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⟩ <;>
        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
  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
  -- Then `b a⁻¹ : K` is in `M⁻¹` but not in `1`.
  refine' ⟨algebraMap A K b * (algebraMap A K a)⁻¹, (mem_inv_iff _).mpr _, _⟩
  · exact coeIdeal_ne_zero.mpr hM0.ne'
  · rintro y₀ hy₀
    obtain ⟨y, h_Iy, rfl⟩ := (mem_coeIdeal _).mp hy₀
    rw [mul_comm, ← mul_assoc, ← RingHom.map_mul]
    have h_yb : y * b ∈ J := by
      apply hle
      rw [Multiset.prod_cons]
      

Proof state:
case a
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
_hb0 : (algebraMap A K) b ≠ 0
y : A
h_Iy : y ∈ M
hy₀ : (algebraMap A K) y ∈ ↑M
⊢ y * b ∈ M * Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)

Next tactic:
exact Submodule.smul_mem_smul h_Iy hbZ
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
  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
  -- Then `b a⁻¹ : K` is in `M⁻¹` but not in `1`.
  refine' ⟨algebraMap A K b * (algebraMap A K a)⁻¹, (mem_inv_iff _).mpr _, _⟩
  · exact coeIdeal_ne_zero.mpr hM0.ne'
  · rintro y₀ hy₀
    obtain ⟨y, h_Iy, rfl⟩ := (mem_coeIdeal _).mp hy₀
    rw [mul_comm, ← mul_assoc, ← RingHom.map_mul]
    have h_yb : y * b ∈ J := by
      apply hle
      rw [Multiset.prod_cons]
      exact Submodule.smul_mem_smul h_Iy hbZ
    

Proof state:
case intro.mk.intro.intro.intro.intro.refine'_2.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
_hb0 : (algebraMap A K) b ≠ 0
y : A
h_Iy : y ∈ M
hy₀ : (algebraMap A K) y ∈ ↑M
h_yb : y * b ∈ J
⊢ (algebraMap A K) (y * b) * ((algebraMap A K) a)⁻¹ ∈ 1

Next tactic:
rw [Ideal.mem_span_singleton'] at h_yb
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
  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
  -- Then `b a⁻¹ : K` is in `M⁻¹` but not in `1`.
  refine' ⟨algebraMap A K b * (algebraMap A K a)⁻¹, (mem_inv_iff _).mpr _, _⟩
  · exact coeIdeal_ne_zero.mpr hM0.ne'
  · rintro y₀ hy₀
    obtain ⟨y, h_Iy, rfl⟩ := (mem_coeIdeal _).mp hy₀
    rw [mul_comm, ← mul_assoc, ← RingHom.map_mul]
    have h_yb : y * b ∈ J := by
      apply hle
      rw [Multiset.prod_cons]
      exact Submodule.smul_mem_smul h_Iy hbZ
    rw [Ideal.mem_span_singleton'] at h_yb
    

Proof state:
case intro.mk.intro.intro.intro.intro.refine'_2.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
_hb0 : (algebraMap A K) b ≠ 0
y : A
h_Iy : y ∈ M
hy₀ : (algebraMap A K) y ∈ ↑M
h_yb : ∃ a_1, a_1 * a = y * b
⊢ (algebraMap A K) (y * b) * ((algebraMap A K) a)⁻¹ ∈ 1

Next tactic:
rcases h_yb with ⟨c, hc⟩
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
  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
  -- Then `b a⁻¹ : K` is in `M⁻¹` but not in `1`.
  refine' ⟨algebraMap A K b * (algebraMap A K a)⁻¹, (mem_inv_iff _).mpr _, _⟩
  · exact coeIdeal_ne_zero.mpr hM0.ne'
  · rintro y₀ hy₀
    obtain ⟨y, h_Iy, rfl⟩ := (mem_coeIdeal _).mp hy₀
    rw [mul_comm, ← mul_assoc, ← RingHom.map_mul]
    have h_yb : y * b ∈ J := by
      apply hle
      rw [Multiset.prod_cons]
      exact Submodule.smul_mem_smul h_Iy hbZ
    rw [Ideal.mem_span_singleton'] at h_yb
    rcases h_yb with ⟨c, hc⟩
    rw [← hc, RingHom.map_mul, mul_assoc, mul_inv_cancel hnz_fa, mul_one]
    apply coe_mem_one
  · 

Proof state:
case intro.mk.intro.intro.intro.intro.refine'_3
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
_hb0 : (algebraMap A K) b ≠ 0
⊢ (algebraMap A K) b * ((algebraMap A K) a)⁻¹ ∉ 1

Next tactic:
refine' mt (mem_one_iff _).mp _
theorem one_mem_inv_coe_ideal {I : Ideal A} (hI : I ≠ ⊥) :
    (1 : K) ∈ (I : FractionalIdeal A⁰ K)⁻¹ := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I : Ideal A
hI : I ≠ ⊥
⊢ 1 ∈ (↑I)⁻¹

Next tactic:
rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI)]
theorem one_mem_inv_coe_ideal {I : Ideal A} (hI : I ≠ ⊥) :
    (1 : K) ∈ (I : FractionalIdeal A⁰ K)⁻¹ := by
  rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI)]
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I : Ideal A
hI : I ≠ ⊥
⊢ ∀ y ∈ ↑I, 1 * y ∈ 1

Next tactic:
intro y hy
theorem one_mem_inv_coe_ideal {I : Ideal A} (hI : I ≠ ⊥) :
    (1 : K) ∈ (I : FractionalIdeal A⁰ K)⁻¹ := by
  rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI)]
  intro y hy
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I : Ideal A
hI : I ≠ ⊥
y : K
hy : y ∈ ↑I
⊢ 1 * y ∈ 1

Next tactic:
rw [one_mul]
theorem one_mem_inv_coe_ideal {I : Ideal A} (hI : I ≠ ⊥) :
    (1 : K) ∈ (I : FractionalIdeal A⁰ K)⁻¹ := by
  rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI)]
  intro y hy
  rw [one_mul]
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I : Ideal A
hI : I ≠ ⊥
y : K
hy : y ∈ ↑I
⊢ y ∈ 1

Next tactic:
exact coeIdeal_le_one hy
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
by_cases hI1 : I = ⊤
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · 

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : I = ⊤
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
rw [hI1, coeIdeal_top, one_mul, inv_one]
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
by_cases hNF : IsField A
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · 

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : IsField A
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
letI := hNF.toField
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; 

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : IsField A
this : Field A := IsField.toField hNF
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
  

Proof state:
case neg.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
J : Ideal A
hJ : ↑J = ↑I * (↑I)⁻¹
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
by_cases hJ0 : J = ⊥
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
  by_cases hJ0 : J = ⊥
  · 

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
J : Ideal A
hJ : ↑J = ↑I * (↑I)⁻¹
hJ0 : J = ⊥
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
subst hJ0
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
  by_cases hJ0 : J = ⊥
  · subst hJ0
    

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
hJ : ↑⊥ = ↑I * (↑I)⁻¹
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
refine' absurd _ hI0
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
  by_cases hJ0 : J = ⊥
  · subst hJ0
    refine' absurd _ hI0
    

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
hJ : ↑⊥ = ↑I * (↑I)⁻¹
⊢ I = ⊥

Next tactic:
rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
  by_cases hJ0 : J = ⊥
  · subst hJ0
    refine' absurd _ hI0
    rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]
    

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
hJ : ↑⊥ = ↑I * (↑I)⁻¹
⊢ ↑I ≤ ↑I * (↑I)⁻¹

Next tactic:
exact coe_ideal_le_self_mul_inv K I