-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
h_aJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
a₁ : R :=
  choose (p := fun (x : R) ↦
    ∃ (aI : Ideal R),
      x ≠ 0 ∧
        I =
          HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
            (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
            ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
    ⋯
J₁ : Ideal R :=
  choose (p := fun (x : Ideal R) ↦
    choose (p := fun (x : R) ↦
          ∃ (aI : Ideal R),
            x ≠ 0 ∧
              I =
                HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                  FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                  (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                  ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
          ⋯ ≠
        0 ∧
      I =
        HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
          (spanSingleton R⁰
              ((algebraMap R K : (a : R) → K)
                  (choose (p := fun (x : R) ↦
                    ∃ (aI : Ideal R),
                      x ≠ 0 ∧
                        I =
                          HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                            FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                            (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ :
                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                            ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                    ⋯))⁻¹ :
            FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
          ((↑x : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
    ⋯
h_a₁J₁ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a₁)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J₁ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
h_a₁_ne_zero : a₁ ≠ 0
h_J₁_ne_zero : J₁ ≠ 0
h_a_ne_zero : Ne (α := Ideal R) (Ideal.span {a}) 0
h_J_ne_zero : J ≠ 0
h_a₁' :
  Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a₁) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (0 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
h_a' :
  Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (0 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
h_aJ₁ :
  Eq.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (↑(J₁ * Ideal.span {a}) : FractionalIdeal R⁰ K)
    (↑(J * Ideal.span {a₁}) : FractionalIdeal R⁰ K)
h_a₁J₂ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a₁)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J₁ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
h_a₁_ne_zero₁ : a₁ ≠ 0
h_J₁_ne_zero₁ : J₁ ≠ 0
h_a_ne_zero₁ : Ne (α := Ideal R) (Ideal.span {a}) 0
⊢ J ≠ 0

Tactic:
rw [h_a₁J₁, ← div_spanSingleton, ← div_spanSingleton, div_eq_div_iff h_a₁' h_a',
    ← coeIdeal_span_singleton, ← coeIdeal_span_singleton, ← coeIdeal_mul, ← coeIdeal_mul] at h_aJ

Premises used: FractionalIdeal.coeIdeal_span_singleton, div_eq_div_iff, FractionalIdeal.coeIdeal_mul, FractionalIdeal.div_spanSingleton