-- Mathlib/RingTheory/FractionalIdeal/Operations.lean
theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
    (hI : IsNoetherian R₁ I) :
    IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K)

Proof state:
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
J : FractionalIdeal R₁⁰ K
hJ : J * spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≤ I
⊢ (↑J).FG

Tactic:
obtain ⟨s, hs⟩ := hI _ hJ

Result:
case neg.intro
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
J : FractionalIdeal R₁⁰ K
hJ : J * spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≤ I
s : Finset K
hs : span R₁ ↑s = ↑(J * spanSingleton R₁⁰ ((algebraMap R₁ K) x))
⊢ (↑J).FG

Proof state:
case pos
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetherian R₁ ↥↑I
hx : x = 0
⊢ IsNoetherian R₁ ↥↑0

Tactic:
exact isNoetherian_zero

Result:
no goals

Proof state:
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
J : FractionalIdeal R₁⁰ K
hJ : J ≤ spanSingleton R₁⁰ ((algebraMap R₁ K) x)⁻¹ * I
⊢ (↑J).FG

Tactic:
rw [← div_spanSingleton, le_div_iff_mul_le h_spanx] at hJ

Result:
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
J : FractionalIdeal R₁⁰ K
hJ : J * spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≤ I
⊢ (↑J).FG

Proof state:
case h
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
J : FractionalIdeal R₁⁰ K
hJ : J * spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≤ I
s : Finset K
hs : span R₁ ↑s = ↑(J * spanSingleton R₁⁰ ((algebraMap R₁ K) x))
⊢ span R₁ ↑(s * {((algebraMap R₁ K) x)⁻¹}) = ↑J

Tactic:
rw [Finset.coe_mul, Finset.coe_singleton, ← span_mul_span, hs, ← coe_spanSingleton R₁⁰, ←
  coe_mul, mul_assoc, spanSingleton_mul_spanSingleton, mul_inv_cancel₀ h_gx, spanSingleton_one,
  mul_one]

Result:
no goals

Proof state:
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
⊢ ∀ J ≤ spanSingleton R₁⁰ ((algebraMap R₁ K) x)⁻¹ * I, (↑J).FG

Tactic:
intro J hJ

Result:
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
J : FractionalIdeal R₁⁰ K
hJ : J ≤ spanSingleton R₁⁰ ((algebraMap R₁ K) x)⁻¹ * I
⊢ (↑J).FG

Proof state:
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetherian R₁ ↥↑I
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
⊢ IsNoetherian R₁ ↥↑(spanSingleton R₁⁰ ((algebraMap R₁ K) x)⁻¹ * I)

Tactic:
have h_spanx : spanSingleton R₁⁰ (algebraMap R₁ K x) ≠ 0 := spanSingleton_ne_zero_iff.mpr h_gx

Result:
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetherian R₁ ↥↑I
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
⊢ IsNoetherian R₁ ↥↑(spanSingleton R₁⁰ ((algebraMap R₁ K) x)⁻¹ * I)

Proof state:
case pos
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetherian R₁ ↥↑I
hx : x = 0
⊢ IsNoetherian R₁ ↥↑(spanSingleton R₁⁰ ((algebraMap R₁ K) x)⁻¹ * I)

Tactic:
rw [hx, RingHom.map_zero, inv_zero, spanSingleton_zero, zero_mul]

Result:
case pos
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetherian R₁ ↥↑I
hx : x = 0
⊢ IsNoetherian R₁ ↥↑0

Proof state:
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetherian R₁ ↥↑I
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
⊢ IsNoetherian R₁ ↥↑(spanSingleton R₁⁰ ((algebraMap R₁ K) x)⁻¹ * I)

Tactic:
rw [isNoetherian_iff] at hI ⊢

Result:
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
⊢ ∀ J ≤ spanSingleton R₁⁰ ((algebraMap R₁ K) x)⁻¹ * I, (↑J).FG

Proof state:
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetherian R₁ ↥↑I
hx : ¬x = 0
⊢ IsNoetherian R₁ ↥↑(spanSingleton R₁⁰ ((algebraMap R₁ K) x)⁻¹ * I)

Tactic:
have h_gx : algebraMap R₁ K x ≠ 0 :=
  mt ((injective_iff_map_eq_zero (algebraMap R₁ K)).mp (IsFractionRing.injective _ _) x) hx

Result:
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetherian R₁ ↥↑I
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
⊢ IsNoetherian R₁ ↥↑(spanSingleton R₁⁰ ((algebraMap R₁ K) x)⁻¹ * I)

Proof state:
case neg.intro
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
J : FractionalIdeal R₁⁰ K
hJ : J * spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≤ I
s : Finset K
hs : span R₁ ↑s = ↑(J * spanSingleton R₁⁰ ((algebraMap R₁ K) x))
⊢ (↑J).FG

Tactic:
use s * {(algebraMap R₁ K x)⁻¹}

Result:
case h
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
J : FractionalIdeal R₁⁰ K
hJ : J * spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≤ I
s : Finset K
hs : span R₁ ↑s = ↑(J * spanSingleton R₁⁰ ((algebraMap R₁ K) x))
⊢ span R₁ ↑(s * {((algebraMap R₁ K) x)⁻¹}) = ↑J