awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
inst_1 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
⊢ 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), n ≠ 0 → ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a))
have refine : ∀ (hIJ : map f I ≤ J)
  (H : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)) (n : ℕ),
  ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
Tactic: refine fun hIJ ↦ ⟨fun H n _ ↦ H, fun H n ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
inst_1 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hIJ : map f I ≤ J
H : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
n : ℕ
⊢ ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
have pos : ∀ (hn : n = 0), ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
inst_1 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hIJ : map f I ≤ J
H : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
n : ℕ
pos : ∀ (hn : n = 0), ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
⊢ ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
have neg : ∀ (hn : ¬n = 0), ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
inst_1 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hIJ : map f I ≤ J
H : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
n : ℕ
hn : n = 0
a : A
ha : a ∈ I
⊢ hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
have pos : hJ.dpow 0 ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow 0 a)
Tactic: rw [hn, hI.dpow_zero ha, hJ.dpow_zero (hIJ (mem_map_of_mem f ha)), map_one]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
inst_1 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hIJ : map f I ≤ J
H : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
n : ℕ
hn : n = 0
a : A
ha : a ∈ I
pos : hJ.dpow 0 ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow 0 a)
⊢ hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
have pos₁ : hJ.dpow 0 ((f : (a : A) → B) a) = (f : (a : A) → B) 1
Tactic: rw [hn, hI.dpow_zero ha, hJ.dpow_zero (hIJ (mem_map_of_mem f ha)), map_one]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
inst_1 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hIJ : map f I ≤ J
H : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
n : ℕ
hn : n = 0
a : A
ha : a ∈ I
pos : hJ.dpow 0 ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow 0 a)
pos₁ : hJ.dpow 0 ((f : (a : A) → B) a) = (f : (a : A) → B) 1
⊢ hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
have pos₂ : Eq.{u_2 + 1} (α := B) (1 : B) ((f : (a : A) → B) 1 : B)
Tactic: rw [hn, hI.dpow_zero ha, hJ.dpow_zero (hIJ (mem_map_of_mem f ha)), map_one]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
inst_1 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hIJ : map f I ≤ J
H : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
n : ℕ
hn : n = 0
a : A
ha : a ∈ I
pos : hJ.dpow 0 ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow 0 a)
pos₁ : hJ.dpow 0 ((f : (a : A) → B) a) = (f : (a : A) → B) 1
pos₂ : Eq.{u_2 + 1} (α := B) (1 : B) ((f : (a : A) → B) 1 : B)
⊢ hJ.dpow n ((f : (a : A) → B) a) = (f : (a : A) → B) (hI.dpow n a)
have pos₃ : Eq.{u_2 + 1} (α := B) 1 1
Tactic: rw [hn, hI.dpow_zero ha, hJ.dpow_zero (hIJ (mem_map_of_mem f ha)), map_one]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
inst_1 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
C : Type u_3
inst_2 : CommSemiring C
K : Ideal C
hK : DividedPowers K
f : A →+* B
g : B →+* C
hg : hJ.IsDPMorphism hK g
hf : hI.IsDPMorphism hJ f
⊢ hI.IsDPMorphism hK (g.comp f)
have refine : ∀ {n : ℕ} (a : A) (ha : a ∈ I), hK.dpow n ((g.comp f : (a : A) → C) a) = (g.comp f : (a : A) → C) (hI.dpow n a)
Tactic: refine ⟨le_trans (map_map f g ▸ map_mono hf.1) hg.1, fun a ha ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
inst_1 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
C : Type u_3
inst_2 : CommSemiring C
K : Ideal C
hK : DividedPowers K
f : A →+* B
g : B →+* C
hg : hJ.IsDPMorphism hK g
hf : hI.IsDPMorphism hJ f
n : ℕ
a : A
ha : a ∈ I
⊢ hK.dpow n ((g.comp f : (a : A) → C) a) = (g.comp f : (a : A) → C) (hI.dpow n a)
have simp : hK.dpow n ((g : (a : B) → C) ((f : (a : A) → B) a : B)) = (g : (a : B) → C) ((f : (a : A) → B) (hI.dpow n a) : B)
Tactic: simp only [RingHom.coe_comp, Function.comp_apply]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommSemiring A
inst_1 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
C : Type u_3
inst_2 : CommSemiring C
K : Ideal C
hK : DividedPowers K
f : A →+* B
g : B →+* C
hg : hJ.IsDPMorphism hK g
hf : hI.IsDPMorphism hJ f
n : ℕ
a : A
ha : a ∈ I
⊢ hK.dpow n ((g : (a : B) → C) ((f : (a : A) → B) a : B)) = (g : (a : B) → C) ((f : (a : A) → B) (hI.dpow n a) : B)
have a₁ : Membership.mem (α := B) J ((f : (a : A) → B) a : B)
Tactic: rw [← hf.2 a ha, hg.2]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
h h' : hI.DPMorphism hJ
hh' :
  Eq (α := (a : A) → B) ((fun (h : hI.DPMorphism hJ) ↦ ((⇑h.toRingHom : (a : A) → B) : (a : A) → B)) h)
    ((fun (h : hI.DPMorphism hJ) ↦ ((⇑h.toRingHom : (a : A) → B) : (a : A) → B)) h')
⊢ h = h'
have mk : ∀ (toRingHom : A →+* B) (ideal_comp : map toRingHom I ≤ J)
  (dpow_comp : ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n ((toRingHom : (a : A) → B) a) = (toRingHom : (a : A) → B) (hI.dpow n a))
  (hh' :
    Eq (α := (a : A) → B)
      ((fun (h : hI.DPMorphism hJ) ↦ ((⇑h.toRingHom : (a : A) → B) : (a : A) → B))
        { toRingHom := toRingHom, ideal_comp := ideal_comp, dpow_comp := @dpow_comp })
      ((fun (h : hI.DPMorphism hJ) ↦ ((⇑h.toRingHom : (a : A) → B) : (a : A) → B)) h')),
  { toRingHom := toRingHom, ideal_comp := ideal_comp, dpow_comp := @dpow_comp : hI.DPMorphism hJ } = h'
Tactic: cases h
Cleared hypotheses: h, hh'
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
h' : hI.DPMorphism hJ
toRingHom : A →+* B
ideal_comp : map toRingHom I ≤ J
dpow_comp : ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n ((toRingHom : (a : A) → B) a) = (toRingHom : (a : A) → B) (hI.dpow n a)
hh' :
  Eq (α := (a : A) → B)
    ((fun (h : hI.DPMorphism hJ) ↦ ((⇑h.toRingHom : (a : A) → B) : (a : A) → B))
      { toRingHom := toRingHom, ideal_comp := ideal_comp, dpow_comp := @dpow_comp })
    ((fun (h : hI.DPMorphism hJ) ↦ ((⇑h.toRingHom : (a : A) → B) : (a : A) → B)) h')
⊢ { toRingHom := toRingHom, ideal_comp := ideal_comp, dpow_comp := @dpow_comp : hI.DPMorphism hJ } = h'
have mk : ∀ (toRingHom_1 : A →+* B) (ideal_comp_1 : map toRingHom_1 I ≤ J)
  (dpow_comp_1 :
    ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n ((toRingHom_1 : (a : A) → B) a) = (toRingHom_1 : (a : A) → B) (hI.dpow n a))
  (hh' :
    Eq (α := (a : A) → B)
      ((fun (h : hI.DPMorphism hJ) ↦ ((⇑h.toRingHom : (a : A) → B) : (a : A) → B))
        { toRingHom := toRingHom, ideal_comp := ideal_comp, dpow_comp := @dpow_comp })
      ((fun (h : hI.DPMorphism hJ) ↦ ((⇑h.toRingHom : (a : A) → B) : (a : A) → B))
        { toRingHom := toRingHom_1, ideal_comp := ideal_comp_1, dpow_comp := @dpow_comp_1 })),
  { toRingHom := toRingHom, ideal_comp := ideal_comp, dpow_comp := @dpow_comp : hI.DPMorphism hJ } =
    { toRingHom := toRingHom_1, ideal_comp := ideal_comp_1, dpow_comp := @dpow_comp_1 }
Tactic: cases h'
Cleared hypotheses: h', hh'
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
toRingHom : A →+* B
ideal_comp : map toRingHom I ≤ J
dpow_comp : ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n ((toRingHom : (a : A) → B) a) = (toRingHom : (a : A) → B) (hI.dpow n a)
toRingHom_1 : A →+* B
ideal_comp_1 : map toRingHom_1 I ≤ J
dpow_comp_1 : ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n ((toRingHom_1 : (a : A) → B) a) = (toRingHom_1 : (a : A) → B) (hI.dpow n a)
hh' :
  Eq (α := (a : A) → B)
    ((fun (h : hI.DPMorphism hJ) ↦ ((⇑h.toRingHom : (a : A) → B) : (a : A) → B))
      { toRingHom := toRingHom, ideal_comp := ideal_comp, dpow_comp := @dpow_comp })
    ((fun (h : hI.DPMorphism hJ) ↦ ((⇑h.toRingHom : (a : A) → B) : (a : A) → B))
      { toRingHom := toRingHom_1, ideal_comp := ideal_comp_1, dpow_comp := @dpow_comp_1 })
⊢ { toRingHom := toRingHom, ideal_comp := ideal_comp, dpow_comp := @dpow_comp : hI.DPMorphism hJ } =
    { toRingHom := toRingHom_1, ideal_comp := ideal_comp_1, dpow_comp := @dpow_comp_1 }
have mk : toRingHom = toRingHom_1
Tactic: congr
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
toRingHom : A →+* B
ideal_comp : map toRingHom I ≤ J
dpow_comp : ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n ((toRingHom : (a : A) → B) a) = (toRingHom : (a : A) → B) (hI.dpow n a)
toRingHom_1 : A →+* B
ideal_comp_1 : map toRingHom_1 I ≤ J
dpow_comp_1 : ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n ((toRingHom_1 : (a : A) → B) a) = (toRingHom_1 : (a : A) → B) (hI.dpow n a)
hh' : Eq (α := (a : A) → B) (⇑toRingHom : (a : A) → B) (⇑toRingHom_1 : (a : A) → B)
⊢ toRingHom = toRingHom_1
have mk : ∀ (x : A), Eq.{u_4 + 1} (α := B) ((toRingHom : (a : A) → B) x : B) ((toRingHom_1 : (a : A) → B) x : B)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
toRingHom : A →+* B
ideal_comp : map toRingHom I ≤ J
dpow_comp : ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n ((toRingHom : (a : A) → B) a) = (toRingHom : (a : A) → B) (hI.dpow n a)
toRingHom_1 : A →+* B
ideal_comp_1 : map toRingHom_1 I ≤ J
dpow_comp_1 : ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n ((toRingHom_1 : (a : A) → B) a) = (toRingHom_1 : (a : A) → B) (hI.dpow n a)
hh' : Eq (α := (a : A) → B) (⇑toRingHom : (a : A) → B) (⇑toRingHom_1 : (a : A) → B)
x : A
⊢ Eq.{u_4 + 1} (α := B) ((toRingHom : (a : A) → B) x : B) ((toRingHom_1 : (a : A) → B) x : B)
have mk : Eq.{u_4 + 1} (α := B) ((toRingHom_1 : (a : A) → B) x : B) ((toRingHom_1 : (a : A) → B) x : B)
Tactic: rw [hh']
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
a b : A
hx :
  Membership.mem (γ := Set A)
    {x : A | x ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n x) = hJ.dpow n ((f : (a : A) → B) x)} a
hy :
  Membership.mem (γ := Set A)
    {x : A | x ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n x) = hJ.dpow n ((f : (a : A) → B) x)} b
⊢ Membership.mem (γ := Set A)
    {x : A | x ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n x) = hJ.dpow n ((f : (a : A) → B) x)} (a + b)
have simp : a + b ∈ I ∧
  ∀ (n : ℕ),
    (f : (a : A) → B) (hI.dpow n (a + b)) =
      hJ.dpow n (HAdd.hAdd (α := B) (β := B) ((f : (a : A) → B) a : B) ((f : (a : A) → B) b : B))
Tactic: simp only [mem_setOf_eq, map_add] at hx hy ⊢
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
a b : A
hx : a ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n a) = hJ.dpow n ((f : (a : A) → B) a)
hy : b ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n b) = hJ.dpow n ((f : (a : A) → B) b)
⊢ a + b ∈ I ∧
    ∀ (n : ℕ),
      (f : (a : A) → B) (hI.dpow n (a + b)) =
        hJ.dpow n (HAdd.hAdd (α := B) (β := B) ((f : (a : A) → B) a : B) ((f : (a : A) → B) b : B))
have refine : ∀ (n : ℕ),
  (f : (a : A) → B) (hI.dpow n (a + b)) =
    hJ.dpow n (HAdd.hAdd (α := B) (β := B) ((f : (a : A) → B) a : B) ((f : (a : A) → B) b : B))
Tactic: refine ⟨I.add_mem hx.1 hy.1, fun n ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
a b : A
hx : a ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n a) = hJ.dpow n ((f : (a : A) → B) a)
hy : b ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n b) = hJ.dpow n ((f : (a : A) → B) b)
n : ℕ
⊢ (f : (a : A) → B) (hI.dpow n (a + b)) =
    hJ.dpow n (HAdd.hAdd (α := B) (β := B) ((f : (a : A) → B) a : B) ((f : (a : A) → B) b : B))
have rw : Eq.{u_4 + 1} (α := B)
  (∑ x ∈ @Finset.antidiagonal _ Nat.instAddMonoid _ n, (f : (a : A) → B) (hI.dpow x.1 a * hI.dpow x.2 b))
  (∑ k ∈ @Finset.antidiagonal _ Nat.instAddMonoid _ n,
    hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b))
Tactic: rw [hI.dpow_add hx.1 hy.1, map_sum,
      hJ.dpow_add (hf (mem_map_of_mem f hx.1)) (hf (mem_map_of_mem f hy.1))]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
a b : A
hx : a ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n a) = hJ.dpow n ((f : (a : A) → B) a)
hy : b ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n b) = hJ.dpow n ((f : (a : A) → B) b)
n : ℕ
⊢ Eq.{u_4 + 1} (α := B)
    (∑ x ∈ @Finset.antidiagonal _ Nat.instAddMonoid _ n, (f : (a : A) → B) (hI.dpow x.1 a * hI.dpow x.2 b))
    (∑ k ∈ @Finset.antidiagonal _ Nat.instAddMonoid _ n,
      hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b))
have h : Eq.{u_4 + 1} (α := (x : ℕ × ℕ) → B) (fun (x : ℕ × ℕ) ↦ (f : (a : A) → B) (hI.dpow x.1 a * hI.dpow x.2 b))
  fun (k : ℕ × ℕ) ↦ hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
Tactic: apply congr_arg
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
a b : A
hx : a ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n a) = hJ.dpow n ((f : (a : A) → B) a)
hy : b ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n b) = hJ.dpow n ((f : (a : A) → B) b)
n : ℕ
⊢ Eq.{u_4 + 1} (α := (x : ℕ × ℕ) → B) (fun (x : ℕ × ℕ) ↦ (f : (a : A) → B) (hI.dpow x.1 a * hI.dpow x.2 b))
    fun (k : ℕ × ℕ) ↦ hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
have h : ∀ (k : ℕ × ℕ),
  (f : (a : A) → B) (hI.dpow k.1 a * hI.dpow k.2 b) =
    hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
Tactic: ext k
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
a b : A
hx : a ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n a) = hJ.dpow n ((f : (a : A) → B) a)
hy : b ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n b) = hJ.dpow n ((f : (a : A) → B) b)
n : ℕ
k : ℕ × ℕ
⊢ (f : (a : A) → B) (hI.dpow k.1 a * hI.dpow k.2 b) =
    hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
have h : HMul.hMul (α := B) (β := B) ((f : (a : A) → B) (hI.dpow k.1 a) : B) ((f : (a : A) → B) (hI.dpow k.2 b) : B) =
  hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
Tactic: rw [map_mul, hx.2, hy.2]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
a b : A
hx : a ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n a) = hJ.dpow n ((f : (a : A) → B) a)
hy : b ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n b) = hJ.dpow n ((f : (a : A) → B) b)
n : ℕ
k : ℕ × ℕ
h :
  HMul.hMul (α := B) (β := B) ((f : (a : A) → B) (hI.dpow k.1 a) : B) ((f : (a : A) → B) (hI.dpow k.2 b) : B) =
    hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
⊢ (f : (a : A) → B) (hI.dpow k.1 a * hI.dpow k.2 b) =
    hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
have h₁ : HMul.hMul (β := B) (hJ.dpow k.1 ((f : (a : A) → B) a)) ((f : (a : A) → B) (hI.dpow k.2 b) : B) =
  hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
Tactic: rw [map_mul, hx.2, hy.2]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
a b : A
hx : a ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n a) = hJ.dpow n ((f : (a : A) → B) a)
hy : b ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n b) = hJ.dpow n ((f : (a : A) → B) b)
n : ℕ
k : ℕ × ℕ
h :
  HMul.hMul (α := B) (β := B) ((f : (a : A) → B) (hI.dpow k.1 a) : B) ((f : (a : A) → B) (hI.dpow k.2 b) : B) =
    hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
h₁ :
  HMul.hMul (β := B) (hJ.dpow k.1 ((f : (a : A) → B) a)) ((f : (a : A) → B) (hI.dpow k.2 b) : B) =
    hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
⊢ (f : (a : A) → B) (hI.dpow k.1 a * hI.dpow k.2 b) =
    hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
have h₂ : hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b) =
  hJ.dpow k.1 ((f : (a : A) → B) a) * hJ.dpow k.2 ((f : (a : A) → B) b)
Tactic: rw [map_mul, hx.2, hy.2]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
⊢ Membership.mem (γ := Set A)
    {x : A | x ∈ I ∧ ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n x) = hJ.dpow n ((f : (a : A) → B) x)} 0
have simp : ∀ (n : ℕ), (f : (a : A) → B) (hI.dpow n 0) = hJ.dpow n 0
Tactic: simp only [mem_setOf_eq, Submodule.zero_mem, map_zero, true_and]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
⊢ (f : (a : A) → B) (hI.dpow 0 0) = hJ.dpow 0 0
have zero : (f : (a : A) → B) 1 = hJ.dpow 0 0
Tactic: rw [hI.dpow_zero I.zero_mem, hJ.dpow_zero J.zero_mem, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
zero : (f : (a : A) → B) 1 = hJ.dpow 0 0
⊢ (f : (a : A) → B) (hI.dpow 0 0) = hJ.dpow 0 0
have zero₁ : Eq.{u_4 + 1} (α := B) ((f : (a : A) → B) 1 : B) (1 : B)
Tactic: rw [hI.dpow_zero I.zero_mem, hJ.dpow_zero J.zero_mem, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.DPMorphism (goal)
Goal:
A_1 : Type u_1
B_1 : Type u_2
inst : CommSemiring A_1
inst_1 : CommSemiring B_1
I_1 : Ideal A_1
J_1 : Ideal B_1
hI_1 : DividedPowers I_1
hJ_1 : DividedPowers J_1
A : Type u_3
B : Type u_4
inst_2 : CommSemiring A
inst_3 : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
zero : (f : (a : A) → B) 1 = hJ.dpow 0 0
zero₁ : Eq.{u_4 + 1} (α := B) ((f : (a : A) → B) 1 : B) (1 : B)
⊢ (f : (a : A) → B) (hI.dpow 0 0) = hJ.dpow 0 0
have zero₂ : Eq.{u_4 + 1} (α := B) 1 1
Tactic: rw [hI.dpow_zero I.zero_mem, hJ.dpow_zero J.zero_mem, map_one]