ^ i = ↑(↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹)
⊢ ∃ ε'', E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''

State before:
case intro.intro
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i✝ = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
i : ℕ
e : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h) ^ i = ↑(↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹)
⊢ ∃ ε'', E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''
Tactic:
use ((ζ ^ (p : ℕ) ^ h) ^ i * ε')
State after:
case h
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i✝ = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
i : ℕ
e : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h) ^ i = ↑(↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹)
⊢ E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ((ζ ^ ↑p ^ h) ^ i * ε')

State before:
case h
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i✝ = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
i : ℕ
e : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h) ^ i = ↑(↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹)
⊢ E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ((ζ ^ ↑p ^ h) ^ i * ε')
Tactic:
rw [map_mul, ← mul_inv_eq_iff_eq_mul]
State after:
case h
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i✝ = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
i : ℕ
e : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h) ^ i = ↑(↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹)
⊢ E * ((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε')⁻¹ = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ((ζ ^ ↑p ^ h) ^ i)

State before:
case h
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i✝ = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
i : ℕ
e : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h) ^ i = ↑(↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹)
⊢ E * ((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε')⁻¹ = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ((ζ ^ ↑p ^ h) ^ i)
Tactic:
ext
State after:
case h.a.h
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i✝ = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
i : ℕ
e : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h) ^ i = ↑(↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹)
⊢ ↑↑(E * ((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε')⁻¹) = ↑↑((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ((ζ ^ ↑p ^ h) ^ i))

State before:
case h.a.h
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i✝ = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
i : ℕ
e : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h) ^ i = ↑(↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹)
⊢ ↑↑(E * ((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε')⁻¹) = ↑↑((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ((ζ ^ ↑p ^ h) ^ i))
Tactic:
simpa using e.symm
State after:
no goals

State before:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
ε'' : (𝓞 k)ˣ
hε'' : E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) (↑ζ ^ ↑p ^ h)) = (algebraMap k K) ↑↑ε'' / (algebraMap k K) ↑↑ε''
⊢ (algebraMap k K) (↑↑ζ ^ ↑p ^ h) = 1
Tactic:
rwa [div_self (by simp)] at hE
State after:
no goals

State before:
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
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
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 (Fin.last (r + 1))
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)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
ε'' : (𝓞 k)ˣ
hε'' : E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) (↑ζ ^ ↑p ^ h)) = (algebraMap k K) ↑↑ε'' / (algebraMap k K) ↑↑ε''
⊢ (algebraMap k K) ↑↑ε'' ≠ 0
Tactic:
simp
State after:
no goals

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]
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

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)
⊢ ∑ 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
Tactic:
simp only [Fin.snoc_castSucc, toMul_zsmul, unit_to_U_zpow, unitlifts_spec, Fin.snoc_last,
  toMul_ofMul, RingHom.toMonoidHom_eq_coe, zpow_neg, unit_to_U_inv, Function.comp_apply,
  unit_to_U_map, smul_zero, neg_zero, add_zero, add_right_eq_self, NE, η, H2, J, N, H]
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)
⊢ ι (Fin.last (r + 1)) • mkG E = 0

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)
⊢ ι (Fin.last (r + 1)) • mkG E = 0
Tactic:
apply_fun mkG at NE_p_pow
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
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)
NE_p_pow : mkG ((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) NE) = mkG (E ^ ↑p)
⊢ ι (Fin.last (r + 1)) • mkG E = 0

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
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)
NE_p_pow : mkG ((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) NE) = mkG (E ^ ↑p)
⊢ ι (Fin.last (r + 1)) • mkG E = 0
Tactic:
simp only [RingHom.toMonoidHom_eq_coe, unit_to_U_map,
  unit_to_U_neg, unit_to_U_pow] at NE_p_pow
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
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)
NE_p_pow : 0 = ↑p • mkG E
⊢ ι (Fin.last (r + 1)) • mkG E = 0

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
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)
NE_p_pow : 0 = ↑p • mkG E
⊢ ι (Fin.last (r + 1)) • mkG E = 0
Tactic:
rw [eq_comm, smul_eq_zero] at NE_p_pow
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
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)
NE_p_pow : ↑p = 0 ∨ mkG E = 0
⊢ ι (Fin.last (r + 1)) • mkG E = 0

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
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)
NE_p_pow : ↑p = 0 ∨ mkG E = 0
⊢ ι (Fin.last (r + 1)) • mkG E = 0
Tactic:
simp only [Nat.cast_eq_zero, PNat.ne_zero, false_or] at NE_p_pow
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
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)
NE_p_pow : mkG E = 0
⊢ ι (Fin.last (r + 1)) • mkG E = 0

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
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)
NE_p_pow : mkG E = 0
⊢ ι (Fin.last (r + 1)) • mkG E = 0
Tactic:
rw [NE_p_pow, smul_zero]
State after:
no goals