awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.IsSubDPIdeal J
hK : hI.IsSubDPIdeal K
⊢ hI.IsSubDPIdeal (J ⊔ K)
have rw : ∀ {n : ℕ},
  n ≠ 0 →
    ∀ s ∈ (↑J : Set A) ∪ (↑K : Set A), Membership.mem (γ := Ideal A) (span ((↑J : Set A) ∪ (↑K : Set A))) (hI.dpow n s)
Tactic: rw [← J.span_eq, ← K.span_eq, ← span_union,
    span_isSubDPIdeal_iff (Set.union_subset_iff.mpr ⟨hJ.1, hK.1⟩)]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.IsSubDPIdeal J
hK : hI.IsSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ (↑J : Set A) ∪ (↑K : Set A)
⊢ Membership.mem (γ := Ideal A) (span ((↑J : Set A) ∪ (↑K : Set A))) (hI.dpow n a)
have inl : ∀ (ha : a ∈ (↑J : Set A)), Membership.mem (γ := Ideal A) (span ((↑J : Set A) ∪ (↑K : Set A))) (hI.dpow n a)
Tactic: rcases ha with ha | ha
Cleared hypotheses: ha
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.IsSubDPIdeal J
hK : hI.IsSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ (↑J : Set A) ∪ (↑K : Set A)
inl : ∀ (ha : a ∈ (↑J : Set A)), Membership.mem (γ := Ideal A) (span ((↑J : Set A) ∪ (↑K : Set A))) (hI.dpow n a)
⊢ Membership.mem (γ := Ideal A) (span ((↑J : Set A) ∪ (↑K : Set A))) (hI.dpow n a)
have inr : ∀ (ha : a ∈ (↑K : Set A)), Membership.mem (γ := Ideal A) (span ((↑J : Set A) ∪ (↑K : Set A))) (hI.dpow n a)
Tactic: rcases ha with ha | ha
Cleared hypotheses: ha
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_3
J : ι → Ideal A
hJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)
⊢ hI.IsSubDPIdeal (iSup J)
have rw : ∀ {n : ℕ},
  n ≠ 0 →
    ∀ (s : A),
      Membership.mem (γ := Set A) (⋃ (i : ι), (↑(J i) : Set A)) s →
        Membership.mem (γ := Ideal A) (span (⋃ (i : ι), (↑(J i) : Set A))) (hI.dpow n s)
Tactic: rw [iSup_eq_span, span_isSubDPIdeal_iff (Set.iUnion_subset_iff.mpr <| fun i ↦ (hJ i).1)]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_3
J : ι → Ideal A
hJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)
⊢ ∀ {n : ℕ},
    n ≠ 0 →
      ∀ (s : A),
        Membership.mem (γ := Set A) (⋃ (i : ι), (↑(J i) : Set A)) s →
          Membership.mem (γ := Ideal A) (span (⋃ (i : ι), (↑(J i) : Set A))) (hI.dpow n s)
have simp_rw : ∀ {n : ℕ},
  n ≠ 0 →
    ∀ (s : A),
      (∃ (i : ι), s ∈ (↑(J i) : Set A)) →
        Membership.mem (γ := Ideal A) (span (⋃ (i : ι), (↑(J i) : Set A))) (hI.dpow n s)
Tactic: simp_rw [Set.mem_iUnion]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_3
J : ι → Ideal A
hJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)
h : Nonempty ι
⊢ hI.IsSubDPIdeal (I ⊓ ⨅ (i : ι), J i)
have inr : ∀ (n : ℕ), n ≠ 0 → ∀ {j : A}, j ∈ I ⊓ ⨅ (i : ι), J i → hI.dpow n j ∈ I ⊓ ⨅ (i : ι), J i
Tactic: refine ⟨fun _ hx ↦ hx.1, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_3
J : ι → Ideal A
hJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)
h : Nonempty ι
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ⊓ ⨅ (i : ι), J i
⊢ hI.dpow n x ∈ I ⊓ ⨅ (i : ι), J i
have inr : ∀ (hx : x ∈ I ∧ ∀ (i : ι), x ∈ J i), hI.dpow n x ∈ I ∧ ∀ (i : ι), hI.dpow n x ∈ J i
Tactic: simp only [Ideal.mem_inf, mem_iInf] at hx ⊢
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_3
J : ι → Ideal A
hJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)
h : IsEmpty ι
⊢ hI.IsSubDPIdeal (I ⊓ ⨅ (i : ι), J i)
have inl : hI.IsSubDPIdeal I
Tactic: simp only [iInf_of_empty, le_top, inf_of_le_left]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (hypothesis)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_3
J : ι → Ideal A
hJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)
⊢ hI.IsSubDPIdeal (I ⊓ ⨅ (i : ι), J i)
have x : IsEmpty ι ∨ Nonempty ι
Tactic: cases isEmpty_or_nonempty ι with
  | inr _ =>
    refine ⟨fun _ hx ↦ hx.1, ?_⟩
    intro n hn x hx
    simp only [Ideal.mem_inf, mem_iInf] at hx ⊢
    exact ⟨hI.dpow_mem hn hx.1, fun i ↦  IsSubDPIdeal.dpow_mem (hJ i) n hn (hx.2 i)⟩
  | inl _ =>
    simp only [iInf_of_empty, le_top, inf_of_le_left]
    exact IsSubDPIdeal.self hI
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
inst_1 : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.IsDPMorphism hJ f
K : Ideal A
hK : hI.IsSubDPIdeal K
⊢ hJ.IsSubDPIdeal (Ideal.map f K)
have rw : ∀ {n : ℕ},
  n ≠ 0 →
    ∀ (s : B),
      Membership.mem (γ := Set B) ((⇑f : (a : A) → B) '' (↑K : Set A)) s →
        Membership.mem (γ := Ideal B) (span ((⇑f : (a : A) → B) '' (↑K : Set A))) (hJ.dpow n s)
Tactic: rw [Ideal.map, span_isSubDPIdeal_iff]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
inst_1 : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.IsDPMorphism hJ f
K : Ideal A
hK : hI.IsSubDPIdeal K
rw :
  ∀ {n : ℕ},
    n ≠ 0 →
      ∀ (s : B),
        Membership.mem (γ := Set B) ((⇑f : (a : A) → B) '' (↑K : Set A)) s →
          Membership.mem (γ := Ideal B) (span ((⇑f : (a : A) → B) '' (↑K : Set A))) (hJ.dpow n s)
⊢ hJ.IsSubDPIdeal (Ideal.map f K)
have rw₁ : (⇑f : (a : A) → B) '' (↑K : Set A) ⊆ (↑J : Set B)
Tactic: rw [Ideal.map, span_isSubDPIdeal_iff]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (hypothesis)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
p q : hI.SubDPIdeal
h :
  Eq.{u_1 + 1} (α := Set A) ((fun (s : hI.SubDPIdeal) ↦ (↑s.carrier : Set A)) p)
    ((fun (s : hI.SubDPIdeal) ↦ (↑s.carrier : Set A)) q)
⊢ p = q
have h₁ : p.carrier = q.carrier
Tactic: rw [SetLike.coe_set_eq] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
p q : hI.SubDPIdeal
h : p.carrier = q.carrier
⊢ p = q
have mk : ∀ (carrier : Ideal A) (isSubideal : carrier ≤ I) (dpow_mem : ∀ (n : ℕ), n ≠ 0 → ∀ j ∈ carrier, hI.dpow n j ∈ carrier)
  (h : { carrier := carrier, isSubideal := isSubideal, dpow_mem := dpow_mem : hI.SubDPIdeal }.carrier = q.carrier),
  { carrier := carrier, isSubideal := isSubideal, dpow_mem := dpow_mem : hI.SubDPIdeal } = q
Tactic: cases p
Cleared hypotheses: p, h
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
q : hI.SubDPIdeal
carrier : Ideal A
isSubideal : carrier ≤ I
dpow_mem : ∀ (n : ℕ), n ≠ 0 → ∀ j ∈ carrier, hI.dpow n j ∈ carrier
h : { carrier := carrier, isSubideal := isSubideal, dpow_mem := dpow_mem : hI.SubDPIdeal }.carrier = q.carrier
⊢ { carrier := carrier, isSubideal := isSubideal, dpow_mem := dpow_mem : hI.SubDPIdeal } = q
have mk : ∀ (carrier_1 : Ideal A) (isSubideal_1 : carrier_1 ≤ I)
  (dpow_mem_1 : ∀ (n : ℕ), n ≠ 0 → ∀ j ∈ carrier_1, hI.dpow n j ∈ carrier_1)
  (h :
    { carrier := carrier, isSubideal := isSubideal, dpow_mem := dpow_mem : hI.SubDPIdeal }.carrier =
      { carrier := carrier_1, isSubideal := isSubideal_1, dpow_mem := dpow_mem_1 : hI.SubDPIdeal }.carrier),
  { carrier := carrier, isSubideal := isSubideal, dpow_mem := dpow_mem : hI.SubDPIdeal } =
    { carrier := carrier_1, isSubideal := isSubideal_1, dpow_mem := dpow_mem_1 }
Tactic: cases q
Cleared hypotheses: q, h
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
x a : A
ha : a ∈ I
b : A
hb : b ∈ J
m : ℕ
hm : m ≠ 0
⊢ hI.dpow m (a • b) ∈ I • J
have smul : b ^ m * hI.dpow m a ∈ J * I
Tactic: rw [smul_eq_mul, smul_eq_mul, mul_comm a b, hI.dpow_mul ha, mul_comm]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
x_1 x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ (m : ℕ) (hm : m ≠ 0), hI.dpow m x ∈ I • J
hy' : ∀ (m : ℕ) (hm : m ≠ 0), hI.dpow m y ∈ I • J
m : ℕ
hm : m ≠ 0
⊢ hI.dpow m (x + y) ∈ I • J
have add : ∑ k ∈ Finset.range (m + 1), hI.dpow k x * hI.dpow (m - k) y ∈ I • J
Tactic: rw [hI.dpow_add' (mul_le_right hx) (mul_le_right hy)]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
x_1 x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ (m : ℕ) (hm : m ≠ 0), hI.dpow m x ∈ I • J
hy' : ∀ (m : ℕ) (hm : m ≠ 0), hI.dpow m y ∈ I • J
m : ℕ
hm : m ≠ 0
⊢ ∑ k ∈ Finset.range (m + 1), hI.dpow k x * hI.dpow (m - k) y ∈ I • J
have add : ∀ c ∈ Finset.range (m + 1), hI.dpow c x * hI.dpow (m - c) y ∈ I • J
Tactic: apply Submodule.sum_mem (I • J)
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
x_1 x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ (m : ℕ) (hm : m ≠ 0), hI.dpow m x ∈ I • J
hy' : ∀ (m : ℕ) (hm : m ≠ 0), hI.dpow m y ∈ I • J
m : ℕ
hm : m ≠ 0
k : ℕ
h : k ∈ Finset.range (m + 1)
⊢ hI.dpow k x * hI.dpow (m - k) y ∈ I • J
have pos : ∀ (hk0 : k = 0), hI.dpow k x * hI.dpow (m - k) y ∈ I • J
Tactic: by_cases hk0 : k = 0
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
x_1 x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ (m : ℕ) (hm : m ≠ 0), hI.dpow m x ∈ I • J
hy' : ∀ (m : ℕ) (hm : m ≠ 0), hI.dpow m y ∈ I • J
m : ℕ
hm : m ≠ 0
k : ℕ
h : k ∈ Finset.range (m + 1)
pos : ∀ (hk0 : k = 0), hI.dpow k x * hI.dpow (m - k) y ∈ I • J
⊢ hI.dpow k x * hI.dpow (m - k) y ∈ I • J
have neg : ∀ (hk0 : ¬k = 0), hI.dpow k x * hI.dpow (m - k) y ∈ I • J
Tactic: by_cases hk0 : k = 0
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
m : ℕ
hm : m ≠ 0
x : A
hx : x ∈ I • J
⊢ hI.dpow m x ∈ I • J
have smul : ∀ (a : A) (ha : a ∈ I) (b : A) (hb : b ∈ J) (m : ℕ) (hm : m ≠ 0), hI.dpow m (a • b) ∈ I • J
Tactic: induction hx using Submodule.smul_induction_on' generalizing m with
    | smul a ha b hb =>
      rw [smul_eq_mul, smul_eq_mul, mul_comm a b, hI.dpow_mul ha, mul_comm]
      exact Submodule.mul_mem_mul (J.pow_mem_of_mem hb m (zero_lt_iff.mpr hm))
        (hI.dpow_mem hm ha)
    | add x hx y hy hx' hy' =>
      rw [hI.dpow_add' (mul_le_right hx) (mul_le_right hy)]
      apply Submodule.sum_mem (I • J)
      intro k _
      by_cases hk0 : k = 0
      · exact hk0 ▸ mul_mem_left (I • J) _ (hy' _ hm)
      · exact mul_mem_right _ (I • J) (hx' k hk0)
Cleared hypotheses: m, hm, hx
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
m : ℕ
hm : m ≠ 0
x : A
hx : x ∈ I • J
smul : ∀ (a : A) (ha : a ∈ I) (b : A) (hb : b ∈ J) (m : ℕ) (hm : m ≠ 0), hI.dpow m (a • b) ∈ I • J
⊢ hI.dpow m x ∈ I • J
have add : ∀ (x : A) (hx : x ∈ I • J) (y : A) (hy : y ∈ I • J) (hx' : ∀ (m : ℕ) (hm : m ≠ 0), hI.dpow m x ∈ I • J)
  (hy' : ∀ (m : ℕ) (hm : m ≠ 0), hI.dpow m y ∈ I • J) (m : ℕ) (hm : m ≠ 0), hI.dpow m (x + y) ∈ I • J
Tactic: induction hx using Submodule.smul_induction_on' generalizing m with
    | smul a ha b hb =>
      rw [smul_eq_mul, smul_eq_mul, mul_comm a b, hI.dpow_mul ha, mul_comm]
      exact Submodule.mul_mem_mul (J.pow_mem_of_mem hb m (zero_lt_iff.mpr hm))
        (hI.dpow_mem hm ha)
    | add x hx y hy hx' hy' =>
      rw [hI.dpow_add' (mul_le_right hx) (mul_le_right hy)]
      apply Submodule.sum_mem (I • J)
      intro k _
      by_cases hk0 : k = 0
      · exact hk0 ▸ mul_mem_left (I • J) _ (hy' _ hm)
      · exact mul_mem_right _ (I • J) (hx' k hk0)
Cleared hypotheses: m, hm, hx
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
x_1 : ℕ
hn : x_1 ≠ 0
x : A
hx : Membership.mem (γ := Ideal A) ⊥ x
⊢ Membership.mem (γ := Ideal A) ⊥ (hI.dpow x_1 x)
have rw : Membership.mem (γ := Ideal A) ⊥ (hI.dpow x_1 0)
Tactic: rw [mem_bot.mp hx, hI.dpow_eval_zero hn, mem_bot]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
x_1 : ℕ
hn : x_1 ≠ 0
x : A
hx : Membership.mem (γ := Ideal A) ⊥ x
rw : Membership.mem (γ := Ideal A) ⊥ (hI.dpow x_1 0)
⊢ Membership.mem (γ := Ideal A) ⊥ (hI.dpow x_1 x)
have rw₁ : Membership.mem (γ := Ideal A) ⊥ 0
Tactic: rw [mem_bot.mp hx, hI.dpow_eval_zero hn, mem_bot]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
x_1 : ℕ
hn : x_1 ≠ 0
x : A
hx : Membership.mem (γ := Ideal A) ⊥ x
rw : Membership.mem (γ := Ideal A) ⊥ (hI.dpow x_1 0)
rw₁ : Membership.mem (γ := Ideal A) ⊥ 0
⊢ Membership.mem (γ := Ideal A) ⊥ (hI.dpow x_1 x)
have rw₂ : Eq.{u_1 + 1} (α := A) 0 0
Tactic: rw [mem_bot.mp hx, hI.dpow_eval_zero hn, mem_bot]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (hypothesis)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
x : A
hx : Membership.mem (γ := Submodule A A) (⨅ s ∈ insert ⊤ S, s.carrier) x
⊢ x ∈ I
have hx₁ : ∀ (i : hI.SubDPIdeal) (i_1 : i ∈ insert ⊤ S), x ∈ i.carrier
Tactic: simp only [mem_iInf] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
x_1 : ℕ
hn : x_1 ≠ 0
x : A
hx : Membership.mem (γ := Ideal A) (⨅ s ∈ insert ⊤ S, s.carrier) x
⊢ Membership.mem (γ := Ideal A) (⨅ s ∈ insert ⊤ S, s.carrier) (hI.dpow x_1 x)
have simp : ∀ (hx : ∀ (i : hI.SubDPIdeal) (i_1 : i ∈ insert ⊤ S), x ∈ i.carrier) (i : hI.SubDPIdeal) (i_1 : i ∈ insert ⊤ S),
  hI.dpow x_1 x ∈ i.carrier
Tactic: simp only [mem_iInf] at hx ⊢
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (hypothesis)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
a : A
J : Ideal A
haJ :
  Membership.mem (γ := Set A)
    ((fun (i : Ideal A) ↦
        (⋃ (_ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) i), (↑i : Set A) :
          Set A))
      J)
    a
⊢ a ∈ (↑I : Set A)
have haJ₁ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) J ∧ a ∈ J
Tactic: rw [Set.mem_iUnion, SetLike.mem_coe, exists_prop] at haJ
Cleared hypotheses: haJ
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
a : A
J : Ideal A
haJ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) J ∧ a ∈ J
⊢ a ∈ (↑I : Set A)
have obtain : ∀ (J' : hI.SubDPIdeal) (hJ' : J' ∈ S ∧ J'.carrier = J), a ∈ (↑I : Set A)
Tactic: obtain ⟨J', hJ'⟩ := (Set.mem_image _ _ _).mp haJ.1
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (hypothesis)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
⊢ hI.IsSubDPIdeal (sSup ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S))
have h : ⋃ (i : Ideal A),
    ⋃ (_ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) i), (↑i : Set A) ⊆
  (↑I : Set A)
Tactic: have h : (⋃ (i : Ideal A) (_ : i ∈ (fun J ↦ J.carrier) '' S), ↑i) ⊆ (I : Set A) := by
        rintro a ⟨-, ⟨J, rfl⟩, haJ⟩
        rw [Set.mem_iUnion, SetLike.mem_coe, exists_prop] at haJ
        obtain ⟨J', hJ'⟩ := (Set.mem_image _ _ _).mp haJ.1
        exact  J'.isSubideal  (hJ'.2 ▸ haJ.2)
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
h :
  ⋃ (i : Ideal A),
      ⋃ (_ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) i), (↑i : Set A) ⊆
    (↑I : Set A)
⊢ hI.IsSubDPIdeal (sSup ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S))
have rw : ∀ {n : ℕ},
  n ≠ 0 →
    ∀ (s : A),
      Membership.mem (γ := Set A)
          (⋃ (i : Ideal A),
            ⋃ (_ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) i), (↑i : Set A))
          s →
        Membership.mem (γ := Ideal A)
          (span
            (⋃ (i : Ideal A),
              ⋃ (_ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) i), (↑i : Set A)))
          (hI.dpow n s)
Tactic: rw [sSup_eq_iSup, Submodule.iSup_eq_span', submodule_span_eq, span_isSubDPIdeal_iff h]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
h :
  ⋃ (i : Ideal A),
      ⋃ (_ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) i), (↑i : Set A) ⊆
    (↑I : Set A)
n : ℕ
hn : n ≠ 0
x : A
J : Ideal A
hJ' : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) J
h' :
  Membership.mem (γ := Set A)
    ((fun (x : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) J) ↦ (↑J : Set A)) hJ')
    x
⊢ Membership.mem (γ := Ideal A)
    (span
      (⋃ (i : Ideal A),
        ⋃ (_ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) i), (↑i : Set A)))
    (hI.dpow n x)
have a : Membership.mem (γ := Set A)
  (⋃ (i : Ideal A),
    ⋃ (_ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) i), (↑i : Set A))
  (hI.dpow n x)
Tactic: apply subset_span
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
h :
  ⋃ (i : Ideal A),
      ⋃ (_ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) i), (↑i : Set A) ⊆
    (↑I : Set A)
n : ℕ
hn : n ≠ 0
x : A
J : Ideal A
hJ' : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) J
h' :
  Membership.mem (γ := Set A)
    ((fun (x : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) J) ↦ (↑J : Set A)) hJ')
    x
⊢ Membership.mem (γ := Set A)
    (⋃ (i : Ideal A),
      ⋃ (_ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) i), (↑i : Set A))
    (hI.dpow n x)
have a : hI.dpow n x ∈ (↑J : Set A)
Tactic: apply Set.mem_biUnion hJ'
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
h :
  ⋃ (i : Ideal A),
      ⋃ (_ : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) i), (↑i : Set A) ⊆
    (↑I : Set A)
n : ℕ
hn : n ≠ 0
x : A
J : Ideal A
hJ' : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) J
h' :
  Membership.mem (γ := Set A)
    ((fun (x : Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S) J) ↦ (↑J : Set A)) hJ')
    x
⊢ hI.dpow n x ∈ (↑J : Set A)
have a : ∀ (K : hI.SubDPIdeal) (hKS : K ∈ S)
  (h' :
    Membership.mem (γ := Set A)
      ((fun
            (x :
              Membership.mem (γ := Set (Ideal A)) ((fun (J : hI.SubDPIdeal) ↦ J.carrier) '' S)
                ((fun (J : hI.SubDPIdeal) ↦ J.carrier) K)) ↦
          (↑((fun (J : hI.SubDPIdeal) ↦ J.carrier) K) : Set A))
        sorry)
      x),
  hI.dpow n x ∈ (↑((fun (J : hI.SubDPIdeal) ↦ J.carrier) K) : Set A)
Tactic: obtain ⟨K, hKS, rfl⟩ := hJ'
Cleared hypotheses: J, hJ', h'
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
⊢ CompleteLattice hI.SubDPIdeal
have refine_1 : ∀ (S : Set hI.SubDPIdeal),
  Eq.{u_1 + 1} (α := { x : Ideal A // x ∈ Set.Iic I })
    ((fun (J : hI.SubDPIdeal) ↦
        (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry : { x : Ideal A // x ∈ Set.Iic I }))
      (sSup S))
    (⨆ a ∈ S,
      (fun (J : hI.SubDPIdeal) ↦
          (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry : { x : Ideal A // x ∈ Set.Iic I }))
        a)
Tactic: refine Function.Injective.completeLattice (fun J : SubDPIdeal hI ↦ (J : Set.Iic I))
    (fun J J' h ↦ by simpa only [SubDPIdeal.ext_iff, Subtype.mk.injEq] using h)
    .rfl .rfl (fun J J' ↦ by rfl) (fun J J' ↦ by rfl) (fun S ↦ ?_) (fun S ↦ ?_) rfl rfl
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
refine_1 :
  ∀ (S : Set hI.SubDPIdeal),
    Eq.{u_1 + 1} (α := { x : Ideal A // x ∈ Set.Iic I })
      ((fun (J : hI.SubDPIdeal) ↦
          (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯ : { x : Ideal A // x ∈ Set.Iic I }))
        (sSup S))
      (⨆ a ∈ S,
        (fun (J : hI.SubDPIdeal) ↦
            (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯ : { x : Ideal A // x ∈ Set.Iic I }))
          a)
⊢ CompleteLattice hI.SubDPIdeal
have refine_2 : ∀ (S : Set hI.SubDPIdeal),
  Eq.{u_1 + 1} (α := { x : Ideal A // x ∈ Set.Iic I })
    ((fun (J : hI.SubDPIdeal) ↦
        (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry : { x : Ideal A // x ∈ Set.Iic I }))
      (sInf S))
    (⨅ a ∈ S,
      (fun (J : hI.SubDPIdeal) ↦
          (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry : { x : Ideal A // x ∈ Set.Iic I }))
        a)
Tactic: refine Function.Injective.completeLattice (fun J : SubDPIdeal hI ↦ (J : Set.Iic I))
    (fun J J' h ↦ by simpa only [SubDPIdeal.ext_iff, Subtype.mk.injEq] using h)
    .rfl .rfl (fun J J' ↦ by rfl) (fun J J' ↦ by rfl) (fun S ↦ ?_) (fun S ↦ ?_) rfl rfl
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
⊢ Eq.{u_1 + 1} (α := { x : Ideal A // x ∈ Set.Iic I })
    ((fun (J : hI.SubDPIdeal) ↦
        (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯ : { x : Ideal A // x ∈ Set.Iic I }))
      (sSup S))
    (sSup
      (Set.range fun (a : hI.SubDPIdeal) ↦
        ⨆ (_ : a ∈ S),
          (fun (J : hI.SubDPIdeal) ↦
              (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯ : { x : Ideal A // x ∈ Set.Iic I }))
            a))
have refine_1 : Eq.{u_1 + 1} (α := Ideal A)
  (↑((fun (J : hI.SubDPIdeal) ↦
          (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry : { x : Ideal A // x ∈ Set.Iic I }))
        (sSup S)) :
    Ideal A)
  (sSup
    (Set.image (α := (↑(Set.Iic I) : Type u_1)) (val (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I))
      (Set.range fun (a : hI.SubDPIdeal) ↦
        ⨆ (_ : a ∈ S),
          (fun (J : hI.SubDPIdeal) ↦
              (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry : { x : Ideal A // x ∈ Set.Iic I }))
            a)))
Tactic: rw [Subtype.ext_iff, Set.Iic.coe_sSup]