awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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 := ⋯ }
⊢ S ≤ (↑⊤ : Set A)

Tactic:
exacts [hS, hKS]
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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)

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]
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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)
a : {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ⊆ (↑K.carrier : Set A)
⊢ span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ K.carrier

Tactic:
rw [span_le]

Premises used: Ideal.span_le
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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)
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
⊢ hI.dpow n x ∈ (↑K.carrier : Set A)

Tactic:
K.dpow_mem n hn x (this hx)

Premises used: DividedPowers.SubDPIdeal.dpow_mem
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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)
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
⊢ hI.dpow n x ∈ (↑K.carrier : Set A)

Tactic:
exact K.dpow_mem n hn x (this hx)

Premises used: DividedPowers.SubDPIdeal.dpow_mem
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
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)
⊢ hI.IsSubDPIdeal (RingHom.ker f ⊓ I)

Tactic:
rw [isSubDPIdeal_inf_iff]

Premises used: DividedPowers.isSubDPIdeal_inf_iff
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
⊢ 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

Premises used: DividedPowers.isDPMorphism_def
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
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)
⊢ 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:
simp only [RingHom.sub_mem_ker_iff, ← hf.2 a ha, ← hf.2 b hb]

Premises used: And.right, RingHom.sub_mem_ker_iff
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
⊢ 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:
congr_arg _

Premises used: congr_arg
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
⊢ 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:
exact congr_arg _

Premises used: congr_arg
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
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
⊢ a ∈ RingHom.ker f.toRingHom ⊓ I → hI.dpow x a ∈ RingHom.ker f.toRingHom ⊓ I

Tactic:
simp only [mem_inf, and_imp, RingHom.mem_ker]

Premises used: and_imp, RingHom.mem_ker, Ideal.mem_inf
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
rw : hJ.dpow x 0 = 0 ∧ hI.dpow x 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:
rw [← f.isDPMorphism.2 a ha', ha]

Premises used: DividedPowers.IsDPMorphism.dpow_comp, DividedPowers.DPMorphism.isDPMorphism
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
⊢ hJ.dpow x 0 = 0 ∧ hI.dpow x a ∈ I

Tactic:
⟨dpow_eval_zero hJ hn, hI.dpow_mem hn ha'⟩

Premises used: DividedPowers.dpow_mem, DividedPowers.dpow_eval_zero, And.intro
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
⊢ hJ.dpow x 0 = 0 ∧ hI.dpow x a ∈ I

Tactic:
exact ⟨dpow_eval_zero hJ hn, hI.dpow_mem hn ha'⟩
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
apply : ∀ (n : ℕ), hI.dpow n (a + b) = 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)

Tactic:
apply And.intro (I.add_mem ha.1 hb.1) (fun n ↦ ?_)

Premises used: Ideal.add_mem, And.left, And.intro
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
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)
⊢ hI.dpow n (a + b) = hI'.dpow n (a + b)

Tactic:
rw [hI.dpow_add ha.1 hb.1, hI'.dpow_add ha.1 hb.1]

Premises used: DividedPowers.dpow_add, And.left
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
rw₁ rw₂ : 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

Tactic:
rw [ha.2, hb.2]

Premises used: And.right
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
⊢ 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:
exact Finset.sum_congr rfl (fun k _ ↦ by rw [ha.2, hb.2])

Premises used: Finset.sum_congr, rfl
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
apply : ∀ (n : ℕ), hI.dpow n 0 = hI'.dpow n 0
⊢ Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} 0

Tactic:
apply And.intro I.zero_mem (fun n ↦ ?_)

Premises used: Ideal.zero_mem, And.intro
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
neg : ∀ (hn : ¬n = 0), hI.dpow n 0 = hI'.dpow n 0
⊢ hI.dpow n 0 = hI'.dpow n 0

Tactic:
by_cases hn : n = 0
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
pos₂ pos₃ : Eq.{u_1 + 1} (α := A) 1 1
⊢ hI.dpow n 0 = hI'.dpow n 0

Tactic:
rw [hn, hI.dpow_zero (zero_mem I), hI'.dpow_zero (zero_mem I)]

Premises used: DividedPowers.dpow_zero, ZeroMemClass.zero_mem
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
neg₁ neg₂ : Eq.{u_1 + 1} (α := A) 0 0
⊢ hI.dpow n 0 = hI'.dpow n 0

Tactic:
rw [hI.dpow_eval_zero hn, hI'.dpow_eval_zero hn]

Premises used: DividedPowers.dpow_eval_zero
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
rw : Membership.mem (γ := Set A) {a : A | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a} (a * x)
⊢ 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]

Premises used: smul_eq_mul
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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
rw₂ rw₃ : a ^ n * hI'.dpow n x = a ^ n * hI'.dpow n x
⊢ hI.dpow n (a * x) = hI'.dpow n (a * x)

Tactic:
rw [hI.dpow_mul hx.1, hI'.dpow_mul hx.1, hx.2]

Premises used: DividedPowers.dpow_mul, And.right, And.left
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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)

Tactic:
exact ⟨I.mul_mem_left a hx.1, (fun n ↦ by rw [hI.dpow_mul hx.1, hI'.dpow_mul hx.1, hx.2])⟩
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
⊢ x ∈ hI.dpEqualizer hI' ↔ x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x

Tactic:
simp [dpEqualizer, Submodule.mem_mk, AddSubmonoid.mem_mk, AddSubsemigroup.mem_mk,
    Set.mem_setOf_eq]

Premises used: AddSubsemigroup.mem_mk, DividedPowers.dpEqualizer, Set.mem_setOf_eq, AddSubmonoid.mem_mk, Submodule.mem_mk
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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)
rw₂ : (↑(m.uniformBell x_1) : A) * hI'.dpow (m * x_1) x = hI'.dpow m (hI'.dpow x_1 x)
rw₃ rw₄ : (↑(m.uniformBell x_1) : A) * hI'.dpow (m * x_1) x = (↑(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)

Tactic:
rw [hI.dpow_comp hn hx.1, hx.2, hx.2, hI'.dpow_comp hn hx.1]

Premises used: DividedPowers.dpow_comp, And.right, And.left
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
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 : hI.dpow m (hI.dpow 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)
rw₂ : (↑(m.uniformBell x_1) : A) * hI'.dpow (m * x_1) x = hI'.dpow m (hI.dpow x_1 x)
rw₃ : (↑(m.uniformBell x_1) : A) * hI'.dpow (m * x_1) x = hI'.dpow m (hI'.dpow x_1 x)
rw₄ rw₅ : (↑(m.uniformBell x_1) : A) * hI'.dpow (m * x_1) x = (↑(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)

Tactic:
rw [← hx.2, hI.dpow_comp hn hx.1, hx.2, hx.2, hI'.dpow_comp hn hx.1]

Premises used: DividedPowers.dpow_comp, And.right, And.left
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst_1 : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.IsDPMorphism hK f
hIK' : hI.IsDPMorphism hK' f
rw : (⇑f : (a : A) → B) '' (↑I : Set A) ⊆ (↑(hK.dpEqualizer hK') : Set B)
⊢ Ideal.map f I ≤ hK.dpEqualizer hK'

Tactic:
rw [Ideal.map, span_le]

Premises used: Ideal.span_le, Ideal.map
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst_1 : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.IsDPMorphism hK f
hIK' : hI.IsDPMorphism hK' f
a : A
ha : a ∈ (↑I : Set A)
n : ℕ
rw : (f : (a : A) → B) (hI.dpow n a) = hK'.dpow n ((f : (a : A) → B) a)
rw₁ rw₂ : Eq.{u_2 + 1} (α := B) ((f : (a : A) → B) (hI.dpow n a) : B) ((f : (a : A) → B) (hI.dpow n a) : B)
⊢ hK.dpow n ((f : (a : A) → B) a) = hK'.dpow n ((f : (a : A) → B) a)

Tactic:
rw [hIK.2 a ha, hIK'.2 a ha]

Premises used: DividedPowers.IsDPMorphism.dpow_comp
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst_1 : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.IsDPMorphism hK f
hIK' : hI.IsDPMorphism hK' f
a : A
ha : a ∈ (↑I : Set A)
⊢ Membership.mem (α := B) (↑(hK.dpEqualizer hK') : Set B) ((f : (a : A) → B) a : B)

Tactic:
exact ⟨hI_le_K (mem_map_of_mem f ha), fun n ↦ by rw [hIK.2 a ha, hIK'.2 a ha]⟩
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A_1 : Type u_1
inst : CommSemiring A_1
I_1 : Ideal A_1
hI_1 hI' : DividedPowers I_1
A : Type u_2
inst_1 : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers (A := A ⧸ J) (Ideal.map (F := A →+* A ⧸ J) (Ideal.Quotient.mk J) I)
φ : hI.DPMorphism hJ
hφ : φ.toRingHom = Ideal.Quotient.mk J
⊢ J ⊓ I ≤ I

Tactic:
simp only [inf_le_right]

Premises used: inf_le_right
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A_1 : Type u_1
inst : CommSemiring A_1
I_1 : Ideal A_1
hI_1 hI' : DividedPowers I_1
A : Type u_2
inst_1 : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers (A := A ⧸ J) (Ideal.map (F := A →+* A ⧸ J) (Ideal.Quotient.mk J) I)
φ : hI.DPMorphism hJ
hφ : φ.toRingHom = Ideal.Quotient.mk J
x : ℕ
hn : x ≠ 0
a : A
h : a ∈ J ⊓ I
haJ : a ∈ (↑J : Set A)
haI : a ∈ (↑I : Set A)
⊢ hI.dpow x a ∈ J ⊓ I

Tactic:
⟨?_, hI.dpow_mem hn haI⟩

Premises used: DividedPowers.dpow_mem, Set.Mem, setOf, Set.inter, And.intro
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A_1 : Type u_1
inst : CommSemiring A_1
I_1 : Ideal A_1
hI_1 hI' : DividedPowers I_1
A : Type u_2
inst_1 : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers (A := A ⧸ J) (Ideal.map (F := A →+* A ⧸ J) (Ideal.Quotient.mk J) I)
φ : hI.DPMorphism hJ
hφ : φ.toRingHom = Ideal.Quotient.mk J
x : ℕ
hn : x ≠ 0
a : A
h : a ∈ J ⊓ I
haJ : a ∈ (↑J : Set A)
haI : a ∈ (↑I : Set A)
refine : hI.dpow x a ∈ (↑J : Set A)
⊢ hI.dpow x a ∈ J ⊓ I

Tactic:
refine ⟨?_, hI.dpow_mem hn haI⟩
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A_1 : Type u_1
inst : CommSemiring A_1
I_1 : Ideal A_1
hI_1 hI' : DividedPowers I_1
A : Type u_2
inst_1 : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers (A := A ⧸ J) (Ideal.map (F := A →+* A ⧸ J) (Ideal.Quotient.mk J) I)
φ : hI.DPMorphism hJ
hφ : φ.toRingHom = Ideal.Quotient.mk J
x : ℕ
hn : x ≠ 0
a : A
h : a ∈ J ⊓ I
haJ : a ∈ (↑J : Set A)
haI : a ∈ (↑I : Set A)
rw : hJ.dpow x ((φ.toRingHom : (a : A) → A ⧸ J) a) = 0
⊢ hI.dpow x a ∈ (↑J : Set A)

Tactic:
rw [SetLike.mem_coe, ← Quotient.eq_zero_iff_mem, ← hφ, ← φ.dpow_comp a haI]

Premises used: DividedPowers.DPMorphism.dpow_comp, Ideal.Quotient.eq_zero_iff_mem, SetLike.mem_coe
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A_1 : Type u_1
inst : CommSemiring A_1
I_1 : Ideal A_1
hI_1 hI' : DividedPowers I_1
A : Type u_2
inst_1 : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers (A := A ⧸ J) (Ideal.map (F := A →+* A ⧸ J) (Ideal.Quotient.mk J) I)
φ : hI.DPMorphism hJ
hφ : φ.toRingHom = Ideal.Quotient.mk J
x : ℕ
hn : x ≠ 0
a : A
h : a ∈ J ⊓ I
haJ : a ∈ (↑J : Set A)
haI : a ∈ (↑I : Set A)
ha0 : Eq (α := A ⧸ J) ((φ.toRingHom : (a : A) → A ⧸ J) a : A ⧸ J) (0 : A ⧸ J)
rw : hJ.dpow x 0 = 0
rw₁ rw₂ : Eq.{u_2 + 1} (α := A ⧸ J) 0 0
⊢ hJ.dpow x ((φ.toRingHom : (a : A) → A ⧸ J) a) = 0

Tactic:
rw [ha0, hJ.dpow_eval_zero hn]

Premises used: DividedPowers.dpow_eval_zero
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A_1 : Type u_1
inst : CommSemiring A_1
I_1 : Ideal A_1
hI_1 hI' : DividedPowers I_1
A : Type u_2
inst_1 : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers (A := A ⧸ J) (Ideal.map (F := A →+* A ⧸ J) (Ideal.Quotient.mk J) I)
φ : hI.DPMorphism hJ
hφ : φ.toRingHom = Ideal.Quotient.mk J
x : ℕ
hn : x ≠ 0
a : A
h : a ∈ J ⊓ I
haJ : a ∈ (↑J : Set A)
haI : a ∈ (↑I : Set A)
suffices : Eq (α := A ⧸ J) ((φ.toRingHom : (a : A) → A ⧸ J) a : A ⧸ J) (0 : A ⧸ J)
⊢ hJ.dpow x ((φ.toRingHom : (a : A) → A ⧸ J) a) = 0

Tactic:
suffices ha0 : φ.toRingHom a = 0 by
      rw [ha0, hJ.dpow_eval_zero hn]
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A_1 : Type u_1
inst : CommSemiring A_1
I_1 : Ideal A_1
hI_1 hI' : DividedPowers I_1
A : Type u_2
inst_1 : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers (A := A ⧸ J) (Ideal.map (F := A →+* A ⧸ J) (Ideal.Quotient.mk J) I)
φ : hI.DPMorphism hJ
hφ : φ.toRingHom = Ideal.Quotient.mk J
x : ℕ
hn : x ≠ 0
a : A
h : a ∈ J ⊓ I
haJ : a ∈ (↑J : Set A)
haI : a ∈ (↑I : Set A)
rw : a ∈ J
⊢ Eq (α := A ⧸ J) ((φ.toRingHom : (a : A) → A ⧸ J) a : A ⧸ J) (0 : A ⧸ J)

Tactic:
rw [hφ, Quotient.eq_zero_iff_mem]

Premises used: Ideal.Quotient.eq_zero_iff_mem
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A_1 : Type u_1
inst : CommSemiring A_1
I_1 : Ideal A_1
hI_1 hI' : DividedPowers I_1
A : Type u_2
inst_1 : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers (A := A ⧸ J) (Ideal.map (F := A →+* A ⧸ J) (Ideal.Quotient.mk J) I)
φ : hI.DPMorphism hJ
hφ : φ.toRingHom = Ideal.Quotient.mk J
x : ℕ
hn : x ≠ 0
a : A
h : a ∈ J ⊓ I
haJ : a ∈ (↑J : Set A)
haI : a ∈ (↑I : Set A)
⊢ a ∈ J

Tactic:
haJ

Premises used: Set.Mem
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A_1 : Type u_1
inst : CommSemiring A_1
I_1 : Ideal A_1
hI_1 hI' : DividedPowers I_1
A : Type u_2
inst_1 : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers (A := A ⧸ J) (Ideal.map (F := A →+* A ⧸ J) (Ideal.Quotient.mk J) I)
φ : hI.DPMorphism hJ
hφ : φ.toRingHom = Ideal.Quotient.mk J
x : ℕ
hn : x ≠ 0
a : A
h : a ∈ J ⊓ I
haJ : a ∈ (↑J : Set A)
haI : a ∈ (↑I : Set A)
⊢ a ∈ J

Tactic:
exact haJ
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A : Type u_1
inst : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst_1 : CommRing B
f : A →+* B
J : Ideal B
hf : Function.Surjective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
hIJ : J = Ideal.map f I
hIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
x : B
hx' : x ∉ J
hnc : ¬∃ (a : ↥I), (f : (a : A) → B) (↑a : A) = x
⊢ dpow hI f n x = 0

Tactic:
rw [dpow, Function.extend_def, dif_neg, Pi.zero_apply]

Premises used: Function.extend_def, Pi.zero_apply, dif_neg, DividedPowers.Quotient.OfSurjective.dpow
-- Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal
Goal:
A : Type u_1
inst : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst_1 : CommRing B
f : A →+* B
J : Ideal B
hf : Function.Surjective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
hIJ : J = Ideal.map f I
hIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
hx' : ((f : (a : A) → B) (↑(Subtype.mk (p := fun (x : A) ↦ x ∈ I) a ha) : A) : B) ∉ J
⊢ False

Tactic:
exact (hIJ ▸ hx') (apply_coe_mem_map f I ⟨a, ha⟩)