Theorem: _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.exists_cube_associated
File: Mathlib/NumberTheory/FLT/Three.lean

Proof steps:
state_before: K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
S' : Solution' hζ
inst✝ : DecidableRel fun a b => a ∣ b
⊢ (∃ X, Associated (X ^ 3) (FermatLastTheoremForThreeGen.Solution.x S)) ∧
    (∃ Y, Associated (Y ^ 3) (FermatLastTheoremForThreeGen.Solution.y S)) ∧
      ∃ Z, Associated (Z ^ 3) (FermatLastTheoremForThreeGen.Solution.z S)
state_after: K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
S' : Solution' hζ
inst✝ : DecidableRel fun a b => a ∣ b
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 h₁ := S.isCoprime_x_z.mul_left S.isCoprime_y_z
state_before: K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
S' : Solution' hζ
inst✝ : DecidableRel fun a b => a ∣ b
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)
state_after: K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
S' : Solution' hζ
inst✝ : DecidableRel fun a b => a ∣ b
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: 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]⟩
state_before: K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
S' : Solution' hζ
inst✝ : DecidableRel fun a b => a ∣ b
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)
state_after: case intro
K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
S' : Solution' hζ
inst✝ : DecidableRel fun a b => a ∣ b
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: obtain ⟨T, h₃⟩ := exists_associated_pow_of_associated_pow_mul h₁ h₂
state_before: case intro
K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
S' : Solution' hζ
inst✝ : DecidableRel fun a b => a ∣ b
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)
state_after: no goals
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)⟩