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
S : Set A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
x y : A
hx :
  Membership.mem (γ := Submodule A A)
    (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) x
hy :
  Membership.mem (γ := Submodule A A)
    (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) y
hx_pow :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n x)
hy_pow :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n y)
n : ℕ
hn : n ≠ 0
m : ℕ
h : m ∈ Finset.range (n + 1)
hm0 : m = 0
⊢ Membership.mem (γ := Submodule A A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (hI.dpow m x * hI.dpow (n - m) y)
have pos : Membership.mem (γ := Submodule A A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
  (hI.dpow 0 x * hI.dpow (n - 0) y)
Tactic: rw [hm0]
-- 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 A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
a x : A
hx :
  Membership.mem (γ := Submodule A A)
    (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) x
hx_pow :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n x)
n : ℕ
hn : n ≠ 0
⊢ Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (hI.dpow n (a • x))
have refine_4 : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
  (a ^ n * hI.dpow n x)
Tactic: rw [smul_eq_mul, hI.dpow_mul (hSI 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 A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
⊢ Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (hI.dpow k z)
have haux : ∀ (n : ℕ),
  n ≠ 0 →
    Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
      (hI.dpow n z)
Tactic: have haux : ∀ (n : ℕ) (_ : n ≠ 0),
      hI.dpow n z ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} := by
    refine Submodule.span_induction ?_ ?_ ?_ ?_ hz
    · -- Elements of S
      rintro y ⟨m, hm, x, hxS, hxy⟩ n hn
      rw [hxy, hI.dpow_comp hm (hS hxS)]
      exact mul_mem_left _ _ (subset_span ⟨n * m, mul_ne_zero hn hm, x, hxS, rfl⟩)
    · -- Zero
      exact fun _ hn ↦ by simp only [hI.dpow_eval_zero hn, zero_mem]
    · intro x y hx hy hx_pow hy_pow n hn
      rw [hI.dpow_add' (hSI hx) (hSI hy)]
      apply Submodule.sum_mem (span _)
      intro m _
      by_cases hm0 : m = 0
      · rw [hm0]; exact (span _).mul_mem_left _ (hy_pow n hn)
      · exact (span _).mul_mem_right _ (hx_pow m hm0)
    · intro a x hx hx_pow n hn
      rw [smul_eq_mul, hI.dpow_mul (hSI hx)]
      exact mul_mem_left (span _) (a ^ n) (hx_pow n hn)
-- 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 A
hS : S ⊆ (↑I : Set A)
J : hI.SubDPIdeal :=
  { carrier := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯,
    dpow_mem := ⋯ }
⊢ Eq.{u_1 + 1} (α := Ideal A)
    (⨅ (s : hI.SubDPIdeal),
      ⨅ (_ : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) s),
        s.carrier)
    (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
have a : LE.le.{u_1} (α := Ideal A)
  (⨅ (s : hI.SubDPIdeal),
    ⨅ (_ : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) s),
      s.carrier)
  (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
Tactic: apply le_antisymm
-- 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 A
hS : S ⊆ (↑I : Set A)
J : hI.SubDPIdeal :=
  { carrier := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯,
    dpow_mem := ⋯ }
a :
  LE.le.{u_1} (α := Ideal A)
    (⨅ (s : hI.SubDPIdeal),
      ⨅ (_ : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) s),
        s.carrier)
    (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
⊢ Eq.{u_1 + 1} (α := Ideal A)
    (⨅ (s : hI.SubDPIdeal),
      ⨅ (_ : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) s),
        s.carrier)
    (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
have a₁ : LE.le.{u_1} (α := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
  (⨅ (s : hI.SubDPIdeal),
    ⨅ (_ : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) s),
      s.carrier)
Tactic: apply le_antisymm
-- 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 A
hS : S ⊆ (↑I : Set A)
J : hI.SubDPIdeal :=
  { carrier := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯,
    dpow_mem := ⋯ }
x : A
hx : x ∈ S
⊢ x = hI.dpow 1 x
have rw : x = x
Tactic: rw [hI.dpow_one (hS 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 A
hS : S ⊆ (↑I : Set A)
J : hI.SubDPIdeal :=
  { carrier := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯,
    dpow_mem := ⋯ }
⊢ LE.le.{u_1} (α := Ideal A)
    (⨅ (s : hI.SubDPIdeal),
      ⨅ (_ : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) s),
        s.carrier)
    (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
have h : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) J
Tactic: have h : J ∈ insert ⊤ {J : hI.SubDPIdeal | S ⊆ ↑J.carrier} :=
      Set.mem_insert_of_mem _
        (fun x hx ↦ subset_span ⟨1, one_ne_zero, x, hx, by rw [hI.dpow_one (hS 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 A
hS : S ⊆ (↑I : Set A)
J : hI.SubDPIdeal :=
  { carrier := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯,
    dpow_mem := ⋯ }
h : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) J
⊢ LE.le.{u_1} (α := Ideal A)
    (⨅ (s : hI.SubDPIdeal),
      ⨅ (_ : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) s),
        s.carrier)
    (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
have a : Eq.{u_1 + 1} (α := Ideal A)
  ((fun (s : hI.SubDPIdeal) ↦
      (⨅ (_ : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) s),
          s.carrier :
        Ideal A))
    J)
  (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
Tactic: refine sInf_le_of_le ⟨J, ?_⟩ (le_refl _)
-- 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 A
hS : S ⊆ (↑I : Set A)
J : hI.SubDPIdeal :=
  { carrier := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯,
    dpow_mem := ⋯ }
⊢ LE.le.{u_1} (α := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (⨅ (s : hI.SubDPIdeal),
      ⨅ (_ : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) s),
        s.carrier)
have a : ∀ (i : hI.SubDPIdeal)
  (j : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) i),
  span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ i.carrier
Tactic: rw [le_iInf₂_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
S : Set A
hS : S ⊆ (↑I : Set A)
J : hI.SubDPIdeal :=
  { carrier := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯,
    dpow_mem := ⋯ }
K : hI.SubDPIdeal
hK : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) K
⊢ S ≤ (↑K : Set A)
have hK₁ : K = ⊤ ∨ S ⊆ (↑K.carrier : Set A)
Tactic: simp only [Set.mem_insert_iff, Set.mem_setOf_eq] at hK
Cleared hypotheses: hK
-- 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 A
hS : S ⊆ (↑I : Set A)
J : hI.SubDPIdeal :=
  { carrier := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯,
    dpow_mem := ⋯ }
K : hI.SubDPIdeal
hK : K = ⊤ ∨ S ⊆ (↑K.carrier : Set A)
⊢ S ≤ (↑K : Set A)
have inl : S ≤ (↑⊤ : Set A)
Tactic: rcases hK with rfl | hKS
Cleared hypotheses: K, hK
-- 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 A
hS : S ⊆ (↑I : Set A)
J : hI.SubDPIdeal :=
  { carrier := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯,
    dpow_mem := ⋯ }
K : hI.SubDPIdeal
hK : K = ⊤ ∨ S ⊆ (↑K.carrier : Set A)
inl : S ≤ (↑⊤ : Set A)
⊢ S ≤ (↑K : Set A)
have inr : ∀ (hKS : S ⊆ (↑K.carrier : Set A)), S ≤ (↑K : Set A)
Tactic: rcases hK with rfl | hKS
Cleared hypotheses: hK
-- 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 A
hS : S ⊆ (↑I : Set A)
J : hI.SubDPIdeal :=
  { carrier := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯,
    dpow_mem := ⋯ }
K : hI.SubDPIdeal
hK : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) K
⊢ span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ K.carrier
have this : S ≤ (↑K : Set A)
Tactic: have : S ≤ K := by
      simp only [Set.mem_insert_iff, Set.mem_setOf_eq] at hK
      rcases hK with rfl | hKS
      exacts [hS, hKS]
-- 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 A
hS : S ⊆ (↑I : Set A)
J : hI.SubDPIdeal :=
  { carrier := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯,
    dpow_mem := ⋯ }
K : hI.SubDPIdeal
hK : Membership.mem (γ := Set hI.SubDPIdeal) (insert ⊤ {J : hI.SubDPIdeal | S ⊆ (↑J.carrier : Set A)}) K
this : S ≤ (↑K : Set A)
⊢ span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ K.carrier
have a : {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ⊆ (↑K.carrier : Set A)
Tactic: rw [span_le]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst_1 : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.IsDPMorphism hJ f
⊢ hI.IsSubDPIdeal (RingHom.ker f ⊓ I)
have rw : ∀ {n : ℕ} {a b : A},
  a ∈ I →
    b ∈ I →
      Membership.mem (γ := Ideal A) (RingHom.ker f) (a - b) →
        Membership.mem (γ := Ideal A) (RingHom.ker f) (hI.dpow n a - hI.dpow n b)
Tactic: rw [isSubDPIdeal_inf_iff]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (hypothesis)
Goal:
A : Type u_1
inst : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst_1 : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.IsDPMorphism hJ f
⊢ ∀ {n : ℕ} {a b : A},
    a ∈ I →
      b ∈ I →
        Membership.mem (γ := Ideal A) (RingHom.ker f) (a - b) →
          Membership.mem (γ := Ideal A) (RingHom.ker f) (hI.dpow n a - hI.dpow n b)
have hf₁ : Ideal.map f I ≤ J ∧ ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
Tactic: simp only [isDPMorphism_def] at hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst_1 : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
⊢ Membership.mem (γ := Ideal A) (RingHom.ker f) (a - b) →
    Membership.mem (γ := Ideal A) (RingHom.ker f) (hI.dpow n a - hI.dpow n b)
have simp : Eq.{u_2 + 1} (α := B) ((f : (a : A) → B) a : B) ((f : (a : A) → B) b : B) →
  hJ.dpow n ((f : (a : A) → B) a) = hJ.dpow n ((f : (a : A) → B) b)
Tactic: simp only [RingHom.sub_mem_ker_iff, ← hf.2 a ha, ← hf.2 b hb]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst_1 : CommRing B
J : Ideal B
hJ : DividedPowers J
f : hI.DPMorphism hJ
x : ℕ
hn : x ≠ 0
a : A
⊢ a ∈ RingHom.ker f.toRingHom ⊓ I → hI.dpow x a ∈ RingHom.ker f.toRingHom ⊓ I
have simp : Eq.{u_2 + 1} (α := B) ((f.toRingHom : (a : A) → B) a : B) (0 : B) →
  a ∈ I → Eq.{u_2 + 1} (α := B) ((f.toRingHom : (a : A) → B) (hI.dpow x a) : B) (0 : B) ∧ hI.dpow x a ∈ I
Tactic: simp only [mem_inf, and_imp, RingHom.mem_ker]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst_1 : CommRing B
J : Ideal B
hJ : DividedPowers J
f : hI.DPMorphism hJ
x : ℕ
hn : x ≠ 0
a : A
ha : Eq.{u_2 + 1} (α := B) ((f.toRingHom : (a : A) → B) a : B) (0 : B)
ha' : a ∈ I
⊢ Eq.{u_2 + 1} (α := B) ((f.toRingHom : (a : A) → B) (hI.dpow x a) : B) (0 : B) ∧ hI.dpow x a ∈ I
have rw : hJ.dpow x 0 = 0 ∧ hI.dpow x a ∈ I
Tactic: rw [← f.isDPMorphism.2 a ha', ha]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
a b : A
ha : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} a
hb : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} b
⊢ Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} (a + b)
have apply : ∀ (n : ℕ), hI.dpow n (a + b) = hI'.dpow n (a + b)
Tactic: apply And.intro (I.add_mem ha.1 hb.1) (fun n ↦ ?_)
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
a b : A
ha : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} a
hb : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} b
n : ℕ
⊢ hI.dpow n (a + b) = hI'.dpow n (a + b)
have rw : Eq.{u_1 + 1} (α := A) (∑ k ∈ @Finset.antidiagonal _ Nat.instAddMonoid _ n, hI.dpow k.1 a * hI.dpow k.2 b)
  (∑ k ∈ @Finset.antidiagonal _ Nat.instAddMonoid _ n, hI'.dpow k.1 a * hI'.dpow k.2 b)
Tactic: rw [hI.dpow_add ha.1 hb.1, hI'.dpow_add ha.1 hb.1]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
a b : A
ha : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} a
hb : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} b
n : ℕ
k : ℕ × ℕ
h : k ∈ @Finset.antidiagonal _ Nat.instAddMonoid _ n
⊢ hI.dpow k.1 a * hI.dpow k.2 b = hI'.dpow k.1 a * hI'.dpow k.2 b
have rw : hI'.dpow k.1 a * hI.dpow k.2 b = hI'.dpow k.1 a * hI'.dpow k.2 b
Tactic: rw [ha.2, hb.2]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
a b : A
ha : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} a
hb : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} b
n : ℕ
k : ℕ × ℕ
h : k ∈ @Finset.antidiagonal _ Nat.instAddMonoid _ n
rw : hI'.dpow k.1 a * hI.dpow k.2 b = hI'.dpow k.1 a * hI'.dpow k.2 b
⊢ hI.dpow k.1 a * hI.dpow k.2 b = hI'.dpow k.1 a * hI'.dpow k.2 b
have rw₁ : hI'.dpow k.1 a * hI'.dpow k.2 b = hI'.dpow k.1 a * hI'.dpow k.2 b
Tactic: rw [ha.2, hb.2]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} 0
have apply : ∀ (n : ℕ), hI.dpow n 0 = hI'.dpow n 0
Tactic: apply And.intro I.zero_mem (fun n ↦ ?_)
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
⊢ hI.dpow n 0 = hI'.dpow n 0
have pos : ∀ (hn : n = 0), hI.dpow n 0 = hI'.dpow n 0
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
pos : ∀ (hn : n = 0), hI.dpow n 0 = hI'.dpow n 0
⊢ hI.dpow n 0 = hI'.dpow n 0
have neg : ∀ (hn : ¬n = 0), hI.dpow n 0 = hI'.dpow n 0
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n = 0
⊢ hI.dpow n 0 = hI'.dpow n 0
have pos : hI.dpow 0 0 = hI'.dpow 0 0
Tactic: rw [hn, hI.dpow_zero (zero_mem I), hI'.dpow_zero (zero_mem I)]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n = 0
pos : hI.dpow 0 0 = hI'.dpow 0 0
⊢ hI.dpow n 0 = hI'.dpow n 0
have pos₁ : 1 = hI'.dpow 0 0
Tactic: rw [hn, hI.dpow_zero (zero_mem I), hI'.dpow_zero (zero_mem I)]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n = 0
pos : hI.dpow 0 0 = hI'.dpow 0 0
pos₁ : 1 = hI'.dpow 0 0
⊢ hI.dpow n 0 = hI'.dpow n 0
have pos₂ : Eq.{u_1 + 1} (α := A) 1 1
Tactic: rw [hn, hI.dpow_zero (zero_mem I), hI'.dpow_zero (zero_mem I)]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : ¬n = 0
⊢ hI.dpow n 0 = hI'.dpow n 0
have neg : 0 = hI'.dpow n 0
Tactic: rw [hI.dpow_eval_zero hn, hI'.dpow_eval_zero hn]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : ¬n = 0
neg : 0 = hI'.dpow n 0
⊢ hI.dpow n 0 = hI'.dpow n 0
have neg₁ : Eq.{u_1 + 1} (α := A) 0 0
Tactic: rw [hI.dpow_eval_zero hn, hI'.dpow_eval_zero hn]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
a x : A
hx : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} x
⊢ Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} (a • x)
have rw : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} (a * x)
Tactic: rw [smul_eq_mul]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
a x : A
hx : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} x
n : ℕ
⊢ hI.dpow n (a * x) = hI'.dpow n (a * x)
have rw : a ^ n * hI.dpow n x = hI'.dpow n (a * x)
Tactic: rw [hI.dpow_mul hx.1, hI'.dpow_mul hx.1, hx.2]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
a x : A
hx : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} x
n : ℕ
rw : a ^ n * hI.dpow n x = hI'.dpow n (a * x)
⊢ hI.dpow n (a * x) = hI'.dpow n (a * x)
have rw₁ : a ^ n * hI.dpow n x = a ^ n * hI'.dpow n x
Tactic: rw [hI.dpow_mul hx.1, hI'.dpow_mul hx.1, hx.2]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
a x : A
hx : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} x
n : ℕ
rw : a ^ n * hI.dpow n x = hI'.dpow n (a * x)
rw₁ : a ^ n * hI.dpow n x = a ^ n * hI'.dpow n x
⊢ hI.dpow n (a * x) = hI'.dpow n (a * x)
have rw₂ : a ^ n * hI'.dpow n x = a ^ n * hI'.dpow n x
Tactic: rw [hI.dpow_mul hx.1, hI'.dpow_mul hx.1, hx.2]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x_1 : ℕ
hn : x_1 ≠ 0
x : A
hx : x ∈ hI.dpEqualizer hI'
m : ℕ
⊢ hI.dpow m (hI.dpow x_1 x) = hI'.dpow m (hI.dpow x_1 x)
have rw : (↑(m.uniformBell x_1) : A) * hI.dpow (m * x_1) x = hI'.dpow m (hI.dpow x_1 x)
Tactic: rw [hI.dpow_comp hn hx.1, hx.2, hx.2, hI'.dpow_comp hn hx.1]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x_1 : ℕ
hn : x_1 ≠ 0
x : A
hx : x ∈ hI.dpEqualizer hI'
m : ℕ
rw : (↑(m.uniformBell x_1) : A) * hI.dpow (m * x_1) x = hI'.dpow m (hI.dpow x_1 x)
⊢ hI.dpow m (hI.dpow x_1 x) = hI'.dpow m (hI.dpow x_1 x)
have rw₁ : (↑(m.uniformBell x_1) : A) * hI'.dpow (m * x_1) x = hI'.dpow m (hI.dpow x_1 x)
Tactic: rw [hI.dpow_comp hn hx.1, hx.2, hx.2, hI'.dpow_comp hn hx.1]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x_1 : ℕ
hn : x_1 ≠ 0
x : A
hx : x ∈ hI.dpEqualizer hI'
m : ℕ
rw : (↑(m.uniformBell x_1) : A) * hI.dpow (m * x_1) x = hI'.dpow m (hI.dpow x_1 x)
rw₁ : (↑(m.uniformBell x_1) : A) * hI'.dpow (m * x_1) x = hI'.dpow m (hI.dpow x_1 x)
⊢ hI.dpow m (hI.dpow x_1 x) = hI'.dpow m (hI.dpow x_1 x)
have rw₂ : (↑(m.uniformBell x_1) : A) * hI'.dpow (m * x_1) x = hI'.dpow m (hI'.dpow x_1 x)
Tactic: rw [hI.dpow_comp hn hx.1, hx.2, hx.2, hI'.dpow_comp hn hx.1]