l3lab/ntp-mathlib huggingface dataset:
/-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
    IsPrincipalIdealRing R := by
  -- Suppose the set of `nonPrincipals` is not empty.
  rw [← nonPrincipals_eq_empty_iff, Set.eq_empty_iff_forall_not_mem]
  intro J hJ
  -- We will show a maximal element `I ∈ nonPrincipals R` (which exists by Zorn) is prime.
  obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partialOrder₀ (nonPrincipals R) nonPrincipals_zorn _ hJ
  have Imax' : ∀ {J}, I < J → J.IsPrincipal := by
    intro J hJ
    by_contra He
    exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm
  by_cases hI1 : I = ⊤
  · subst hI1
    exact Ibad top_isPrincipal
  -- Let `x y : R` with `x * y ∈ I` and suppose WLOG `y ∉ I`.
  refine' Ibad (H I ⟨hI1, fun {x y} hxy => or_iff_not_imp_right.mpr fun hy => _⟩)
  obtain ⟨a, ha⟩ : (I ⊔ span {y}).IsPrincipal :=
    Imax' (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))
  -- Then `x ∈ I.colon (span {y})`, which is equal to `I` if it's not principal.
  suffices He : ¬(I.colon (span {y})).IsPrincipal
  · rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
        Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]
    exact Ideal.mem_colon_singleton.2 hxy
  -- So suppose for the sake of contradiction that both `I ⊔ span {y}` and `I.colon (span {y})`
  -- are principal.
  rintro ⟨b, hb⟩
  -- We will show `I` is generated by `a * b`.
  refine (nonPrincipals_def _).1 Ibad ⟨a * b, ?_⟩
  refine'
    le_antisymm (α := Ideal R) (fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _
  · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
    have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
    erw [ha, mem_span_singleton'] at hisup this
    obtain ⟨v, rfl⟩ := this
    

Proof state:
case He.mk.intro.refine'_1.intro
R : Type u
M : Type v
inst✝ : CommRing R
H : ∀ (P : Ideal R), IsPrime P → IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
Ibad : I ∈ nonPrincipals R
Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I
Imax' : ∀ {J : Ideal R}, I < J → IsPrincipal J
hI1 : ¬I = ⊤
x a b i : R
hi : i ∈ I
hisup : ∃ a_1, a_1 * a = i
v : R
hxy : x * (v * a) ∈ I
hy : v * a ∉ I
ha : I ⊔ Ideal.span {v * a} = Submodule.span R {a}
hb : colon I (Ideal.span {v * a}) = Submodule.span R {b}
⊢ i ∈ Submodule.span R {a * b}

Next tactic:
obtain ⟨u, rfl⟩ := hisup
/-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
    IsPrincipalIdealRing R := by
  -- Suppose the set of `nonPrincipals` is not empty.
  rw [← nonPrincipals_eq_empty_iff, Set.eq_empty_iff_forall_not_mem]
  intro J hJ
  -- We will show a maximal element `I ∈ nonPrincipals R` (which exists by Zorn) is prime.
  obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partialOrder₀ (nonPrincipals R) nonPrincipals_zorn _ hJ
  have Imax' : ∀ {J}, I < J → J.IsPrincipal := by
    intro J hJ
    by_contra He
    exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm
  by_cases hI1 : I = ⊤
  · subst hI1
    exact Ibad top_isPrincipal
  -- Let `x y : R` with `x * y ∈ I` and suppose WLOG `y ∉ I`.
  refine' Ibad (H I ⟨hI1, fun {x y} hxy => or_iff_not_imp_right.mpr fun hy => _⟩)
  obtain ⟨a, ha⟩ : (I ⊔ span {y}).IsPrincipal :=
    Imax' (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))
  -- Then `x ∈ I.colon (span {y})`, which is equal to `I` if it's not principal.
  suffices He : ¬(I.colon (span {y})).IsPrincipal
  · rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
        Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]
    exact Ideal.mem_colon_singleton.2 hxy
  -- So suppose for the sake of contradiction that both `I ⊔ span {y}` and `I.colon (span {y})`
  -- are principal.
  rintro ⟨b, hb⟩
  -- We will show `I` is generated by `a * b`.
  refine (nonPrincipals_def _).1 Ibad ⟨a * b, ?_⟩
  refine'
    le_antisymm (α := Ideal R) (fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _
  · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
    have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
    erw [ha, mem_span_singleton'] at hisup this
    obtain ⟨v, rfl⟩ := this
    obtain ⟨u, rfl⟩ := hisup
    

Proof state:
case He.mk.intro.refine'_1.intro.intro
R : Type u
M : Type v
inst✝ : CommRing R
H : ∀ (P : Ideal R), IsPrime P → IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
Ibad : I ∈ nonPrincipals R
Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I
Imax' : ∀ {J : Ideal R}, I < J → IsPrincipal J
hI1 : ¬I = ⊤
x a b v : R
hxy : x * (v * a) ∈ I
hy : v * a ∉ I
ha : I ⊔ Ideal.span {v * a} = Submodule.span R {a}
hb : colon I (Ideal.span {v * a}) = Submodule.span R {b}
u : R
hi : u * a ∈ I
⊢ u * a ∈ Submodule.span R {a * b}

Next tactic:
have hucolon : u ∈ I.colon (span {v * a}) := by
      rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
      exact mul_mem_right _ _ hi
/-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
    IsPrincipalIdealRing R := by
  -- Suppose the set of `nonPrincipals` is not empty.
  rw [← nonPrincipals_eq_empty_iff, Set.eq_empty_iff_forall_not_mem]
  intro J hJ
  -- We will show a maximal element `I ∈ nonPrincipals R` (which exists by Zorn) is prime.
  obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partialOrder₀ (nonPrincipals R) nonPrincipals_zorn _ hJ
  have Imax' : ∀ {J}, I < J → J.IsPrincipal := by
    intro J hJ
    by_contra He
    exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm
  by_cases hI1 : I = ⊤
  · subst hI1
    exact Ibad top_isPrincipal
  -- Let `x y : R` with `x * y ∈ I` and suppose WLOG `y ∉ I`.
  refine' Ibad (H I ⟨hI1, fun {x y} hxy => or_iff_not_imp_right.mpr fun hy => _⟩)
  obtain ⟨a, ha⟩ : (I ⊔ span {y}).IsPrincipal :=
    Imax' (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))
  -- Then `x ∈ I.colon (span {y})`, which is equal to `I` if it's not principal.
  suffices He : ¬(I.colon (span {y})).IsPrincipal
  · rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
        Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]
    exact Ideal.mem_colon_singleton.2 hxy
  -- So suppose for the sake of contradiction that both `I ⊔ span {y}` and `I.colon (span {y})`
  -- are principal.
  rintro ⟨b, hb⟩
  -- We will show `I` is generated by `a * b`.
  refine (nonPrincipals_def _).1 Ibad ⟨a * b, ?_⟩
  refine'
    le_antisymm (α := Ideal R) (fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _
  · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
    have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
    erw [ha, mem_span_singleton'] at hisup this
    obtain ⟨v, rfl⟩ := this
    obtain ⟨u, rfl⟩ := hisup
    have hucolon : u ∈ I.colon (span {v * a}) := by
      

Proof state:
R : Type u
M : Type v
inst✝ : CommRing R
H : ∀ (P : Ideal R), IsPrime P → IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
Ibad : I ∈ nonPrincipals R
Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I
Imax' : ∀ {J : Ideal R}, I < J → IsPrincipal J
hI1 : ¬I = ⊤
x a b v : R
hxy : x * (v * a) ∈ I
hy : v * a ∉ I
ha : I ⊔ Ideal.span {v * a} = Submodule.span R {a}
hb : colon I (Ideal.span {v * a}) = Submodule.span R {b}
u : R
hi : u * a ∈ I
⊢ u ∈ colon I (Ideal.span {v * a})

Next tactic:
rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
/-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
    IsPrincipalIdealRing R := by
  -- Suppose the set of `nonPrincipals` is not empty.
  rw [← nonPrincipals_eq_empty_iff, Set.eq_empty_iff_forall_not_mem]
  intro J hJ
  -- We will show a maximal element `I ∈ nonPrincipals R` (which exists by Zorn) is prime.
  obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partialOrder₀ (nonPrincipals R) nonPrincipals_zorn _ hJ
  have Imax' : ∀ {J}, I < J → J.IsPrincipal := by
    intro J hJ
    by_contra He
    exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm
  by_cases hI1 : I = ⊤
  · subst hI1
    exact Ibad top_isPrincipal
  -- Let `x y : R` with `x * y ∈ I` and suppose WLOG `y ∉ I`.
  refine' Ibad (H I ⟨hI1, fun {x y} hxy => or_iff_not_imp_right.mpr fun hy => _⟩)
  obtain ⟨a, ha⟩ : (I ⊔ span {y}).IsPrincipal :=
    Imax' (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))
  -- Then `x ∈ I.colon (span {y})`, which is equal to `I` if it's not principal.
  suffices He : ¬(I.colon (span {y})).IsPrincipal
  · rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
        Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]
    exact Ideal.mem_colon_singleton.2 hxy
  -- So suppose for the sake of contradiction that both `I ⊔ span {y}` and `I.colon (span {y})`
  -- are principal.
  rintro ⟨b, hb⟩
  -- We will show `I` is generated by `a * b`.
  refine (nonPrincipals_def _).1 Ibad ⟨a * b, ?_⟩
  refine'
    le_antisymm (α := Ideal R) (fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _
  · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
    have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
    erw [ha, mem_span_singleton'] at hisup this
    obtain ⟨v, rfl⟩ := this
    obtain ⟨u, rfl⟩ := hisup
    have hucolon : u ∈ I.colon (span {v * a}) := by
      rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
      

Proof state:
R : Type u
M : Type v
inst✝ : CommRing R
H : ∀ (P : Ideal R), IsPrime P → IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
Ibad : I ∈ nonPrincipals R
Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I
Imax' : ∀ {J : Ideal R}, I < J → IsPrincipal J
hI1 : ¬I = ⊤
x a b v : R
hxy : x * (v * a) ∈ I
hy : v * a ∉ I
ha : I ⊔ Ideal.span {v * a} = Submodule.span R {a}
hb : colon I (Ideal.span {v * a}) = Submodule.span R {b}
u : R
hi : u * a ∈ I
⊢ u * a * v ∈ I

Next tactic:
exact mul_mem_right _ _ hi
/-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
    IsPrincipalIdealRing R := by
  -- Suppose the set of `nonPrincipals` is not empty.
  rw [← nonPrincipals_eq_empty_iff, Set.eq_empty_iff_forall_not_mem]
  intro J hJ
  -- We will show a maximal element `I ∈ nonPrincipals R` (which exists by Zorn) is prime.
  obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partialOrder₀ (nonPrincipals R) nonPrincipals_zorn _ hJ
  have Imax' : ∀ {J}, I < J → J.IsPrincipal := by
    intro J hJ
    by_contra He
    exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm
  by_cases hI1 : I = ⊤
  · subst hI1
    exact Ibad top_isPrincipal
  -- Let `x y : R` with `x * y ∈ I` and suppose WLOG `y ∉ I`.
  refine' Ibad (H I ⟨hI1, fun {x y} hxy => or_iff_not_imp_right.mpr fun hy => _⟩)
  obtain ⟨a, ha⟩ : (I ⊔ span {y}).IsPrincipal :=
    Imax' (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))
  -- Then `x ∈ I.colon (span {y})`, which is equal to `I` if it's not principal.
  suffices He : ¬(I.colon (span {y})).IsPrincipal
  · rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
        Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]
    exact Ideal.mem_colon_singleton.2 hxy
  -- So suppose for the sake of contradiction that both `I ⊔ span {y}` and `I.colon (span {y})`
  -- are principal.
  rintro ⟨b, hb⟩
  -- We will show `I` is generated by `a * b`.
  refine (nonPrincipals_def _).1 Ibad ⟨a * b, ?_⟩
  refine'
    le_antisymm (α := Ideal R) (fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _
  · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
    have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
    erw [ha, mem_span_singleton'] at hisup this
    obtain ⟨v, rfl⟩ := this
    obtain ⟨u, rfl⟩ := hisup
    have hucolon : u ∈ I.colon (span {v * a}) := by
      rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
      exact mul_mem_right _ _ hi
    

Proof state:
case He.mk.intro.refine'_1.intro.intro
R : Type u
M : Type v
inst✝ : CommRing R
H : ∀ (P : Ideal R), IsPrime P → IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
Ibad : I ∈ nonPrincipals R
Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I
Imax' : ∀ {J : Ideal R}, I < J → IsPrincipal J
hI1 : ¬I = ⊤
x a b v : R
hxy : x * (v * a) ∈ I
hy : v * a ∉ I
ha : I ⊔ Ideal.span {v * a} = Submodule.span R {a}
hb : colon I (Ideal.span {v * a}) = Submodule.span R {b}
u : R
hi : u * a ∈ I
hucolon : u ∈ colon I (Ideal.span {v * a})
⊢ u * a ∈ Submodule.span R {a * b}

Next tactic:
erw [hb, mem_span_singleton'] at hucolon
/-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
    IsPrincipalIdealRing R := by
  -- Suppose the set of `nonPrincipals` is not empty.
  rw [← nonPrincipals_eq_empty_iff, Set.eq_empty_iff_forall_not_mem]
  intro J hJ
  -- We will show a maximal element `I ∈ nonPrincipals R` (which exists by Zorn) is prime.
  obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partialOrder₀ (nonPrincipals R) nonPrincipals_zorn _ hJ
  have Imax' : ∀ {J}, I < J → J.IsPrincipal := by
    intro J hJ
    by_contra He
    exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm
  by_cases hI1 : I = ⊤
  · subst hI1
    exact Ibad top_isPrincipal
  -- Let `x y : R` with `x * y ∈ I` and suppose WLOG `y ∉ I`.
  refine' Ibad (H I ⟨hI1, fun {x y} hxy => or_iff_not_imp_right.mpr fun hy => _⟩)
  obtain ⟨a, ha⟩ : (I ⊔ span {y}).IsPrincipal :=
    Imax' (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))
  -- Then `x ∈ I.colon (span {y})`, which is equal to `I` if it's not principal.
  suffices He : ¬(I.colon (span {y})).IsPrincipal
  · rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
        Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]
    exact Ideal.mem_colon_singleton.2 hxy
  -- So suppose for the sake of contradiction that both `I ⊔ span {y}` and `I.colon (span {y})`
  -- are principal.
  rintro ⟨b, hb⟩
  -- We will show `I` is generated by `a * b`.
  refine (nonPrincipals_def _).1 Ibad ⟨a * b, ?_⟩
  refine'
    le_antisymm (α := Ideal R) (fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _
  · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
    have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
    erw [ha, mem_span_singleton'] at hisup this
    obtain ⟨v, rfl⟩ := this
    obtain ⟨u, rfl⟩ := hisup
    have hucolon : u ∈ I.colon (span {v * a}) := by
      rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
      exact mul_mem_right _ _ hi
    erw [hb, mem_span_singleton'] at hucolon
    

Proof state:
case He.mk.intro.refine'_1.intro.intro
R : Type u
M : Type v
inst✝ : CommRing R
H : ∀ (P : Ideal R), IsPrime P → IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
Ibad : I ∈ nonPrincipals R
Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I
Imax' : ∀ {J : Ideal R}, I < J → IsPrincipal J
hI1 : ¬I = ⊤
x a b v : R
hxy : x * (v * a) ∈ I
hy : v * a ∉ I
ha : I ⊔ Ideal.span {v * a} = Submodule.span R {a}
hb : colon I (Ideal.span {v * a}) = Submodule.span R {b}
u : R
hi : u * a ∈ I
hucolon : ∃ a, a * b = u
⊢ u * a ∈ Submodule.span R {a * b}

Next tactic:
obtain ⟨z, rfl⟩ := hucolon
/-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
    IsPrincipalIdealRing R := by
  -- Suppose the set of `nonPrincipals` is not empty.
  rw [← nonPrincipals_eq_empty_iff, Set.eq_empty_iff_forall_not_mem]
  intro J hJ
  -- We will show a maximal element `I ∈ nonPrincipals R` (which exists by Zorn) is prime.
  obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partialOrder₀ (nonPrincipals R) nonPrincipals_zorn _ hJ
  have Imax' : ∀ {J}, I < J → J.IsPrincipal := by
    intro J hJ
    by_contra He
    exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm
  by_cases hI1 : I = ⊤
  · subst hI1
    exact Ibad top_isPrincipal
  -- Let `x y : R` with `x * y ∈ I` and suppose WLOG `y ∉ I`.
  refine' Ibad (H I ⟨hI1, fun {x y} hxy => or_iff_not_imp_right.mpr fun hy => _⟩)
  obtain ⟨a, ha⟩ : (I ⊔ span {y}).IsPrincipal :=
    Imax' (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))
  -- Then `x ∈ I.colon (span {y})`, which is equal to `I` if it's not principal.
  suffices He : ¬(I.colon (span {y})).IsPrincipal
  · rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
        Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]
    exact Ideal.mem_colon_singleton.2 hxy
  -- So suppose for the sake of contradiction that both `I ⊔ span {y}` and `I.colon (span {y})`
  -- are principal.
  rintro ⟨b, hb⟩
  -- We will show `I` is generated by `a * b`.
  refine (nonPrincipals_def _).1 Ibad ⟨a * b, ?_⟩
  refine'
    le_antisymm (α := Ideal R) (fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _
  · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
    have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
    erw [ha, mem_span_singleton'] at hisup this
    obtain ⟨v, rfl⟩ := this
    obtain ⟨u, rfl⟩ := hisup
    have hucolon : u ∈ I.colon (span {v * a}) := by
      rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
      exact mul_mem_right _ _ hi
    erw [hb, mem_span_singleton'] at hucolon
    obtain ⟨z, rfl⟩ := hucolon
    

Proof state:
case He.mk.intro.refine'_1.intro.intro.intro
R : Type u
M : Type v
inst✝ : CommRing R
H : ∀ (P : Ideal R), IsPrime P → IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
Ibad : I ∈ nonPrincipals R
Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I
Imax' : ∀ {J : Ideal R}, I < J → IsPrincipal J
hI1 : ¬I = ⊤
x a b v : R
hxy : x * (v * a) ∈ I
hy : v * a ∉ I
ha : I ⊔ Ideal.span {v * a} = Submodule.span R {a}
hb : colon I (Ideal.span {v * a}) = Submodule.span R {b}
z : R
hi : z * b * a ∈ I
⊢ z * b * a ∈ Submodule.span R {a * b}

Next tactic:
exact mem_span_singleton'.2 ⟨z, by ring⟩
/-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
    IsPrincipalIdealRing R := by
  -- Suppose the set of `nonPrincipals` is not empty.
  rw [← nonPrincipals_eq_empty_iff, Set.eq_empty_iff_forall_not_mem]
  intro J hJ
  -- We will show a maximal element `I ∈ nonPrincipals R` (which exists by Zorn) is prime.
  obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partialOrder₀ (nonPrincipals R) nonPrincipals_zorn _ hJ
  have Imax' : ∀ {J}, I < J → J.IsPrincipal := by
    intro J hJ
    by_contra He
    exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm
  by_cases hI1 : I = ⊤
  · subst hI1
    exact Ibad top_isPrincipal
  -- Let `x y : R` with `x * y ∈ I` and suppose WLOG `y ∉ I`.
  refine' Ibad (H I ⟨hI1, fun {x y} hxy => or_iff_not_imp_right.mpr fun hy => _⟩)
  obtain ⟨a, ha⟩ : (I ⊔ span {y}).IsPrincipal :=
    Imax' (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))
  -- Then `x ∈ I.colon (span {y})`, which is equal to `I` if it's not principal.
  suffices He : ¬(I.colon (span {y})).IsPrincipal
  · rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
        Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]
    exact Ideal.mem_colon_singleton.2 hxy
  -- So suppose for the sake of contradiction that both `I ⊔ span {y}` and `I.colon (span {y})`
  -- are principal.
  rintro ⟨b, hb⟩
  -- We will show `I` is generated by `a * b`.
  refine (nonPrincipals_def _).1 Ibad ⟨a * b, ?_⟩
  refine'
    le_antisymm (α := Ideal R) (fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _
  · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
    have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
    erw [ha, mem_span_singleton'] at hisup this
    obtain ⟨v, rfl⟩ := this
    obtain ⟨u, rfl⟩ := hisup
    have hucolon : u ∈ I.colon (span {v * a}) := by
      rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
      exact mul_mem_right _ _ hi
    erw [hb, mem_span_singleton'] at hucolon
    obtain ⟨z, rfl⟩ := hucolon
    exact mem_span_singleton'.2 ⟨z, by 

Proof state:
R : Type u
M : Type v
inst✝ : CommRing R
H : ∀ (P : Ideal R), IsPrime P → IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
Ibad : I ∈ nonPrincipals R
Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I
Imax' : ∀ {J : Ideal R}, I < J → IsPrincipal J
hI1 : ¬I = ⊤
x a b v : R
hxy : x * (v * a) ∈ I
hy : v * a ∉ I
ha : I ⊔ Ideal.span {v * a} = Submodule.span R {a}
hb : colon I (Ideal.span {v * a}) = Submodule.span R {b}
z : R
hi : z * b * a ∈ I
⊢ z * (a * b) = z * b * a

Next tactic:
ring
/-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
    IsPrincipalIdealRing R := by
  -- Suppose the set of `nonPrincipals` is not empty.
  rw [← nonPrincipals_eq_empty_iff, Set.eq_empty_iff_forall_not_mem]
  intro J hJ
  -- We will show a maximal element `I ∈ nonPrincipals R` (which exists by Zorn) is prime.
  obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partialOrder₀ (nonPrincipals R) nonPrincipals_zorn _ hJ
  have Imax' : ∀ {J}, I < J → J.IsPrincipal := by
    intro J hJ
    by_contra He
    exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm
  by_cases hI1 : I = ⊤
  · subst hI1
    exact Ibad top_isPrincipal
  -- Let `x y : R` with `x * y ∈ I` and suppose WLOG `y ∉ I`.
  refine' Ibad (H I ⟨hI1, fun {x y} hxy => or_iff_not_imp_right.mpr fun hy => _⟩)
  obtain ⟨a, ha⟩ : (I ⊔ span {y}).IsPrincipal :=
    Imax' (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))
  -- Then `x ∈ I.colon (span {y})`, which is equal to `I` if it's not principal.
  suffices He : ¬(I.colon (span {y})).IsPrincipal
  · rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
        Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]
    exact Ideal.mem_colon_singleton.2 hxy
  -- So suppose for the sake of contradiction that both `I ⊔ span {y}` and `I.colon (span {y})`
  -- are principal.
  rintro ⟨b, hb⟩
  -- We will show `I` is generated by `a * b`.
  refine (nonPrincipals_def _).1 Ibad ⟨a * b, ?_⟩
  refine'
    le_antisymm (α := Ideal R) (fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _
  · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
    have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
    erw [ha, mem_span_singleton'] at hisup this
    obtain ⟨v, rfl⟩ := this
    obtain ⟨u, rfl⟩ := hisup
    have hucolon : u ∈ I.colon (span {v * a}) := by
      rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
      exact mul_mem_right _ _ hi
    erw [hb, mem_span_singleton'] at hucolon
    obtain ⟨z, rfl⟩ := hucolon
    exact mem_span_singleton'.2 ⟨z, by ring⟩
  · 

Proof state:
case He.mk.intro.refine'_2
R : Type u
M : Type v
inst✝ : CommRing R
H : ∀ (P : Ideal R), IsPrime P → IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
Ibad : I ∈ nonPrincipals R
Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I
Imax' : ∀ {J : Ideal R}, I < J → IsPrincipal J
hI1 : ¬I = ⊤
x y : R
hxy : x * y ∈ I
hy : y ∉ I
a : R
ha : I ⊔ Ideal.span {y} = Submodule.span R {a}
b : R
hb : colon I (Ideal.span {y}) = Submodule.span R {b}
⊢ Ideal.span {a} * Ideal.span {b} ≤ I

Next tactic:
rw [← Ideal.submodule_span_eq, ← ha, Ideal.sup_mul, sup_le_iff,
      span_singleton_mul_span_singleton, mul_comm y, Ideal.span_singleton_le_iff_mem]