Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
a₁ : R := choose ⋯
J₁ : Ideal R := choose ⋯
h_aJ : ↑(J₁ * Ideal.span {a}) = ↑(J * Ideal.span {a₁})
h_a₁J₁ : I = spanSingleton R⁰ ((algebraMap R K) a₁)⁻¹ * ↑J₁
h_a₁_ne_zero : a₁ ≠ 0
h_J₁_ne_zero : J₁ ≠ 0
h_a_ne_zero : Ideal.span {a} ≠ 0
h_J_ne_zero : J ≠ 0
h_a₁' : spanSingleton R⁰ ((algebraMap R K) a₁) ≠ 0
h_a' : spanSingleton R⁰ ((algebraMap R K) a) ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
⊢ Associates.mk (Ideal.span {choose ⋯}) ≠ 0

pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
a₁ : R := choose ⋯
J₁ : Ideal R := choose ⋯
h_aJ : ↑(J₁ * Ideal.span {a}) = ↑(J * Ideal.span {a₁})
h_a₁J₁ : I = spanSingleton R⁰ ((algebraMap R K) a₁)⁻¹ * ↑J₁
h_a₁_ne_zero : a₁ ≠ 0
h_J₁_ne_zero : J₁ ≠ 0
h_a_ne_zero : Ideal.span {a} ≠ 0
h_J_ne_zero : J ≠ 0
h_a₁' : spanSingleton R⁰ ((algebraMap R K) a₁) ≠ 0
h_a' : spanSingleton R⁰ ((algebraMap R K) a) ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
⊢ Associates.mk (choose ⋯) ≠ 0

pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
a₁ : R := choose ⋯
J₁ : Ideal R := choose ⋯
h_aJ : ↑(J₁ * Ideal.span {a}) = ↑(J * Ideal.span {a₁})
h_a₁J₁ : I = spanSingleton R⁰ ((algebraMap R K) a₁)⁻¹ * ↑J₁
h_a₁_ne_zero : a₁ ≠ 0
h_J₁_ne_zero : J₁ ≠ 0
h_a_ne_zero : Ideal.span {a} ≠ 0
h_J_ne_zero : J ≠ 0
h_a₁' : spanSingleton R⁰ ((algebraMap R K) a₁) ≠ 0
h_a' : spanSingleton R⁰ ((algebraMap R K) a) ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
⊢ Associates.mk (Ideal.span {a}) ≠ 0

Tactic:
· rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
    exact h_a₁_ne_zero

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
a₁ : R := choose ⋯
J₁ : Ideal R := choose ⋯
h_aJ : ↑(J₁ * Ideal.span {a}) = ↑(J * Ideal.span {a₁})
h_a₁J₁ : I = spanSingleton R⁰ ((algebraMap R K) a₁)⁻¹ * ↑J₁
h_a₁_ne_zero : a₁ ≠ 0
h_J₁_ne_zero : J₁ ≠ 0
h_a_ne_zero : Ideal.span {a} ≠ 0
h_J_ne_zero : J ≠ 0
h_a₁' : spanSingleton R⁰ ((algebraMap R K) a₁) ≠ 0
h_a' : spanSingleton R⁰ ((algebraMap R K) a) ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
⊢ Associates.mk (choose ⋯) ≠ 0

R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
a₁ : R := choose ⋯
J₁ : Ideal R := choose ⋯
h_aJ : ↑(J₁ * Ideal.span {a}) = ↑(J * Ideal.span {a₁})
h_a₁J₁ : I = spanSingleton R⁰ ((algebraMap R K) a₁)⁻¹ * ↑J₁
h_a₁_ne_zero : a₁ ≠ 0
h_J₁_ne_zero : J₁ ≠ 0
h_a_ne_zero : Ideal.span {a} ≠ 0
h_J_ne_zero : J ≠ 0
h_a₁' : spanSingleton R⁰ ((algebraMap R K) a₁) ≠ 0
h_a' : spanSingleton R⁰ ((algebraMap R K) a) ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
⊢ Associates.mk (Ideal.span {a}) ≠ 0