l3lab/ntp-mathlib huggingface dataset:
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        

Proof state:
case intro.intro.intro.intro.refine_1
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
this : ∀ {M : Ideal A}, IsMaximal M → ∃ x ∈ (↑M)⁻¹, x ∉ 1
M : Ideal A
hM : IsMaximal M
hIM : I ≤ M
hM0 : M ≠ ⊥
x : K
hxM : x ∈ (↑M)⁻¹
hx1 : x ∉ 1
⊢ ↑I ≠ 0

Next tactic:
rw [coeIdeal_ne_zero]
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        

Proof state:
case intro.intro.intro.intro.refine_2
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
this : ∀ {M : Ideal A}, IsMaximal M → ∃ x ∈ (↑M)⁻¹, x ∉ 1
M : Ideal A
hM : IsMaximal M
hIM : I ≤ M
hM0 : M ≠ ⊥
x : K
hxM : x ∈ (↑M)⁻¹
hx1 : x ∉ 1
⊢ ↑M ≠ 0

Next tactic:
rw [coeIdeal_ne_zero]
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> 

Proof state:
case intro.intro.intro.intro.refine_1
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
this : ∀ {M : Ideal A}, IsMaximal M → ∃ x ∈ (↑M)⁻¹, x ∉ 1
M : Ideal A
hM : IsMaximal M
hIM : I ≤ M
hM0 : M ≠ ⊥
x : K
hxM : x ∈ (↑M)⁻¹
hx1 : x ∉ 1
⊢ I ≠ ⊥

Next tactic:
assumption
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> 

Proof state:
case intro.intro.intro.intro.refine_2
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
this : ∀ {M : Ideal A}, IsMaximal M → ∃ x ∈ (↑M)⁻¹, x ∉ 1
M : Ideal A
hM : IsMaximal M
hIM : I ≤ M
hM0 : M ≠ ⊥
x : K
hxM : x ∈ (↑M)⁻¹
hx1 : x ∉ 1
⊢ M ≠ ⊥

Next tactic:
assumption
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
⊢ ∀ {M : Ideal A}, IsMaximal M → ∃ x ∈ (↑M)⁻¹, x ∉ 1

Next tactic:
intro M hM
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
  intro M hM
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
M : Ideal A
hM : IsMaximal M
⊢ ∃ x ∈ (↑M)⁻¹, x ∉ 1

Next tactic:
obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
  intro M hM
  obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
  

Proof state:
case intro.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
M : Ideal A
hM : IsMaximal M
a : A
haM : a ∈ M
ha0 : { val := a, property := haM } ≠ 0
⊢ ∃ x ∈ (↑M)⁻¹, x ∉ 1

Next tactic:
replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
  intro M hM
  obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
  replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0
  

Proof state:
case intro.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
M : Ideal A
hM : IsMaximal M
a : A
haM : a ∈ M
ha0 : a ≠ 0
⊢ ∃ x ∈ (↑M)⁻¹, x ∉ 1

Next tactic:
let J : Ideal A := Ideal.span {a}
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
  intro M hM
  obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
  replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0
  let J : Ideal A := Ideal.span {a}
  

Proof state:
case intro.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
M : Ideal A
hM : IsMaximal M
a : A
haM : a ∈ M
ha0 : a ≠ 0
J : Ideal A := span {a}
⊢ ∃ x ∈ (↑M)⁻¹, x ∉ 1

Next tactic:
have hJ0 : J ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp ha0
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
  intro M hM
  obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
  replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0
  let J : Ideal A := Ideal.span {a}
  have hJ0 : J ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp ha0
  

Proof state:
case intro.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
M : Ideal A
hM : IsMaximal M
a : A
haM : a ∈ M
ha0 : a ≠ 0
J : Ideal A := span {a}
hJ0 : J ≠ ⊥
⊢ ∃ x ∈ (↑M)⁻¹, x ∉ 1

Next tactic:
have hJM : J ≤ M := Ideal.span_le.mpr (Set.singleton_subset_iff.mpr haM)
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
  intro M hM
  obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
  replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0
  let J : Ideal A := Ideal.span {a}
  have hJ0 : J ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp ha0
  have hJM : J ≤ M := Ideal.span_le.mpr (Set.singleton_subset_iff.mpr haM)
  

Proof state:
case intro.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
M : Ideal A
hM : IsMaximal M
a : A
haM : a ∈ M
ha0 : a ≠ 0
J : Ideal A := span {a}
hJ0 : J ≠ ⊥
hJM : J ≤ M
⊢ ∃ x ∈ (↑M)⁻¹, x ∉ 1

Next tactic:
have hM0 : ⊥ < M := M.bot_lt_of_maximal hNF
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
  intro M hM
  obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
  replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0
  let J : Ideal A := Ideal.span {a}
  have hJ0 : J ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp ha0
  have hJM : J ≤ M := Ideal.span_le.mpr (Set.singleton_subset_iff.mpr haM)
  have hM0 : ⊥ < M := M.bot_lt_of_maximal hNF
  -- Then we can find a product of prime (hence maximal) ideals contained in `J`,
  -- such that removing element `M` from the product is not contained in `J`.
  

Proof state:
case intro.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
M : Ideal A
hM : IsMaximal M
a : A
haM : a ∈ M
ha0 : a ≠ 0
J : Ideal A := span {a}
hJ0 : J ≠ ⊥
hJM : J ≤ M
hM0 : ⊥ < M
⊢ ∃ x ∈ (↑M)⁻¹, x ∉ 1

Next tactic:
obtain ⟨Z, hle, hnle⟩ := exists_multiset_prod_cons_le_and_prod_not_le hNF hJ0 hJM
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
  intro M hM
  obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
  replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0
  let J : Ideal A := Ideal.span {a}
  have hJ0 : J ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp ha0
  have hJM : J ≤ M := Ideal.span_le.mpr (Set.singleton_subset_iff.mpr haM)
  have hM0 : ⊥ < M := M.bot_lt_of_maximal hNF
  -- Then we can find a product of prime (hence maximal) ideals contained in `J`,
  -- such that removing element `M` from the product is not contained in `J`.
  obtain ⟨Z, hle, hnle⟩ := exists_multiset_prod_cons_le_and_prod_not_le hNF hJ0 hJM
  -- Choose an element `b` of the product that is not in `J`.
  

Proof state:
case intro.mk.intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
M : Ideal A
hM : IsMaximal M
a : A
haM : a ∈ M
ha0 : a ≠ 0
J : Ideal A := span {a}
hJ0 : J ≠ ⊥
hJM : J ≤ M
hM0 : ⊥ < M
Z : Multiset (PrimeSpectrum A)
hle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J
hnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J
⊢ ∃ x ∈ (↑M)⁻¹, x ∉ 1

Next tactic:
obtain ⟨b, hbZ, hbJ⟩ := SetLike.not_le_iff_exists.mp hnle
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
  intro M hM
  obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
  replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0
  let J : Ideal A := Ideal.span {a}
  have hJ0 : J ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp ha0
  have hJM : J ≤ M := Ideal.span_le.mpr (Set.singleton_subset_iff.mpr haM)
  have hM0 : ⊥ < M := M.bot_lt_of_maximal hNF
  -- Then we can find a product of prime (hence maximal) ideals contained in `J`,
  -- such that removing element `M` from the product is not contained in `J`.
  obtain ⟨Z, hle, hnle⟩ := exists_multiset_prod_cons_le_and_prod_not_le hNF hJ0 hJM
  -- Choose an element `b` of the product that is not in `J`.
  obtain ⟨b, hbZ, hbJ⟩ := SetLike.not_le_iff_exists.mp hnle
  

Proof state:
case intro.mk.intro.intro.intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
M : Ideal A
hM : IsMaximal M
a : A
haM : a ∈ M
ha0 : a ≠ 0
J : Ideal A := span {a}
hJ0 : J ≠ ⊥
hJM : J ≤ M
hM0 : ⊥ < M
Z : Multiset (PrimeSpectrum A)
hle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J
hnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J
b : A
hbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)
hbJ : b ∉ J
⊢ ∃ x ∈ (↑M)⁻¹, x ∉ 1

Next tactic:
have hnz_fa : algebraMap A K a ≠ 0 :=
    mt ((injective_iff_map_eq_zero _).mp (IsFractionRing.injective A K) a) ha0
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
  intro M hM
  obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
  replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0
  let J : Ideal A := Ideal.span {a}
  have hJ0 : J ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp ha0
  have hJM : J ≤ M := Ideal.span_le.mpr (Set.singleton_subset_iff.mpr haM)
  have hM0 : ⊥ < M := M.bot_lt_of_maximal hNF
  -- Then we can find a product of prime (hence maximal) ideals contained in `J`,
  -- such that removing element `M` from the product is not contained in `J`.
  obtain ⟨Z, hle, hnle⟩ := exists_multiset_prod_cons_le_and_prod_not_le hNF hJ0 hJM
  -- Choose an element `b` of the product that is not in `J`.
  obtain ⟨b, hbZ, hbJ⟩ := SetLike.not_le_iff_exists.mp hnle
  have hnz_fa : algebraMap A K a ≠ 0 :=
    mt ((injective_iff_map_eq_zero _).mp (IsFractionRing.injective A K) a) ha0
  

Proof state:
case intro.mk.intro.intro.intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
M : Ideal A
hM : IsMaximal M
a : A
haM : a ∈ M
ha0 : a ≠ 0
J : Ideal A := span {a}
hJ0 : J ≠ ⊥
hJM : J ≤ M
hM0 : ⊥ < M
Z : Multiset (PrimeSpectrum A)
hle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J
hnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J
b : A
hbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)
hbJ : b ∉ J
hnz_fa : (algebraMap A K) a ≠ 0
⊢ ∃ x ∈ (↑M)⁻¹, x ∉ 1

Next tactic:
have _hb0 : algebraMap A K b ≠ 0 :=
    mt ((injective_iff_map_eq_zero _).mp (IsFractionRing.injective A K) b) fun h =>
      hbJ <| h.symm ▸ J.zero_mem