-- Mathlib/NumberTheory/FLT/Three.lean
private lemma exists_cube_associated :
    (∃ X, Associated (X ^ 3) S.x) ∧ (∃ Y, Associated (Y ^ 3) S.y) ∧
      ∃ Z, Associated (Z ^ 3) S.z

Proof state:
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ (∃ X, Associated (X ^ 3) (FermatLastTheoremForThreeGen.Solution.x S)) ∧
    (∃ Y, Associated (Y ^ 3) (FermatLastTheoremForThreeGen.Solution.y S)) ∧
      ∃ Z, Associated (Z ^ 3) (FermatLastTheoremForThreeGen.Solution.z S)

Tactic:
classical
have h₁ := S.isCoprime_x_z.mul_left S.isCoprime_y_z
have h₂ : Associated (S.w ^ 3) (S.x * S.y * S.z) :=
⟨S.u, by rw [x_mul_y_mul_z_eq_u_mul_w_cube S, mul_comm]⟩
obtain ⟨T, h₃⟩ := exists_associated_pow_of_associated_pow_mul h₁ h₂
exact ⟨exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y h₃,
exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y.symm (mul_comm _ S.x ▸ h₃),
exists_associated_pow_of_associated_pow_mul h₁.symm (mul_comm _ S.z ▸ h₂)⟩

Result:
no goals

Proof state:
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ (∃ X, Associated (X ^ 3) (FermatLastTheoremForThreeGen.Solution.x S)) ∧
    (∃ Y, Associated (Y ^ 3) (FermatLastTheoremForThreeGen.Solution.y S)) ∧
      ∃ Z, Associated (Z ^ 3) (FermatLastTheoremForThreeGen.Solution.z S)

Tactic:
have h₁ := S.isCoprime_x_z.mul_left S.isCoprime_y_z

Result:
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h₁ :
  IsCoprime (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S)
    (FermatLastTheoremForThreeGen.Solution.z S)
⊢ (∃ X, Associated (X ^ 3) (FermatLastTheoremForThreeGen.Solution.x S)) ∧
    (∃ Y, Associated (Y ^ 3) (FermatLastTheoremForThreeGen.Solution.y S)) ∧
      ∃ Z, Associated (Z ^ 3) (FermatLastTheoremForThreeGen.Solution.z S)

Proof state:
case intro
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h₁ :
  IsCoprime (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S)
    (FermatLastTheoremForThreeGen.Solution.z S)
h₂ :
  Associated (FermatLastTheoremForThreeGen.Solution.w S ^ 3)
    (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S *
      FermatLastTheoremForThreeGen.Solution.z S)
T : 𝓞 K
h₃ : Associated (T ^ 3) (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S)
⊢ (∃ X, Associated (X ^ 3) (FermatLastTheoremForThreeGen.Solution.x S)) ∧
    (∃ Y, Associated (Y ^ 3) (FermatLastTheoremForThreeGen.Solution.y S)) ∧
      ∃ Z, Associated (Z ^ 3) (FermatLastTheoremForThreeGen.Solution.z S)

Tactic:
exact ⟨exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y h₃,
  exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y.symm (mul_comm _ S.x ▸ h₃),
  exists_associated_pow_of_associated_pow_mul h₁.symm (mul_comm _ S.z ▸ h₂)⟩

Result:
no goals

Proof state:
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h₁ :
  IsCoprime (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S)
    (FermatLastTheoremForThreeGen.Solution.z S)
h₂ :
  Associated (FermatLastTheoremForThreeGen.Solution.w S ^ 3)
    (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S *
      FermatLastTheoremForThreeGen.Solution.z S)
⊢ (∃ X, Associated (X ^ 3) (FermatLastTheoremForThreeGen.Solution.x S)) ∧
    (∃ Y, Associated (Y ^ 3) (FermatLastTheoremForThreeGen.Solution.y S)) ∧
      ∃ Z, Associated (Z ^ 3) (FermatLastTheoremForThreeGen.Solution.z S)

Tactic:
obtain ⟨T, h₃⟩ := exists_associated_pow_of_associated_pow_mul h₁ h₂

Result:
case intro
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h₁ :
  IsCoprime (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S)
    (FermatLastTheoremForThreeGen.Solution.z S)
h₂ :
  Associated (FermatLastTheoremForThreeGen.Solution.w S ^ 3)
    (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S *
      FermatLastTheoremForThreeGen.Solution.z S)
T : 𝓞 K
h₃ : Associated (T ^ 3) (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S)
⊢ (∃ X, Associated (X ^ 3) (FermatLastTheoremForThreeGen.Solution.x S)) ∧
    (∃ Y, Associated (Y ^ 3) (FermatLastTheoremForThreeGen.Solution.y S)) ∧
      ∃ Z, Associated (Z ^ 3) (FermatLastTheoremForThreeGen.Solution.z S)

Proof state:
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h₁ :
  IsCoprime (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S)
    (FermatLastTheoremForThreeGen.Solution.z S)
⊢ (∃ X, Associated (X ^ 3) (FermatLastTheoremForThreeGen.Solution.x S)) ∧
    (∃ Y, Associated (Y ^ 3) (FermatLastTheoremForThreeGen.Solution.y S)) ∧
      ∃ Z, Associated (Z ^ 3) (FermatLastTheoremForThreeGen.Solution.z S)

Tactic:
have h₂ : Associated (S.w ^ 3) (S.x * S.y * S.z) :=
  ⟨S.u, by rw [x_mul_y_mul_z_eq_u_mul_w_cube S, mul_comm]⟩

Result:
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h₁ :
  IsCoprime (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S)
    (FermatLastTheoremForThreeGen.Solution.z S)
h₂ :
  Associated (FermatLastTheoremForThreeGen.Solution.w S ^ 3)
    (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S *
      FermatLastTheoremForThreeGen.Solution.z S)
⊢ (∃ X, Associated (X ^ 3) (FermatLastTheoremForThreeGen.Solution.x S)) ∧
    (∃ Y, Associated (Y ^ 3) (FermatLastTheoremForThreeGen.Solution.y S)) ∧
      ∃ Z, Associated (Z ^ 3) (FermatLastTheoremForThreeGen.Solution.z S)

Proof state:
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h₁ :
  IsCoprime (FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S)
    (FermatLastTheoremForThreeGen.Solution.z S)
⊢ FermatLastTheoremForThreeGen.Solution.w S ^ 3 * ↑S.u =
    FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S *
      FermatLastTheoremForThreeGen.Solution.z S

Tactic:
rw [x_mul_y_mul_z_eq_u_mul_w_cube S, mul_comm]

Result:
no goals