[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
h' : Submodule.IsPrincipal (maximalIdeal R)
I : Ideal R
hI : I ≠ ⊥
⊢ ∃ n, I = maximalIdeal R ^ n
[PROOFSTEP]
classical
obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
by_cases hI' : I = ⊤
· use 0; rw [pow_zero, hI', Ideal.one_eq_top]
have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r => (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
have : x ≠ 0 := by
  rintro rfl
  apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
  simp [hx]
have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r :=
  by
  intro r hr₁ hr₂
  obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
  have : ∀ b ∈ f, Associated x b := by
    intro b hb
    exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
  clear hr₁ hr₂ hf₁
  induction' f using Multiset.induction with fa fs fh
  · exact (hf₂ rfl).elim
  rcases eq_or_ne fs ∅ with (rfl | hf')
  · use 1
    rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
    exact this _ (Multiset.mem_cons_self _ _)
  · obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
    use n + 1
    rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]
    exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn
have : ∃ n : ℕ, x ^ n ∈ I :=
  by
  obtain ⟨r, hr₁, hr₂⟩ : ∃ r : R, r ∈ I ∧ r ≠ 0 := by by_contra' h; apply hI; rw [eq_bot_iff]; exact h
  obtain ⟨n, u, rfl⟩ := H' r hr₂ (le_maximalIdeal hI' hr₁)
  use n
  rwa [← I.unit_mul_mem_iff_mem u.isUnit, mul_comm]
use Nat.find this
apply le_antisymm
· change ∀ s ∈ I, s ∈ _
  by_contra hI''
  push_neg at hI''
  obtain ⟨s, hs₁, hs₂⟩ := hI''
  apply hs₂
  by_cases hs₃ : s = 0; · rw [hs₃]; exact zero_mem _
  obtain ⟨n, u, rfl⟩ := H' s hs₃ (le_maximalIdeal hI' hs₁)
  rw [mul_comm, Ideal.unit_mul_mem_iff_mem _ u.isUnit] at hs₁ ⊢
  apply Ideal.pow_le_pow (Nat.find_min' this hs₁)
  apply Ideal.pow_mem_pow
  exact (H _).mpr (dvd_refl _)
· rw [hx, Ideal.span_singleton_pow, Ideal.span_le, Set.singleton_subset_iff]
  exact Nat.find_spec this
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
h' : Submodule.IsPrincipal (maximalIdeal R)
I : Ideal R
hI : I ≠ ⊥
⊢ ∃ n, I = maximalIdeal R ^ n
[PROOFSTEP]
obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
[GOAL]
case mk.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
⊢ ∃ n, I = maximalIdeal R ^ n
[PROOFSTEP]
by_cases hI' : I = ⊤
[GOAL]
case pos
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : I = ⊤
⊢ ∃ n, I = maximalIdeal R ^ n
[PROOFSTEP]
use 0
[GOAL]
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : I = ⊤
⊢ I = maximalIdeal R ^ 0
[PROOFSTEP]
rw [pow_zero, hI', Ideal.one_eq_top]
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
⊢ ∃ n, I = maximalIdeal R ^ n
[PROOFSTEP]
have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r => (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
⊢ ∃ n, I = maximalIdeal R ^ n
[PROOFSTEP]
have : x ≠ 0 := by
  rintro rfl
  apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
  simp [hx]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
⊢ x ≠ 0
[PROOFSTEP]
rintro rfl
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
hI' : ¬I = ⊤
hx : maximalIdeal R = Ideal.span {0}
H : ∀ (r : R), ¬IsUnit r ↔ 0 ∣ r
⊢ False
[PROOFSTEP]
apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
hI' : ¬I = ⊤
hx : maximalIdeal R = Ideal.span {0}
H : ∀ (r : R), ¬IsUnit r ↔ 0 ∣ r
⊢ maximalIdeal R = ⊥
[PROOFSTEP]
simp [hx]
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
⊢ ∃ n, I = maximalIdeal R ^ n
[PROOFSTEP]
have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
⊢ ∃ n, I = maximalIdeal R ^ n
[PROOFSTEP]
have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r :=
  by
  intro r hr₁ hr₂
  obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
  have : ∀ b ∈ f, Associated x b := by
    intro b hb
    exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
  clear hr₁ hr₂ hf₁
  induction' f using Multiset.induction with fa fs fh
  · exact (hf₂ rfl).elim
  rcases eq_or_ne fs ∅ with (rfl | hf')
  · use 1
    rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
    exact this _ (Multiset.mem_cons_self _ _)
  · obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
    use n + 1
    rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]
    exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
⊢ ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
[PROOFSTEP]
intro r hr₁ hr₂
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
r : R
hr₁ : r ≠ 0
hr₂ : r ∈ nonunits R
⊢ ∃ n, Associated (x ^ n) r
[PROOFSTEP]
obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
[GOAL]
case intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₁ : ∀ (b : R), b ∈ f → Irreducible b
hf₂ : f ≠ ∅
hr₁ : Multiset.prod f ≠ 0
hr₂ : Multiset.prod f ∈ nonunits R
⊢ ∃ n, Associated (x ^ n) (Multiset.prod f)
[PROOFSTEP]
have : ∀ b ∈ f, Associated x b := by
  intro b hb
  exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₁ : ∀ (b : R), b ∈ f → Irreducible b
hf₂ : f ≠ ∅
hr₁ : Multiset.prod f ≠ 0
hr₂ : Multiset.prod f ∈ nonunits R
⊢ ∀ (b : R), b ∈ f → Associated x b
[PROOFSTEP]
intro b hb
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₁ : ∀ (b : R), b ∈ f → Irreducible b
hf₂ : f ≠ ∅
hr₁ : Multiset.prod f ≠ 0
hr₂ : Multiset.prod f ∈ nonunits R
b : R
hb : b ∈ f
⊢ Associated x b
[PROOFSTEP]
exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
[GOAL]
case intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₁ : ∀ (b : R), b ∈ f → Irreducible b
hf₂ : f ≠ ∅
hr₁ : Multiset.prod f ≠ 0
hr₂ : Multiset.prod f ∈ nonunits R
this : ∀ (b : R), b ∈ f → Associated x b
⊢ ∃ n, Associated (x ^ n) (Multiset.prod f)
[PROOFSTEP]
clear hr₁ hr₂ hf₁
[GOAL]
case intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₂ : f ≠ ∅
this : ∀ (b : R), b ∈ f → Associated x b
⊢ ∃ n, Associated (x ^ n) (Multiset.prod f)
[PROOFSTEP]
induction' f using Multiset.induction with fa fs fh
[GOAL]
case intro.intro.intro.empty
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝¹ : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₂✝ : f ≠ ∅
this✝ : ∀ (b : R), b ∈ f → Associated x b
hf₂ : 0 ≠ ∅
this : ∀ (b : R), b ∈ 0 → Associated x b
⊢ ∃ n, Associated (x ^ n) (Multiset.prod 0)
[PROOFSTEP]
exact (hf₂ rfl).elim
[GOAL]
case intro.intro.intro.cons
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝¹ : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₂✝ : f ≠ ∅
this✝ : ∀ (b : R), b ∈ f → Associated x b
fa : R
fs : Multiset R
fh : fs ≠ ∅ → (∀ (b : R), b ∈ fs → Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod fs)
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ (b : R), b ∈ fa ::ₘ fs → Associated x b
⊢ ∃ n, Associated (x ^ n) (Multiset.prod (fa ::ₘ fs))
[PROOFSTEP]
rcases eq_or_ne fs ∅ with (rfl | hf')
[GOAL]
case intro.intro.intro.cons.inl
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝¹ : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₂✝ : f ≠ ∅
this✝ : ∀ (b : R), b ∈ f → Associated x b
fa : R
fh : ∅ ≠ ∅ → (∀ (b : R), b ∈ ∅ → Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod ∅)
hf₂ : fa ::ₘ ∅ ≠ ∅
this : ∀ (b : R), b ∈ fa ::ₘ ∅ → Associated x b
⊢ ∃ n, Associated (x ^ n) (Multiset.prod (fa ::ₘ ∅))
[PROOFSTEP]
use 1
[GOAL]
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝¹ : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₂✝ : f ≠ ∅
this✝ : ∀ (b : R), b ∈ f → Associated x b
fa : R
fh : ∅ ≠ ∅ → (∀ (b : R), b ∈ ∅ → Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod ∅)
hf₂ : fa ::ₘ ∅ ≠ ∅
this : ∀ (b : R), b ∈ fa ::ₘ ∅ → Associated x b
⊢ Associated (x ^ 1) (Multiset.prod (fa ::ₘ ∅))
[PROOFSTEP]
rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
[GOAL]
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝¹ : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₂✝ : f ≠ ∅
this✝ : ∀ (b : R), b ∈ f → Associated x b
fa : R
fh : ∅ ≠ ∅ → (∀ (b : R), b ∈ ∅ → Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod ∅)
hf₂ : fa ::ₘ ∅ ≠ ∅
this : ∀ (b : R), b ∈ fa ::ₘ ∅ → Associated x b
⊢ Associated x fa
[PROOFSTEP]
exact this _ (Multiset.mem_cons_self _ _)
[GOAL]
case intro.intro.intro.cons.inr
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝¹ : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₂✝ : f ≠ ∅
this✝ : ∀ (b : R), b ∈ f → Associated x b
fa : R
fs : Multiset R
fh : fs ≠ ∅ → (∀ (b : R), b ∈ fs → Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod fs)
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ (b : R), b ∈ fa ::ₘ fs → Associated x b
hf' : fs ≠ ∅
⊢ ∃ n, Associated (x ^ n) (Multiset.prod (fa ::ₘ fs))
[PROOFSTEP]
obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
[GOAL]
case intro.intro.intro.cons.inr.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝¹ : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₂✝ : f ≠ ∅
this✝ : ∀ (b : R), b ∈ f → Associated x b
fa : R
fs : Multiset R
fh : fs ≠ ∅ → (∀ (b : R), b ∈ fs → Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod fs)
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ (b : R), b ∈ fa ::ₘ fs → Associated x b
hf' : fs ≠ ∅
n : ℕ
hn : Associated (x ^ n) (Multiset.prod fs)
⊢ ∃ n, Associated (x ^ n) (Multiset.prod (fa ::ₘ fs))
[PROOFSTEP]
use n + 1
[GOAL]
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝¹ : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₂✝ : f ≠ ∅
this✝ : ∀ (b : R), b ∈ f → Associated x b
fa : R
fs : Multiset R
fh : fs ≠ ∅ → (∀ (b : R), b ∈ fs → Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod fs)
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ (b : R), b ∈ fa ::ₘ fs → Associated x b
hf' : fs ≠ ∅
n : ℕ
hn : Associated (x ^ n) (Multiset.prod fs)
⊢ Associated (x ^ (n + 1)) (Multiset.prod (fa ::ₘ fs))
[PROOFSTEP]
rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]
[GOAL]
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝¹ : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₂✝ : f ≠ ∅
this✝ : ∀ (b : R), b ∈ f → Associated x b
fa : R
fs : Multiset R
fh : fs ≠ ∅ → (∀ (b : R), b ∈ fs → Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod fs)
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ (b : R), b ∈ fa ::ₘ fs → Associated x b
hf' : fs ≠ ∅
n : ℕ
hn : Associated (x ^ n) (Multiset.prod fs)
⊢ Associated (x * x ^ n) (fa * Multiset.prod fs)
[PROOFSTEP]
exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
⊢ ∃ n, I = maximalIdeal R ^ n
[PROOFSTEP]
have : ∃ n : ℕ, x ^ n ∈ I :=
  by
  obtain ⟨r, hr₁, hr₂⟩ : ∃ r : R, r ∈ I ∧ r ≠ 0 := by by_contra' h; apply hI; rw [eq_bot_iff]; exact h
  obtain ⟨n, u, rfl⟩ := H' r hr₂ (le_maximalIdeal hI' hr₁)
  use n
  rwa [← I.unit_mul_mem_iff_mem u.isUnit, mul_comm]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
⊢ ∃ n, x ^ n ∈ I
[PROOFSTEP]
obtain ⟨r, hr₁, hr₂⟩ : ∃ r : R, r ∈ I ∧ r ≠ 0 := by by_contra' h; apply hI; rw [eq_bot_iff]; exact h
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
⊢ ∃ r, r ∈ I ∧ r ≠ 0
[PROOFSTEP]
by_contra' h
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h✝ : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
h : ∀ (r : R), r ∈ I → r = 0
⊢ False
[PROOFSTEP]
apply hI
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h✝ : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
h : ∀ (r : R), r ∈ I → r = 0
⊢ I = ⊥
[PROOFSTEP]
rw [eq_bot_iff]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h✝ : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
h : ∀ (r : R), r ∈ I → r = 0
⊢ I ≤ ⊥
[PROOFSTEP]
exact h
[GOAL]
case intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
r : R
hr₁ : r ∈ I
hr₂ : r ≠ 0
⊢ ∃ n, x ^ n ∈ I
[PROOFSTEP]
obtain ⟨n, u, rfl⟩ := H' r hr₂ (le_maximalIdeal hI' hr₁)
[GOAL]
case intro.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
n : ℕ
u : Rˣ
hr₁ : x ^ n * ↑u ∈ I
hr₂ : x ^ n * ↑u ≠ 0
⊢ ∃ n, x ^ n ∈ I
[PROOFSTEP]
use n
[GOAL]
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
n : ℕ
u : Rˣ
hr₁ : x ^ n * ↑u ∈ I
hr₂ : x ^ n * ↑u ≠ 0
⊢ x ^ n ∈ I
[PROOFSTEP]
rwa [← I.unit_mul_mem_iff_mem u.isUnit, mul_comm]
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
⊢ ∃ n, I = maximalIdeal R ^ n
[PROOFSTEP]
use Nat.find this
[GOAL]
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
⊢ I = maximalIdeal R ^ Nat.find this
[PROOFSTEP]
apply le_antisymm
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
⊢ I ≤ maximalIdeal R ^ Nat.find this
[PROOFSTEP]
change ∀ s ∈ I, s ∈ _
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
⊢ ∀ (s : R), s ∈ I → s ∈ maximalIdeal R ^ Nat.find this
[PROOFSTEP]
by_contra hI''
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
hI'' : ¬∀ (s : R), s ∈ I → s ∈ maximalIdeal R ^ Nat.find this
⊢ False
[PROOFSTEP]
push_neg at hI''
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
hI'' : ∃ s, s ∈ I ∧ ¬s ∈ maximalIdeal R ^ Nat.find this
⊢ False
[PROOFSTEP]
obtain ⟨s, hs₁, hs₂⟩ := hI''
[GOAL]
case h.a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
s : R
hs₁ : s ∈ I
hs₂ : ¬s ∈ maximalIdeal R ^ Nat.find this
⊢ False
[PROOFSTEP]
apply hs₂
[GOAL]
case h.a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
s : R
hs₁ : s ∈ I
hs₂ : ¬s ∈ maximalIdeal R ^ Nat.find this
⊢ s ∈ maximalIdeal R ^ Nat.find this
[PROOFSTEP]
by_cases hs₃ : s = 0
[GOAL]
case pos
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
s : R
hs₁ : s ∈ I
hs₂ : ¬s ∈ maximalIdeal R ^ Nat.find this
hs₃ : s = 0
⊢ s ∈ maximalIdeal R ^ Nat.find this
[PROOFSTEP]
rw [hs₃]
[GOAL]
case pos
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
s : R
hs₁ : s ∈ I
hs₂ : ¬s ∈ maximalIdeal R ^ Nat.find this
hs₃ : s = 0
⊢ 0 ∈ maximalIdeal R ^ Nat.find this
[PROOFSTEP]
exact zero_mem _
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
s : R
hs₁ : s ∈ I
hs₂ : ¬s ∈ maximalIdeal R ^ Nat.find this
hs₃ : ¬s = 0
⊢ s ∈ maximalIdeal R ^ Nat.find this
[PROOFSTEP]
obtain ⟨n, u, rfl⟩ := H' s hs₃ (le_maximalIdeal hI' hs₁)
[GOAL]
case neg.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
n : ℕ
u : Rˣ
hs₁ : x ^ n * ↑u ∈ I
hs₂ : ¬x ^ n * ↑u ∈ maximalIdeal R ^ Nat.find this
hs₃ : ¬x ^ n * ↑u = 0
⊢ x ^ n * ↑u ∈ maximalIdeal R ^ Nat.find this
[PROOFSTEP]
rw [mul_comm, Ideal.unit_mul_mem_iff_mem _ u.isUnit] at hs₁ ⊢
[GOAL]
case neg.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
n : ℕ
u : Rˣ
hs₁ : x ^ n ∈ I
hs₂ : ¬x ^ n * ↑u ∈ maximalIdeal R ^ Nat.find this
hs₃ : ¬x ^ n * ↑u = 0
⊢ x ^ n ∈ maximalIdeal R ^ Nat.find this
[PROOFSTEP]
apply Ideal.pow_le_pow (Nat.find_min' this hs₁)
[GOAL]
case neg.intro.intro.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
n : ℕ
u : Rˣ
hs₁ : x ^ n ∈ I
hs₂ : ¬x ^ n * ↑u ∈ maximalIdeal R ^ Nat.find this
hs₃ : ¬x ^ n * ↑u = 0
⊢ x ^ n ∈ maximalIdeal R ^ n
[PROOFSTEP]
apply Ideal.pow_mem_pow
[GOAL]
case neg.intro.intro.a.hx
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
n : ℕ
u : Rˣ
hs₁ : x ^ n ∈ I
hs₂ : ¬x ^ n * ↑u ∈ maximalIdeal R ^ Nat.find this
hs₃ : ¬x ^ n * ↑u = 0
⊢ x ∈ maximalIdeal R
[PROOFSTEP]
exact (H _).mpr (dvd_refl _)
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
⊢ maximalIdeal R ^ Nat.find this ≤ I
[PROOFSTEP]
rw [hx, Ideal.span_singleton_pow, Ideal.span_le, Set.singleton_subset_iff]
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
this : ∃ n, x ^ n ∈ I
⊢ x ^ Nat.find this ∈ ↑I
[PROOFSTEP]
exact Nat.find_spec this
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
classical
by_cases ne_bot : maximalIdeal R = ⊥
· rw [ne_bot]; infer_instance
obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ maximalIdeal R, a ≠ (0 : R) := by by_contra' h'; apply ne_bot; rwa [eq_bot_iff]
have hle : Ideal.span { a } ≤ maximalIdeal R := by rwa [Ideal.span_le, Set.singleton_subset_iff]
have : (Ideal.span { a }).radical = maximalIdeal R :=
  by
  rw [Ideal.radical_eq_sInf]
  apply le_antisymm
  · exact sInf_le ⟨hle, inferInstance⟩
  · refine' le_sInf fun I hI => (eq_maximalIdeal <| hI.2.isMaximal (fun e => ha₂ _)).ge
    rw [← Ideal.span_singleton_eq_bot, eq_bot_iff, ← e]; exact hI.1
have : ∃ n, maximalIdeal R ^ n ≤ Ideal.span { a } := by rw [← this]; apply Ideal.exists_radical_pow_le_of_fg;
  exact IsNoetherian.noetherian _
cases' hn : Nat.find this with n
· have := Nat.find_spec this
  rw [hn, pow_zero, Ideal.one_eq_top] at this
  exact (Ideal.IsMaximal.ne_top inferInstance (eq_top_iff.mpr <| this.trans hle)).elim
obtain ⟨b, hb₁, hb₂⟩ : ∃ b ∈ maximalIdeal R ^ n, ¬b ∈ Ideal.span { a } := by by_contra' h'; rw [Nat.find_eq_iff] at hn ;
  exact hn.2 n n.lt_succ_self fun x hx => h' x hx
have hb₃ : ∀ m ∈ maximalIdeal R, ∃ k : R, k * a = b * m :=
  by
  intro m hm; rw [← Ideal.mem_span_singleton']; apply Nat.find_spec this
  rw [hn, pow_succ']; exact Ideal.mul_mem_mul hb₁ hm
have hb₄ : b ≠ 0 := by rintro rfl; apply hb₂; exact zero_mem _
let K := FractionRing R
let x : K := algebraMap R K b / algebraMap R K a
let M := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
have ha₃ : algebraMap R K a ≠ 0 := IsFractionRing.to_map_eq_zero_iff.not.mpr ha₂
by_cases hx : ∀ y ∈ M, x * y ∈ M
· have := isIntegral_of_smul_mem_submodule M ?_ ?_ x hx
  · obtain ⟨y, e⟩ := IsIntegrallyClosed.algebraMap_eq_of_integral this
    refine' (hb₂ (Ideal.mem_span_singleton'.mpr ⟨y, _⟩)).elim
    apply IsFractionRing.injective R K
    rw [map_mul, e, div_mul_cancel _ ha₃]
  · rw [Submodule.ne_bot_iff]; refine' ⟨_, ⟨a, ha₁, rfl⟩, _⟩
    exact (IsFractionRing.to_map_eq_zero_iff (K := K)).not.mpr ha₂
  · apply Submodule.FG.map; exact IsNoetherian.noetherian _
· have : (M.map (DistribMulAction.toLinearMap R K x)).comap (Algebra.linearMap R K) = ⊤ :=
    by
    by_contra h; apply hx
    rintro m' ⟨m, hm, rfl : algebraMap R K m = m'⟩
    obtain ⟨k, hk⟩ := hb₃ m hm
    have hk' : x * algebraMap R K m = algebraMap R K k := by
      rw [← mul_div_right_comm, ← map_mul, ← hk, map_mul, mul_div_cancel _ ha₃]
    exact ⟨k, le_maximalIdeal h ⟨_, ⟨_, hm, rfl⟩, hk'⟩, hk'.symm⟩
  obtain ⟨y, hy₁, hy₂⟩ : ∃ y ∈ maximalIdeal R, b * y = a :=
    by
    rw [Ideal.eq_top_iff_one, Submodule.mem_comap] at this
    obtain ⟨_, ⟨y, hy, rfl⟩, hy' : x * algebraMap R K y = algebraMap R K 1⟩ := this
    rw [map_one, ← mul_div_right_comm, div_eq_one_iff_eq ha₃, ← map_mul] at hy'
    exact ⟨y, hy, IsFractionRing.injective R K hy'⟩
  refine' ⟨⟨y, _⟩⟩
  apply le_antisymm
  · intro m hm; obtain ⟨k, hk⟩ := hb₃ m hm; rw [← hy₂, mul_comm, mul_assoc] at hk
    rw [← mul_left_cancel₀ hb₄ hk, mul_comm]; exact Ideal.mem_span_singleton'.mpr ⟨_, rfl⟩
  · rwa [Submodule.span_le, Set.singleton_subset_iff]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
by_cases ne_bot : maximalIdeal R = ⊥
[GOAL]
case pos
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : maximalIdeal R = ⊥
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
rw [ne_bot]
[GOAL]
case pos
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : maximalIdeal R = ⊥
⊢ Submodule.IsPrincipal ⊥
[PROOFSTEP]
infer_instance
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ maximalIdeal R, a ≠ (0 : R) := by by_contra' h'; apply ne_bot; rwa [eq_bot_iff]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
⊢ ∃ a, a ∈ maximalIdeal R ∧ a ≠ 0
[PROOFSTEP]
by_contra' h'
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
h' : ∀ (a : R), a ∈ maximalIdeal R → a = 0
⊢ False
[PROOFSTEP]
apply ne_bot
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
h' : ∀ (a : R), a ∈ maximalIdeal R → a = 0
⊢ maximalIdeal R = ⊥
[PROOFSTEP]
rwa [eq_bot_iff]
[GOAL]
case neg.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
have hle : Ideal.span { a } ≤ maximalIdeal R := by rwa [Ideal.span_le, Set.singleton_subset_iff]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
⊢ Ideal.span {a} ≤ maximalIdeal R
[PROOFSTEP]
rwa [Ideal.span_le, Set.singleton_subset_iff]
[GOAL]
case neg.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
have : (Ideal.span { a }).radical = maximalIdeal R :=
  by
  rw [Ideal.radical_eq_sInf]
  apply le_antisymm
  · exact sInf_le ⟨hle, inferInstance⟩
  · refine' le_sInf fun I hI => (eq_maximalIdeal <| hI.2.isMaximal (fun e => ha₂ _)).ge
    rw [← Ideal.span_singleton_eq_bot, eq_bot_iff, ← e]; exact hI.1
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
⊢ Ideal.radical (Ideal.span {a}) = maximalIdeal R
[PROOFSTEP]
rw [Ideal.radical_eq_sInf]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
⊢ sInf {J | Ideal.span {a} ≤ J ∧ Ideal.IsPrime J} = maximalIdeal R
[PROOFSTEP]
apply le_antisymm
[GOAL]
case a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
⊢ sInf {J | Ideal.span {a} ≤ J ∧ Ideal.IsPrime J} ≤ maximalIdeal R
[PROOFSTEP]
exact sInf_le ⟨hle, inferInstance⟩
[GOAL]
case a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
⊢ maximalIdeal R ≤ sInf {J | Ideal.span {a} ≤ J ∧ Ideal.IsPrime J}
[PROOFSTEP]
refine' le_sInf fun I hI => (eq_maximalIdeal <| hI.2.isMaximal (fun e => ha₂ _)).ge
[GOAL]
case a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
I : Ideal R
hI : I ∈ {J | Ideal.span {a} ≤ J ∧ Ideal.IsPrime J}
e : I = ⊥
⊢ a = 0
[PROOFSTEP]
rw [← Ideal.span_singleton_eq_bot, eq_bot_iff, ← e]
[GOAL]
case a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
I : Ideal R
hI : I ∈ {J | Ideal.span {a} ≤ J ∧ Ideal.IsPrime J}
e : I = ⊥
⊢ Ideal.span {a} ≤ I
[PROOFSTEP]
exact hI.1
[GOAL]
case neg.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this : Ideal.radical (Ideal.span {a}) = maximalIdeal R
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
have : ∃ n, maximalIdeal R ^ n ≤ Ideal.span { a } := by rw [← this]; apply Ideal.exists_radical_pow_le_of_fg;
  exact IsNoetherian.noetherian _
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this : Ideal.radical (Ideal.span {a}) = maximalIdeal R
⊢ ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
[PROOFSTEP]
rw [← this]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this : Ideal.radical (Ideal.span {a}) = maximalIdeal R
⊢ ∃ n, Ideal.radical (Ideal.span {a}) ^ n ≤ Ideal.span {a}
[PROOFSTEP]
apply Ideal.exists_radical_pow_le_of_fg
[GOAL]
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this : Ideal.radical (Ideal.span {a}) = maximalIdeal R
⊢ Ideal.FG (Ideal.radical (Ideal.span {a}))
[PROOFSTEP]
exact IsNoetherian.noetherian _
[GOAL]
case neg.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
cases' hn : Nat.find this with n
[GOAL]
case neg.intro.intro.zero
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
hn : Nat.find this = Nat.zero
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
have := Nat.find_spec this
[GOAL]
case neg.intro.intro.zero
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
hn : Nat.find this✝ = Nat.zero
this : maximalIdeal R ^ Nat.find this✝ ≤ Ideal.span {a}
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
rw [hn, pow_zero, Ideal.one_eq_top] at this
[GOAL]
case neg.intro.intro.zero
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
hn : Nat.find this✝ = Nat.zero
this : ⊤ ≤ Ideal.span {a}
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
exact (Ideal.IsMaximal.ne_top inferInstance (eq_top_iff.mpr <| this.trans hle)).elim
[GOAL]
case neg.intro.intro.succ
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
obtain ⟨b, hb₁, hb₂⟩ : ∃ b ∈ maximalIdeal R ^ n, ¬b ∈ Ideal.span { a } := by by_contra' h'; rw [Nat.find_eq_iff] at hn ;
  exact hn.2 n n.lt_succ_self fun x hx => h' x hx
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
⊢ ∃ b, b ∈ maximalIdeal R ^ n ∧ ¬b ∈ Ideal.span {a}
[PROOFSTEP]
by_contra' h'
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
h' : ∀ (b : R), b ∈ maximalIdeal R ^ n → b ∈ Ideal.span {a}
⊢ False
[PROOFSTEP]
rw [Nat.find_eq_iff] at hn
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn :
  maximalIdeal R ^ Nat.succ n ≤ Ideal.span {a} ∧ ∀ (n_1 : ℕ), n_1 < Nat.succ n → ¬maximalIdeal R ^ n_1 ≤ Ideal.span {a}
h' : ∀ (b : R), b ∈ maximalIdeal R ^ n → b ∈ Ideal.span {a}
⊢ False
[PROOFSTEP]
exact hn.2 n n.lt_succ_self fun x hx => h' x hx
[GOAL]
case neg.intro.intro.succ.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
have hb₃ : ∀ m ∈ maximalIdeal R, ∃ k : R, k * a = b * m :=
  by
  intro m hm; rw [← Ideal.mem_span_singleton']; apply Nat.find_spec this
  rw [hn, pow_succ']; exact Ideal.mul_mem_mul hb₁ hm
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
⊢ ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
[PROOFSTEP]
intro m hm
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
m : R
hm : m ∈ maximalIdeal R
⊢ ∃ k, k * a = b * m
[PROOFSTEP]
rw [← Ideal.mem_span_singleton']
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
m : R
hm : m ∈ maximalIdeal R
⊢ b * m ∈ Ideal.span {a}
[PROOFSTEP]
apply Nat.find_spec this
[GOAL]
case a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
m : R
hm : m ∈ maximalIdeal R
⊢ b * m ∈ maximalIdeal R ^ Nat.find this
[PROOFSTEP]
rw [hn, pow_succ']
[GOAL]
case a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
m : R
hm : m ∈ maximalIdeal R
⊢ b * m ∈ maximalIdeal R ^ n * maximalIdeal R
[PROOFSTEP]
exact Ideal.mul_mem_mul hb₁ hm
[GOAL]
case neg.intro.intro.succ.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
have hb₄ : b ≠ 0 := by rintro rfl; apply hb₂; exact zero_mem _
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
⊢ b ≠ 0
[PROOFSTEP]
rintro rfl
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
hb₁ : 0 ∈ maximalIdeal R ^ n
hb₂ : ¬0 ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = 0 * m
⊢ False
[PROOFSTEP]
apply hb₂
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
hb₁ : 0 ∈ maximalIdeal R ^ n
hb₂ : ¬0 ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = 0 * m
⊢ 0 ∈ Ideal.span {a}
[PROOFSTEP]
exact zero_mem _
[GOAL]
case neg.intro.intro.succ.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
let K := FractionRing R
[GOAL]
case neg.intro.intro.succ.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
let x : K := algebraMap R K b / algebraMap R K a
[GOAL]
case neg.intro.intro.succ.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
let M := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
[GOAL]
case neg.intro.intro.succ.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
have ha₃ : algebraMap R K a ≠ 0 := IsFractionRing.to_map_eq_zero_iff.not.mpr ha₂
[GOAL]
case neg.intro.intro.succ.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
by_cases hx : ∀ y ∈ M, x * y ∈ M
[GOAL]
case pos
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ∀ (y : K), y ∈ M → x * y ∈ M
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
have := isIntegral_of_smul_mem_submodule M ?_ ?_ x hx
[GOAL]
case pos.refine_3
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ∀ (y : K), y ∈ M → x * y ∈ M
this : IsIntegral R x
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
obtain ⟨y, e⟩ := IsIntegrallyClosed.algebraMap_eq_of_integral this
[GOAL]
case pos.refine_3.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ∀ (y : K), y ∈ M → x * y ∈ M
this : IsIntegral R x
y : R
e : ↑(algebraMap R (FractionRing R)) y = x
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
refine' (hb₂ (Ideal.mem_span_singleton'.mpr ⟨y, _⟩)).elim
[GOAL]
case pos.refine_3.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ∀ (y : K), y ∈ M → x * y ∈ M
this : IsIntegral R x
y : R
e : ↑(algebraMap R (FractionRing R)) y = x
⊢ y * a = b
[PROOFSTEP]
apply IsFractionRing.injective R K
[GOAL]
case pos.refine_3.intro.a
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ∀ (y : K), y ∈ M → x * y ∈ M
this : IsIntegral R x
y : R
e : ↑(algebraMap R (FractionRing R)) y = x
⊢ ↑(algebraMap R K) (y * a) = ↑(algebraMap R K) b
[PROOFSTEP]
rw [map_mul, e, div_mul_cancel _ ha₃]
[GOAL]
case pos.refine_1
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ∀ (y : K), y ∈ M → x * y ∈ M
⊢ M ≠ ⊥
[PROOFSTEP]
rw [Submodule.ne_bot_iff]
[GOAL]
case pos.refine_1
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ∀ (y : K), y ∈ M → x * y ∈ M
⊢ ∃ x, x ∈ M ∧ x ≠ 0
[PROOFSTEP]
refine' ⟨_, ⟨a, ha₁, rfl⟩, _⟩
[GOAL]
case pos.refine_1
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ∀ (y : K), y ∈ M → x * y ∈ M
⊢ ↑(Algebra.linearMap R K) a ≠ 0
[PROOFSTEP]
exact (IsFractionRing.to_map_eq_zero_iff (K := K)).not.mpr ha₂
[GOAL]
case pos.refine_2
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ∀ (y : K), y ∈ M → x * y ∈ M
⊢ Submodule.FG M
[PROOFSTEP]
apply Submodule.FG.map
[GOAL]
case pos.refine_2.hs
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ∀ (y : K), y ∈ M → x * y ∈ M
⊢ Submodule.FG (maximalIdeal R)
[PROOFSTEP]
exact IsNoetherian.noetherian _
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
have : (M.map (DistribMulAction.toLinearMap R K x)).comap (Algebra.linearMap R K) = ⊤ :=
  by
  by_contra h; apply hx
  rintro m' ⟨m, hm, rfl : algebraMap R K m = m'⟩
  obtain ⟨k, hk⟩ := hb₃ m hm
  have hk' : x * algebraMap R K m = algebraMap R K k := by
    rw [← mul_div_right_comm, ← map_mul, ← hk, map_mul, mul_div_cancel _ ha₃]
  exact ⟨k, le_maximalIdeal h ⟨_, ⟨_, hm, rfl⟩, hk'⟩, hk'.symm⟩
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
⊢ Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
[PROOFSTEP]
by_contra h
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
h : ¬Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
⊢ False
[PROOFSTEP]
apply hx
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
h : ¬Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
⊢ ∀ (y : K), y ∈ M → x * y ∈ M
[PROOFSTEP]
rintro m' ⟨m, hm, rfl : algebraMap R K m = m'⟩
[GOAL]
case intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
h : ¬Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
m : R
hm : m ∈ ↑(maximalIdeal R)
⊢ x * ↑(algebraMap R K) m ∈ M
[PROOFSTEP]
obtain ⟨k, hk⟩ := hb₃ m hm
[GOAL]
case intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
h : ¬Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
m : R
hm : m ∈ ↑(maximalIdeal R)
k : R
hk : k * a = b * m
⊢ x * ↑(algebraMap R K) m ∈ M
[PROOFSTEP]
have hk' : x * algebraMap R K m = algebraMap R K k := by
  rw [← mul_div_right_comm, ← map_mul, ← hk, map_mul, mul_div_cancel _ ha₃]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
h : ¬Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
m : R
hm : m ∈ ↑(maximalIdeal R)
k : R
hk : k * a = b * m
⊢ x * ↑(algebraMap R K) m = ↑(algebraMap R K) k
[PROOFSTEP]
rw [← mul_div_right_comm, ← map_mul, ← hk, map_mul, mul_div_cancel _ ha₃]
[GOAL]
case intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
h : ¬Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
m : R
hm : m ∈ ↑(maximalIdeal R)
k : R
hk : k * a = b * m
hk' : x * ↑(algebraMap R K) m = ↑(algebraMap R K) k
⊢ x * ↑(algebraMap R K) m ∈ M
[PROOFSTEP]
exact ⟨k, le_maximalIdeal h ⟨_, ⟨_, hm, rfl⟩, hk'⟩, hk'.symm⟩
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
this : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
obtain ⟨y, hy₁, hy₂⟩ : ∃ y ∈ maximalIdeal R, b * y = a :=
  by
  rw [Ideal.eq_top_iff_one, Submodule.mem_comap] at this
  obtain ⟨_, ⟨y, hy, rfl⟩, hy' : x * algebraMap R K y = algebraMap R K 1⟩ := this
  rw [map_one, ← mul_div_right_comm, div_eq_one_iff_eq ha₃, ← map_mul] at hy'
  exact ⟨y, hy, IsFractionRing.injective R K hy'⟩
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
this : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
⊢ ∃ y, y ∈ maximalIdeal R ∧ b * y = a
[PROOFSTEP]
rw [Ideal.eq_top_iff_one, Submodule.mem_comap] at this
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
this : ↑(Algebra.linearMap R K) 1 ∈ Submodule.map (DistribMulAction.toLinearMap R K x) M
⊢ ∃ y, y ∈ maximalIdeal R ∧ b * y = a
[PROOFSTEP]
obtain ⟨_, ⟨y, hy, rfl⟩, hy' : x * algebraMap R K y = algebraMap R K 1⟩ := this
[GOAL]
case intro.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
y : R
hy : y ∈ ↑(maximalIdeal R)
hy' : x * ↑(algebraMap R K) y = ↑(algebraMap R K) 1
⊢ ∃ y, y ∈ maximalIdeal R ∧ b * y = a
[PROOFSTEP]
rw [map_one, ← mul_div_right_comm, div_eq_one_iff_eq ha₃, ← map_mul] at hy'
[GOAL]
case intro.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
y : R
hy : y ∈ ↑(maximalIdeal R)
hy' : ↑(algebraMap R K) (b * y) = ↑(algebraMap R K) a
⊢ ∃ y, y ∈ maximalIdeal R ∧ b * y = a
[PROOFSTEP]
exact ⟨y, hy, IsFractionRing.injective R K hy'⟩
[GOAL]
case neg.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
this : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
y : R
hy₁ : y ∈ maximalIdeal R
hy₂ : b * y = a
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
refine' ⟨⟨y, _⟩⟩
[GOAL]
case neg.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
this : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
y : R
hy₁ : y ∈ maximalIdeal R
hy₂ : b * y = a
⊢ maximalIdeal R = Submodule.span R {y}
[PROOFSTEP]
apply le_antisymm
[GOAL]
case neg.intro.intro.a
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
this : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
y : R
hy₁ : y ∈ maximalIdeal R
hy₂ : b * y = a
⊢ maximalIdeal R ≤ Submodule.span R {y}
[PROOFSTEP]
intro m hm
[GOAL]
case neg.intro.intro.a
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
this : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
y : R
hy₁ : y ∈ maximalIdeal R
hy₂ : b * y = a
m : R
hm : m ∈ maximalIdeal R
⊢ m ∈ Submodule.span R {y}
[PROOFSTEP]
obtain ⟨k, hk⟩ := hb₃ m hm
[GOAL]
case neg.intro.intro.a.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
this : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
y : R
hy₁ : y ∈ maximalIdeal R
hy₂ : b * y = a
m : R
hm : m ∈ maximalIdeal R
k : R
hk : k * a = b * m
⊢ m ∈ Submodule.span R {y}
[PROOFSTEP]
rw [← hy₂, mul_comm, mul_assoc] at hk
[GOAL]
case neg.intro.intro.a.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
this : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
y : R
hy₁ : y ∈ maximalIdeal R
hy₂ : b * y = a
m : R
hm : m ∈ maximalIdeal R
k : R
hk : b * (y * k) = b * m
⊢ m ∈ Submodule.span R {y}
[PROOFSTEP]
rw [← mul_left_cancel₀ hb₄ hk, mul_comm]
[GOAL]
case neg.intro.intro.a.intro
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
this : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
y : R
hy₁ : y ∈ maximalIdeal R
hy₂ : b * y = a
m : R
hm : m ∈ maximalIdeal R
k : R
hk : b * (y * k) = b * m
⊢ k * y ∈ Submodule.span R {y}
[PROOFSTEP]
exact Ideal.mem_span_singleton'.mpr ⟨_, rfl⟩
[GOAL]
case neg.intro.intro.a
R : Type u_1
inst✝⁶ : CommRing R
K✝ : Type u_2
inst✝⁵ : Field K✝
inst✝⁴ : Algebra R K✝
inst✝³ : IsFractionRing R K✝
inst✝² : LocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : Ideal.radical (Ideal.span {a}) = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = Nat.succ n
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : ¬b ∈ Ideal.span {a}
hb₃ : ∀ (m : R), m ∈ maximalIdeal R → ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := ↑(algebraMap R K) b / ↑(algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : ↑(algebraMap R K) a ≠ 0
hx : ¬∀ (y : K), y ∈ M → x * y ∈ M
this : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = ⊤
y : R
hy₁ : y ∈ maximalIdeal R
hy₂ : b * y = a
⊢ Submodule.span R {y} ≤ maximalIdeal R
[PROOFSTEP]
rwa [Submodule.span_le, Set.singleton_subset_iff]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), FiniteDimensional.finrank (ResidueField R) (CotangentSpace R) = 1,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
have ne_bot := Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), FiniteDimensional.finrank (ResidueField R) (CotangentSpace R) = 1,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
classical
rw [finrank_eq_one_iff']
tfae_have 1 → 2
· intro; infer_instance
tfae_have 2 → 1
· intro
  haveI := IsBezout.toGCDDomain R
  haveI : UniqueFactorizationMonoid R := ufm_of_gcd_of_wfDvdMonoid
  apply DiscreteValuationRing.of_ufd_of_unique_irreducible
  · obtain ⟨x, hx₁, hx₂⟩ := Ring.exists_not_isUnit_of_not_isField h
    obtain ⟨p, hp₁, -⟩ := WfDvdMonoid.exists_irreducible_factor hx₂ hx₁
    exact ⟨p, hp₁⟩
  · exact ValuationRing.unique_irreducible
tfae_have 1 → 4
· intro H
  exact ⟨inferInstance, ((DiscreteValuationRing.iff_pid_with_one_nonzero_prime R).mp H).2⟩
tfae_have 4 → 3
· rintro ⟨h₁, h₂⟩;
  exact
    { h₁ with
      maximalOfPrime := fun hI hI' =>
        ExistsUnique.unique h₂ ⟨ne_bot, inferInstance⟩ ⟨hI, hI'⟩ ▸ maximalIdeal.isMaximal R, }
tfae_have 3 → 5
· intro h; exact maximalIdeal_isPrincipal_of_isDedekindDomain R
tfae_have 5 → 6
· rintro ⟨x, hx⟩
  have : x ∈ maximalIdeal R := by rw [hx]; exact Submodule.subset_span (Set.mem_singleton x)
  let x' : maximalIdeal R := ⟨x, this⟩
  use Submodule.Quotient.mk x'
  constructor
  swap
  · intro e
    rw [Submodule.Quotient.mk_eq_zero] at e
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    apply Submodule.eq_bot_of_le_smul_of_le_jacobson_bot (maximalIdeal R)
    · exact ⟨{ x }, (Finset.coe_singleton x).symm ▸ hx.symm⟩
    · conv_lhs => rw [hx]
      rw [Submodule.mem_smul_top_iff] at e
      rwa [Submodule.span_le, Set.singleton_subset_iff]
    · rw [LocalRing.jacobson_eq_maximalIdeal (⊥ : Ideal R) bot_ne_top]
  · refine' fun w => Quotient.inductionOn' w fun y => _
    obtain ⟨y, hy⟩ := y
    rw [hx, Submodule.mem_span_singleton] at hy
    obtain ⟨a, rfl⟩ := hy
    exact ⟨Ideal.Quotient.mk _ a, rfl⟩
tfae_have 6 → 5
· rintro ⟨x, hx, hx'⟩
  induction x using Quotient.inductionOn' with
  | h x => ?_
  use x
  apply le_antisymm
  swap; · rw [Submodule.span_le, Set.singleton_subset_iff]; exact x.prop
  have h₁ :
    (Ideal.span {(x : R)} : Ideal R) ⊔ maximalIdeal R ≤ Ideal.span {(x : R)} ⊔ maximalIdeal R • maximalIdeal R :=
    by
    refine' sup_le le_sup_left _
    rintro m hm
    obtain ⟨c, hc⟩ := hx' (Submodule.Quotient.mk ⟨m, hm⟩)
    induction c using Quotient.inductionOn' with
    | h c => ?_
    rw [← sub_sub_cancel (c * x) m]
    apply sub_mem _ _
    · refine' Ideal.mem_sup_left (Ideal.mem_span_singleton'.mpr ⟨c, rfl⟩)
    · have := (Submodule.Quotient.eq _).mp hc
      rw [Submodule.mem_smul_top_iff] at this
      exact Ideal.mem_sup_right this
  have h₂ : maximalIdeal R ≤ (⊥ : Ideal R).jacobson :=
    by
    rw [LocalRing.jacobson_eq_maximalIdeal]
    exact bot_ne_top
  have := Submodule.smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson (IsNoetherian.noetherian _) h₂ h₁
  rw [Submodule.bot_smul, sup_bot_eq] at this
  rw [← sup_eq_left, eq_comm]
  exact le_sup_left.antisymm (h₁.trans <| le_of_eq this)
tfae_have 5 → 7
· exact exists_maximalIdeal_pow_eq_of_principal R h
tfae_have 7 → 2
· rw [ValuationRing.iff_ideal_total]
  intro H
  constructor
  intro I J
  let _ := Classical.decEq (Ideal R)
  by_cases hI : I = ⊥; · subst hI; left; exact bot_le
  by_cases hJ : J = ⊥; · subst hJ; right; exact bot_le
  obtain ⟨n, rfl⟩ := H I hI
  obtain ⟨m, rfl⟩ := H J hJ
  cases' le_total m n with h' h'
  · left; exact Ideal.pow_le_pow h'
  · right; exact Ideal.pow_le_pow h'
tfae_finish
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), FiniteDimensional.finrank (ResidueField R) (CotangentSpace R) = 1,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
rw [finrank_eq_one_iff']
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
tfae_have 1 → 2
[GOAL]
case tfae_1_to_2
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
⊢ DiscreteValuationRing R → ValuationRing R
[PROOFSTEP]
intro
[GOAL]
case tfae_1_to_2
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
✝ : DiscreteValuationRing R
⊢ ValuationRing R
[PROOFSTEP]
infer_instance
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
tfae_have 2 → 1
[GOAL]
case tfae_2_to_1
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
⊢ ValuationRing R → DiscreteValuationRing R
[PROOFSTEP]
intro
[GOAL]
case tfae_2_to_1
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
✝ : ValuationRing R
⊢ DiscreteValuationRing R
[PROOFSTEP]
haveI := IsBezout.toGCDDomain R
[GOAL]
case tfae_2_to_1
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
✝ : ValuationRing R
this : GCDMonoid R
⊢ DiscreteValuationRing R
[PROOFSTEP]
haveI : UniqueFactorizationMonoid R := ufm_of_gcd_of_wfDvdMonoid
[GOAL]
case tfae_2_to_1
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
✝ : ValuationRing R
this✝ : GCDMonoid R
this : UniqueFactorizationMonoid R
⊢ DiscreteValuationRing R
[PROOFSTEP]
apply DiscreteValuationRing.of_ufd_of_unique_irreducible
[GOAL]
case tfae_2_to_1.h₁
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
✝ : ValuationRing R
this✝ : GCDMonoid R
this : UniqueFactorizationMonoid R
⊢ ∃ p, Irreducible p
[PROOFSTEP]
obtain ⟨x, hx₁, hx₂⟩ := Ring.exists_not_isUnit_of_not_isField h
[GOAL]
case tfae_2_to_1.h₁.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
✝ : ValuationRing R
this✝ : GCDMonoid R
this : UniqueFactorizationMonoid R
x : R
hx₁ : x ≠ 0
hx₂ : ¬IsUnit x
⊢ ∃ p, Irreducible p
[PROOFSTEP]
obtain ⟨p, hp₁, -⟩ := WfDvdMonoid.exists_irreducible_factor hx₂ hx₁
[GOAL]
case tfae_2_to_1.h₁.intro.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
✝ : ValuationRing R
this✝ : GCDMonoid R
this : UniqueFactorizationMonoid R
x : R
hx₁ : x ≠ 0
hx₂ : ¬IsUnit x
p : R
hp₁ : Irreducible p
⊢ ∃ p, Irreducible p
[PROOFSTEP]
exact ⟨p, hp₁⟩
[GOAL]
case tfae_2_to_1.h₂
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
✝ : ValuationRing R
this✝ : GCDMonoid R
this : UniqueFactorizationMonoid R
⊢ ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
[PROOFSTEP]
exact ValuationRing.unique_irreducible
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
tfae_have 1 → 4
[GOAL]
case tfae_1_to_4
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
⊢ DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
[PROOFSTEP]
intro H
[GOAL]
case tfae_1_to_4
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
H : DiscreteValuationRing R
⊢ IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
[PROOFSTEP]
exact ⟨inferInstance, ((DiscreteValuationRing.iff_pid_with_one_nonzero_prime R).mp H).2⟩
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
tfae_have 4 → 3
[GOAL]
case tfae_4_to_3
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
⊢ (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
[PROOFSTEP]
rintro ⟨h₁, h₂⟩
[GOAL]
case tfae_4_to_3.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
h₁ : IsIntegrallyClosed R
h₂ : ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
⊢ IsDedekindDomain R
[PROOFSTEP]
exact
  { h₁ with
    maximalOfPrime := fun hI hI' =>
      ExistsUnique.unique h₂ ⟨ne_bot, inferInstance⟩ ⟨hI, hI'⟩ ▸ maximalIdeal.isMaximal R, }
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
tfae_have 3 → 5
[GOAL]
case tfae_3_to_5
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
⊢ IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
intro h
[GOAL]
case tfae_3_to_5
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h✝ : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
h : IsDedekindDomain R
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
exact maximalIdeal_isPrincipal_of_isDedekindDomain R
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
tfae_have 5 → 6
[GOAL]
case tfae_5_to_6
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
⊢ Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
[PROOFSTEP]
rintro ⟨x, hx⟩
[GOAL]
case tfae_5_to_6.mk.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
⊢ ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
[PROOFSTEP]
have : x ∈ maximalIdeal R := by rw [hx]; exact Submodule.subset_span (Set.mem_singleton x)
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
⊢ x ∈ maximalIdeal R
[PROOFSTEP]
rw [hx]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
⊢ x ∈ Submodule.span R {x}
[PROOFSTEP]
exact Submodule.subset_span (Set.mem_singleton x)
[GOAL]
case tfae_5_to_6.mk.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
⊢ ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
[PROOFSTEP]
let x' : maximalIdeal R := ⟨x, this⟩
[GOAL]
case tfae_5_to_6.mk.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
⊢ ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
[PROOFSTEP]
use Submodule.Quotient.mk x'
[GOAL]
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
⊢ ∃ _n, ∀ (w : CotangentSpace R), ∃ c, c • Submodule.Quotient.mk x' = w
[PROOFSTEP]
constructor
[GOAL]
case h.h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
⊢ ∀ (w : CotangentSpace R), ∃ c, c • Submodule.Quotient.mk x' = w
case h.w
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
⊢ Submodule.Quotient.mk x' ≠ 0
[PROOFSTEP]
swap
[GOAL]
case h.w
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
⊢ Submodule.Quotient.mk x' ≠ 0
[PROOFSTEP]
intro e
[GOAL]
case h.w
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
e : Submodule.Quotient.mk x' = 0
⊢ False
[PROOFSTEP]
rw [Submodule.Quotient.mk_eq_zero] at e
[GOAL]
case h.w
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
e : x' ∈ maximalIdeal R • ⊤
⊢ False
[PROOFSTEP]
apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
[GOAL]
case h.w
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
e : x' ∈ maximalIdeal R • ⊤
⊢ maximalIdeal R = ⊥
[PROOFSTEP]
apply Submodule.eq_bot_of_le_smul_of_le_jacobson_bot (maximalIdeal R)
[GOAL]
case h.w.hN
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
e : x' ∈ maximalIdeal R • ⊤
⊢ Submodule.FG (maximalIdeal R)
[PROOFSTEP]
exact ⟨{ x }, (Finset.coe_singleton x).symm ▸ hx.symm⟩
[GOAL]
case h.w.hIN
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
e : x' ∈ maximalIdeal R • ⊤
⊢ maximalIdeal R ≤ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
conv_lhs => rw [hx]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
e : x' ∈ maximalIdeal R • ⊤
| maximalIdeal R
[PROOFSTEP]
rw [hx]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
e : x' ∈ maximalIdeal R • ⊤
| maximalIdeal R
[PROOFSTEP]
rw [hx]
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
e : x' ∈ maximalIdeal R • ⊤
| maximalIdeal R
[PROOFSTEP]
rw [hx]
[GOAL]
case h.w.hIN
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
e : x' ∈ maximalIdeal R • ⊤
⊢ Submodule.span R {x} ≤ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
rw [Submodule.mem_smul_top_iff] at e
[GOAL]
case h.w.hIN
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
e : ↑x' ∈ maximalIdeal R • maximalIdeal R
⊢ Submodule.span R {x} ≤ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
rwa [Submodule.span_le, Set.singleton_subset_iff]
[GOAL]
case h.w.hIjac
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
e : x' ∈ maximalIdeal R • ⊤
⊢ maximalIdeal R ≤ Ideal.jacobson ⊥
[PROOFSTEP]
rw [LocalRing.jacobson_eq_maximalIdeal (⊥ : Ideal R) bot_ne_top]
[GOAL]
case h.h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
⊢ ∀ (w : CotangentSpace R), ∃ c, c • Submodule.Quotient.mk x' = w
[PROOFSTEP]
refine' fun w => Quotient.inductionOn' w fun y => _
[GOAL]
case h.h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
w : CotangentSpace R
y : { x // x ∈ maximalIdeal R }
⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' y
[PROOFSTEP]
obtain ⟨y, hy⟩ := y
[GOAL]
case h.h.mk
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
w : CotangentSpace R
y : R
hy : y ∈ maximalIdeal R
⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' { val := y, property := hy }
[PROOFSTEP]
rw [hx, Submodule.mem_span_singleton] at hy
[GOAL]
case h.h.mk
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
w : CotangentSpace R
y : R
hy✝ : y ∈ maximalIdeal R
hy : ∃ a, a • x = y
⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' { val := y, property := hy✝ }
[PROOFSTEP]
obtain ⟨a, rfl⟩ := hy
[GOAL]
case h.h.mk.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
x : R
hx : maximalIdeal R = Submodule.span R {x}
this : x ∈ maximalIdeal R
x' : { x // x ∈ maximalIdeal R } := { val := x, property := this }
w : CotangentSpace R
a : R
hy : a • x ∈ maximalIdeal R
⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' { val := a • x, property := hy }
[PROOFSTEP]
exact ⟨Ideal.Quotient.mk _ a, rfl⟩
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
tfae_have 6 → 5
[GOAL]
case tfae_6_to_5
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
⊢ (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
rintro ⟨x, hx, hx'⟩
[GOAL]
case tfae_6_to_5.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : CotangentSpace R
hx : x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • x = w
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
induction x using Quotient.inductionOn' with
| h x => ?_
[GOAL]
case tfae_6_to_5.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : CotangentSpace R
hx : x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • x = w
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
induction x using Quotient.inductionOn' with
| h x => ?_
[GOAL]
case tfae_6_to_5.intro.intro.h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
⊢ Submodule.IsPrincipal (maximalIdeal R)
[PROOFSTEP]
use x
[GOAL]
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
⊢ maximalIdeal R = Submodule.span R {↑x}
[PROOFSTEP]
apply le_antisymm
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
⊢ maximalIdeal R ≤ Submodule.span R {↑x}
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
⊢ Submodule.span R {↑x} ≤ maximalIdeal R
[PROOFSTEP]
swap
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
⊢ Submodule.span R {↑x} ≤ maximalIdeal R
[PROOFSTEP]
rw [Submodule.span_le, Set.singleton_subset_iff]
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
⊢ ↑x ∈ ↑(maximalIdeal R)
[PROOFSTEP]
exact x.prop
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
⊢ maximalIdeal R ≤ Submodule.span R {↑x}
[PROOFSTEP]
have h₁ : (Ideal.span {(x : R)} : Ideal R) ⊔ maximalIdeal R ≤ Ideal.span {(x : R)} ⊔ maximalIdeal R • maximalIdeal R :=
  by
  refine' sup_le le_sup_left _
  rintro m hm
  obtain ⟨c, hc⟩ := hx' (Submodule.Quotient.mk ⟨m, hm⟩)
  induction c using Quotient.inductionOn' with
  | h c => ?_
  rw [← sub_sub_cancel (c * x) m]
  apply sub_mem _ _
  · refine' Ideal.mem_sup_left (Ideal.mem_span_singleton'.mpr ⟨c, rfl⟩)
  · have := (Submodule.Quotient.eq _).mp hc
    rw [Submodule.mem_smul_top_iff] at this
    exact Ideal.mem_sup_right this
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
⊢ Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
refine' sup_le le_sup_left _
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
⊢ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
rintro m hm
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
m : R
hm : m ∈ maximalIdeal R
⊢ m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
obtain ⟨c, hc⟩ := hx' (Submodule.Quotient.mk ⟨m, hm⟩)
[GOAL]
case intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
m : R
hm : m ∈ maximalIdeal R
c : ResidueField R
hc : c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }
⊢ m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
induction c using Quotient.inductionOn' with
| h c => ?_
[GOAL]
case intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
m : R
hm : m ∈ maximalIdeal R
c : ResidueField R
hc : c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }
⊢ m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
induction c using Quotient.inductionOn' with
| h c => ?_
[GOAL]
case intro.h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
m : R
hm : m ∈ maximalIdeal R
c : R
hc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }
⊢ m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
rw [← sub_sub_cancel (c * x) m]
[GOAL]
case intro.h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
m : R
hm : m ∈ maximalIdeal R
c : R
hc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }
⊢ c * ↑x - (c * ↑x - m) ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
apply sub_mem _ _
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
m : R
hm : m ∈ maximalIdeal R
c : R
hc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }
⊢ c * ↑x ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
refine' Ideal.mem_sup_left (Ideal.mem_span_singleton'.mpr ⟨c, rfl⟩)
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
m : R
hm : m ∈ maximalIdeal R
c : R
hc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }
⊢ c * ↑x - m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
have := (Submodule.Quotient.eq _).mp hc
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
m : R
hm : m ∈ maximalIdeal R
c : R
hc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }
this : (fun x x_1 => x • x_1) c x - { val := m, property := hm } ∈ maximalIdeal R • ⊤
⊢ c * ↑x - m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
rw [Submodule.mem_smul_top_iff] at this
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
m : R
hm : m ∈ maximalIdeal R
c : R
hc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }
this : ↑((fun x x_1 => x • x_1) c x - { val := m, property := hm }) ∈ maximalIdeal R • maximalIdeal R
⊢ c * ↑x - m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
[PROOFSTEP]
exact Ideal.mem_sup_right this
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
h₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
⊢ maximalIdeal R ≤ Submodule.span R {↑x}
[PROOFSTEP]
have h₂ : maximalIdeal R ≤ (⊥ : Ideal R).jacobson :=
  by
  rw [LocalRing.jacobson_eq_maximalIdeal]
  exact bot_ne_top
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
h₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
⊢ maximalIdeal R ≤ Ideal.jacobson ⊥
[PROOFSTEP]
rw [LocalRing.jacobson_eq_maximalIdeal]
[GOAL]
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
h₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
⊢ ⊥ ≠ ⊤
[PROOFSTEP]
exact bot_ne_top
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
h₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
h₂ : maximalIdeal R ≤ Ideal.jacobson ⊥
⊢ maximalIdeal R ≤ Submodule.span R {↑x}
[PROOFSTEP]
have := Submodule.smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson (IsNoetherian.noetherian _) h₂ h₁
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
h₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
h₂ : maximalIdeal R ≤ Ideal.jacobson ⊥
this : Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R = Ideal.span {↑x} ⊔ ⊥ • maximalIdeal R
⊢ maximalIdeal R ≤ Submodule.span R {↑x}
[PROOFSTEP]
rw [Submodule.bot_smul, sup_bot_eq] at this
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
h₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
h₂ : maximalIdeal R ≤ Ideal.jacobson ⊥
this : Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R = Ideal.span {↑x}
⊢ maximalIdeal R ≤ Submodule.span R {↑x}
[PROOFSTEP]
rw [← sup_eq_left, eq_comm]
[GOAL]
case h.a
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
x : { x // x ∈ maximalIdeal R }
hx : Quotient.mk'' x ≠ 0
hx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w
h₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R
h₂ : maximalIdeal R ≤ Ideal.jacobson ⊥
this : Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R = Ideal.span {↑x}
⊢ Submodule.span R {↑x} = Submodule.span R {↑x} ⊔ maximalIdeal R
[PROOFSTEP]
exact le_sup_left.antisymm (h₁.trans <| le_of_eq this)
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
tfae_have 5 → 7
[GOAL]
case tfae_5_to_7
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
⊢ Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
[PROOFSTEP]
exact exists_maximalIdeal_pow_eq_of_principal R h
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
tfae_have 7 → 2
[GOAL]
case tfae_7_to_2
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
⊢ (∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n) → ValuationRing R
[PROOFSTEP]
rw [ValuationRing.iff_ideal_total]
[GOAL]
case tfae_7_to_2
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
⊢ (∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n) → IsTotal (Ideal R) fun x x_1 => x ≤ x_1
[PROOFSTEP]
intro H
[GOAL]
case tfae_7_to_2
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
⊢ IsTotal (Ideal R) fun x x_1 => x ≤ x_1
[PROOFSTEP]
constructor
[GOAL]
case tfae_7_to_2.total
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
⊢ ∀ (a b : Ideal R), a ≤ b ∨ b ≤ a
[PROOFSTEP]
intro I J
[GOAL]
case tfae_7_to_2.total
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
I J : Ideal R
⊢ I ≤ J ∨ J ≤ I
[PROOFSTEP]
let _ := Classical.decEq (Ideal R)
[GOAL]
case tfae_7_to_2.total
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
I J : Ideal R
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
⊢ I ≤ J ∨ J ≤ I
[PROOFSTEP]
by_cases hI : I = ⊥
[GOAL]
case pos
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
I J : Ideal R
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
hI : I = ⊥
⊢ I ≤ J ∨ J ≤ I
[PROOFSTEP]
subst hI
[GOAL]
case pos
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
J : Ideal R
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
⊢ ⊥ ≤ J ∨ J ≤ ⊥
[PROOFSTEP]
left
[GOAL]
case pos.h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
J : Ideal R
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
⊢ ⊥ ≤ J
[PROOFSTEP]
exact bot_le
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
I J : Ideal R
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
hI : ¬I = ⊥
⊢ I ≤ J ∨ J ≤ I
[PROOFSTEP]
by_cases hJ : J = ⊥
[GOAL]
case pos
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
I J : Ideal R
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
hI : ¬I = ⊥
hJ : J = ⊥
⊢ I ≤ J ∨ J ≤ I
[PROOFSTEP]
subst hJ
[GOAL]
case pos
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
I : Ideal R
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
hI : ¬I = ⊥
⊢ I ≤ ⊥ ∨ ⊥ ≤ I
[PROOFSTEP]
right
[GOAL]
case pos.h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
I : Ideal R
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
hI : ¬I = ⊥
⊢ ⊥ ≤ I
[PROOFSTEP]
exact bot_le
[GOAL]
case neg
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
I J : Ideal R
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
hI : ¬I = ⊥
hJ : ¬J = ⊥
⊢ I ≤ J ∨ J ≤ I
[PROOFSTEP]
obtain ⟨n, rfl⟩ := H I hI
[GOAL]
case neg.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
J : Ideal R
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
hJ : ¬J = ⊥
n : ℕ
hI : ¬maximalIdeal R ^ n = ⊥
⊢ maximalIdeal R ^ n ≤ J ∨ J ≤ maximalIdeal R ^ n
[PROOFSTEP]
obtain ⟨m, rfl⟩ := H J hJ
[GOAL]
case neg.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
n : ℕ
hI : ¬maximalIdeal R ^ n = ⊥
m : ℕ
hJ : ¬maximalIdeal R ^ m = ⊥
⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m ∨ maximalIdeal R ^ m ≤ maximalIdeal R ^ n
[PROOFSTEP]
cases' le_total m n with h' h'
[GOAL]
case neg.intro.intro.inl
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
n : ℕ
hI : ¬maximalIdeal R ^ n = ⊥
m : ℕ
hJ : ¬maximalIdeal R ^ m = ⊥
h' : m ≤ n
⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m ∨ maximalIdeal R ^ m ≤ maximalIdeal R ^ n
[PROOFSTEP]
left
[GOAL]
case neg.intro.intro.inl.h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
n : ℕ
hI : ¬maximalIdeal R ^ n = ⊥
m : ℕ
hJ : ¬maximalIdeal R ^ m = ⊥
h' : m ≤ n
⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m
[PROOFSTEP]
exact Ideal.pow_le_pow h'
[GOAL]
case neg.intro.intro.inr
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
n : ℕ
hI : ¬maximalIdeal R ^ n = ⊥
m : ℕ
hJ : ¬maximalIdeal R ^ m = ⊥
h' : n ≤ m
⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m ∨ maximalIdeal R ^ m ≤ maximalIdeal R ^ n
[PROOFSTEP]
right
[GOAL]
case neg.intro.intro.inr.h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
H : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
x✝ : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
n : ℕ
hI : ¬maximalIdeal R ^ n = ⊥
m : ℕ
hJ : ¬maximalIdeal R ^ m = ⊥
h' : n ≤ m
⊢ maximalIdeal R ^ m ≤ maximalIdeal R ^ n
[PROOFSTEP]
exact Ideal.pow_le_pow h'
[GOAL]
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
ne_bot : maximalIdeal R ≠ ⊥
tfae_1_to_2 : DiscreteValuationRing R → ValuationRing R
tfae_2_to_1 : ValuationRing R → DiscreteValuationRing R
tfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P
tfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R
tfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w
tfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)
tfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n
tfae_7_to_2 : (∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n) → ValuationRing R
⊢ List.TFAE
    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,
      Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,
      ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]
[PROOFSTEP]
tfae_finish