awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
h' : Module.Finite R M
hM : IsTorsion R M
I : Type u
fI : Fintype I
w : DecidableEq I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h : DirectSum.IsInternal (S := Submodule R M) fun (i : I) ↦ torsionBy R M (p i ^ e i)
⊢ ∀ (i : I),
    ∃ (d : ℕ) (k : Fin d → ℕ),
      Nonempty
        (LinearEquiv.{u, u, v, u} (RingHom.id R) (↥(torsionBy R M (p i ^ e i)))
          (⨁ (j : Fin d), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ k j)))

Tactic:
have :
    ∀ i,
      ∃ (d : ℕ) (k : Fin d → ℕ),
        Nonempty <| torsionBy R M (p i ^ e i) ≃ₗ[R] ⨁ j, R ⧸ R ∙ p i ^ k j := by
    exact fun i =>
      torsion_by_prime_power_decomposition.{u, v} (hp i)
        ((isTorsion'_powers_iff <| p i).mpr fun x => ⟨e i, smul_torsionBy _ _⟩)
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
h' : Module.Finite R M
hM : IsTorsion R M
I : Type u
fI : Fintype I
w : DecidableEq I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h : DirectSum.IsInternal (S := Submodule R M) fun (i : I) ↦ torsionBy R M (p i ^ e i)
this :
  ∀ (i : I),
    ∃ (d : ℕ) (k : Fin d → ℕ),
      Nonempty
        (LinearEquiv.{u, u, v, u} (RingHom.id R) (↥(torsionBy R M (p i ^ e i)))
          (⨁ (j : Fin d), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ k j)))
⊢ ∃ (ι : Type u) (x : Fintype ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
    Nonempty (M ≃ₗ[R] ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))

Tactic:
⟨Σ i, Fin (this i).choose, inferInstance, fun ⟨i, _⟩ => p i, fun ⟨i, _⟩ => hp i, fun ⟨i, j⟩ =>
      (this i).choose_spec.choose j,
      ⟨(LinearEquiv.ofBijective (DirectSum.coeLinearMap _) h).symm.trans <|
          (DFinsupp.mapRange.linearEquiv fun i => (this i).choose_spec.choose_spec.some).trans <|
            (DirectSum.sigmaLcurryEquiv R).symm.trans
              (DFinsupp.mapRange.linearEquiv fun i => quotEquivOfEq _ _ ?_)⟩⟩

Premises used: Exists.choose_spec, LinearEquiv.symm, _private.0.Module.equiv_directSum_of_isTorsion.match_1, Irreducible, inferInstance, Nonempty.some, Sigma.fst, Nonempty.intro, DirectSum.sigmaLcurryEquiv, HPow.hPow, Nat, Exists.intro, Submodule.span, DFinsupp.mapRange.linearEquiv, Exists.choose, Submodule.quotEquivOfEq, DirectSum.coeLinearMap, Singleton.singleton, Sigma.snd, Fin, LinearEquiv.trans, Sigma, LinearEquiv.ofBijective, _private.0.Module.equiv_directSum_of_isTorsion.match_2
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
h' : Module.Finite R M
obtain :
  ∀ (I : Type u) (fI : Fintype I) (p : I → R) (hp : ∀ (i : I), Irreducible (p i)) (e : I → ℕ)
    (h :
      LinearEquiv.{u, u, v, u} (RingHom.id R) (↥(torsion R M))
        (⨁ (i : I), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))),
    ∃ (n : ℕ) (ι : Type u) (x : Fintype ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
      Nonempty (M ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))
⊢ ∃ (n : ℕ) (ι : Type u) (x : Fintype ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
    Nonempty (M ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))

Tactic:
obtain ⟨I, fI, p, hp, e, ⟨h⟩⟩ :=
    equiv_directSum_of_isTorsion.{u, v} (@torsion_isTorsion R M _ _ _)
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
h' : Module.Finite R M
I : Type u
fI : Fintype I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h :
  LinearEquiv.{u, u, v, u} (RingHom.id R) (↥(torsion R M))
    (⨁ (i : I), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))
obtain :
  ∀ (n : ℕ)
    (g :
      LinearEquiv.{u, u, v, u} (RingHom.id R) (HasQuotient.Quotient M (B := Submodule R M) (torsion R M)) (Fin n →₀ R)),
    ∃ (n : ℕ) (ι : Type u) (x : Fintype ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
      Nonempty (M ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))
⊢ ∃ (n : ℕ) (ι : Type u) (x : Fintype ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
    Nonempty (M ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))

Tactic:
obtain ⟨n, ⟨g⟩⟩ := @Module.basisOfFiniteTypeTorsionFree' R _ (M ⧸ torsion R M) _ _ _ _ _ _

Premises used: Module.basisOfFiniteTypeTorsionFree'
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
h' : Module.Finite R M
I : Type u
fI : Fintype I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h :
  LinearEquiv.{u, u, v, u} (RingHom.id R) (↥(torsion R M))
    (⨁ (i : I), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))
n : ℕ
g : LinearEquiv.{u, u, v, u} (RingHom.id R) (HasQuotient.Quotient M (B := Submodule R M) (torsion R M)) (Fin n →₀ R)
obtain :
  ∀ (f : HasQuotient.Quotient M (B := Submodule R M) (torsion R M) →ₗ[R] M)
    (hf : mkQ (torsion R M) ∘ₗ f = LinearMap.id),
    ∃ (n : ℕ) (ι : Type u) (x : Fintype ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
      Nonempty (M ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))
⊢ ∃ (n : ℕ) (ι : Type u) (x : Fintype ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
    Nonempty (M ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))

Tactic:
obtain ⟨f, hf⟩ := Module.projective_lifting_property _ LinearMap.id (torsion R M).mkQ_surjective

Premises used: LinearMap.id, Submodule.torsion, Module.projective_lifting_property, Submodule.mkQ_surjective
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
h' : Module.Finite R M
I : Type u
fI : Fintype I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h :
  LinearEquiv.{u, u, v, u} (RingHom.id R) (↥(torsion R M))
    (⨁ (i : I), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))
n : ℕ
g : LinearEquiv.{u, u, v, u} (RingHom.id R) (HasQuotient.Quotient M (B := Submodule R M) (torsion R M)) (Fin n →₀ R)
f : HasQuotient.Quotient M (B := Submodule R M) (torsion R M) →ₗ[R] M
hf : mkQ (torsion R M) ∘ₗ f = LinearMap.id
⊢ ∃ (n : ℕ) (ι : Type u) (x : Fintype ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
    Nonempty (M ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))

Tactic:
⟨n, I, fI, p, hp, e,
      ⟨(lequivProdOfRightSplitExact (torsion R M).injective_subtype ?_ hf).symm.trans <|
          (h.prodCongr g).trans <| LinearEquiv.prodComm.{u, u} R _ (Fin n →₀ R) ⟩⟩

Premises used: LinearEquiv.symm, LinearEquiv.prodCongr, Submodule.torsion, Nonempty.intro, Submodule.injective_subtype, LinearEquiv.prodComm, Exists.intro, Fin, LinearEquiv.trans, Finsupp, lequivProdOfRightSplitExact
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
h' : Module.Finite R M
I : Type u
fI : Fintype I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h :
  LinearEquiv.{u, u, v, u} (RingHom.id R) (↥(torsion R M))
    (⨁ (i : I), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))
n : ℕ
g : LinearEquiv.{u, u, v, u} (RingHom.id R) (HasQuotient.Quotient M (B := Submodule R M) (torsion R M)) (Fin n →₀ R)
f : HasQuotient.Quotient M (B := Submodule R M) (torsion R M) →ₗ[R] M
hf : mkQ (torsion R M) ∘ₗ f = LinearMap.id
R₁ :
  Eq (α := Submodule R M)
    (LinearMap.range (R := R) (M := ↥(torsion R M)) (τ₁₂ := RingHom.id R) (Submodule.subtype (torsion R M)))
    (LinearMap.ker (M₂ := HasQuotient.Quotient M (B := Submodule R M) (torsion R M)) (mkQ (torsion R M)))
⊢ ∃ (n : ℕ) (ι : Type u) (x : Fintype ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
    Nonempty (M ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))

Tactic:
refine
    ⟨n, I, fI, p, hp, e,
      ⟨(lequivProdOfRightSplitExact (torsion R M).injective_subtype ?_ hf).symm.trans <|
          (h.prodCongr g).trans <| LinearEquiv.prodComm.{u, u} R _ (Fin n →₀ R) ⟩⟩
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
h' : Module.Finite R M
I : Type u
fI : Fintype I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h :
  LinearEquiv.{u, u, v, u} (RingHom.id R) (↥(torsion R M))
    (⨁ (i : I), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ e i))
n : ℕ
g : LinearEquiv.{u, u, v, u} (RingHom.id R) (HasQuotient.Quotient M (B := Submodule R M) (torsion R M)) (Fin n →₀ R)
f : HasQuotient.Quotient M (B := Submodule R M) (torsion R M) →ₗ[R] M
hf : mkQ (torsion R M) ∘ₗ f = LinearMap.id
rw :
  Eq (α := Submodule R M) (torsion R M)
    (LinearMap.ker (M₂ := HasQuotient.Quotient M (B := Submodule R M) (torsion R M)) (mkQ (torsion R M)))
rw₁ rw₂ : Eq (α := Submodule R M) (torsion R M) (torsion R M)
⊢ Eq (α := Submodule R M)
    (LinearMap.range (R := R) (M := ↥(torsion R M)) (τ₁₂ := RingHom.id R) (Submodule.subtype (torsion R M)))
    (LinearMap.ker (M₂ := HasQuotient.Quotient M (B := Submodule R M) (torsion R M)) (mkQ (torsion R M)))

Tactic:
rw [range_subtype, ker_mkQ]

Premises used: Submodule.ker_mkQ, Submodule.range_subtype
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
have :
  ∀ (m : ℕ) (ι : Type u) (w : Fintype ι) (p : ι → R) (irr : ∀ (i : ι), Irreducible (p i)) (n : ι → ℕ)
    (e : M ≃ₗ[R] (Fin m →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)),
    ∃ (x : M), Eq (α := Submodule R R) (ker (toSpanSingleton R M x)) (annihilator R M)
⊢ ∃ (x : M), Eq (α := Submodule R R) (ker (toSpanSingleton R M x)) (annihilator R M)

Tactic:
have ⟨m, ι, _, p, irr, n, ⟨e⟩⟩ := equiv_free_prod_directSum (R := R) (M := M)
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
m : ℕ
ι : Type u
w : Fintype ι
p : ι → R
irr : ∀ (i : ι), Irreducible (p i)
n : ι → ℕ
e : M ≃ₗ[R] (Fin m →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)
⊢ ∃ (x : M), Eq (α := Submodule R R) (ker (toSpanSingleton R M x)) (annihilator R M)

Tactic:
⟨e.symm (Finsupp.equivFunOnFinite.symm fun _ ↦ 1, DFinsupp.equivFunOnFintype.symm
    fun _ ↦ mkQ _ 1), le_antisymm (fun x h ↦ ?_) fun x h ↦ mem_annihilator.mp h _⟩

Premises used: Module.mem_annihilator, LinearEquiv.symm, Finsupp.equivFunOnFinite, Prod.mk, DFinsupp.equivFunOnFintype, le_antisymm, HPow.hPow, Exists.intro, Submodule.span, Iff.mp, Submodule.mkQ, Equiv.symm, Singleton.singleton, OfNat.ofNat, DFunLike.coe
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
m : ℕ
ι : Type u
w : Fintype ι
p : ι → R
irr : ∀ (i : ι), Irreducible (p i)
n : ι → ℕ
e : M ≃ₗ[R] (Fin m →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)
refine :
  ∀ (x : R)
    (h :
      Membership.mem (γ := Submodule R R)
        (ker
          (toSpanSingleton R M
            ((e.symm : (a : (Fin m →₀ R) × ⨁ (i : ι), R ⧸ R ∙ p i ^ n i) → M)
              ((Equiv.symm Finsupp.equivFunOnFinite : (a : Fin m → R) → Fin m →₀ R) fun (x : Fin m) ↦ 1,
                (Equiv.symm
                      (DFinsupp.equivFunOnFintype (β := fun (i : ι) ↦
                        HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))) :
                    (a : (i : ι) → R ⧸ R ∙ p i ^ n i) → Π₀ (i : ι), R ⧸ R ∙ p i ^ n i)
                  fun (x : ι) ↦ (mkQ (R ∙ p x ^ n x) : (a : R) → R ⧸ R ∙ p x ^ n x) 1))))
        x),
    Membership.mem (γ := Submodule R R) (annihilator R M) x
⊢ ∃ (x : M), Eq (α := Submodule R R) (ker (toSpanSingleton R M x)) (annihilator R M)

Tactic:
refine ⟨e.symm (Finsupp.equivFunOnFinite.symm fun _ ↦ 1, DFinsupp.equivFunOnFintype.symm
    fun _ ↦ mkQ _ 1), le_antisymm (fun x h ↦ ?_) fun x h ↦ mem_annihilator.mp h _⟩
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
m : ℕ
ι : Type u
w : Fintype ι
p : ι → R
irr : ∀ (i : ι), Irreducible (p i)
n : ι → ℕ
e : M ≃ₗ[R] (Fin m →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)
x : R
h :
  Membership.mem (γ := Submodule R R)
    (ker
      (toSpanSingleton R M
        ((e.symm : (a : (Fin m →₀ R) × ⨁ (i : ι), R ⧸ R ∙ p i ^ n i) → M)
          ((Equiv.symm Finsupp.equivFunOnFinite : (a : Fin m → R) → Fin m →₀ R) fun (x : Fin m) ↦ 1,
            (Equiv.symm
                  (DFinsupp.equivFunOnFintype (β := fun (i : ι) ↦
                    HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))) :
                (a : (i : ι) → R ⧸ R ∙ p i ^ n i) → Π₀ (i : ι), R ⧸ R ∙ p i ^ n i)
              fun (x : ι) ↦ (mkQ (R ∙ p x ^ n x) : (a : R) → R ⧸ R ∙ p x ^ n x) 1))))
    x
⊢ (∀ (a : Fin m),
      Eq (α := R)
        (((HSMul.hSMul (β := (Fin m →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)) x
                  ((Equiv.symm Finsupp.equivFunOnFinite : (a : Fin m → R) → Fin m →₀ R) fun (x : Fin m) ↦ 1,
                    (Equiv.symm
                          (DFinsupp.equivFunOnFintype (β := fun (i : ι) ↦
                            HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))) :
                        (a : (i : ι) → R ⧸ R ∙ p i ^ n i) → Π₀ (i : ι), R ⧸ R ∙ p i ^ n i)
                      fun (x : ι) ↦ (mkQ (R ∙ p x ^ n x) : (a : R) → R ⧸ R ∙ p x ^ n x) 1)).1 :
              (a : Fin m) → R)
            a :
          R)
        ((0.1 : (a : Fin m) → R) a : R)) ∧
    ∀ (i : ι),
      ((HSMul.hSMul (β := (Fin m →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)) x
                ((Equiv.symm Finsupp.equivFunOnFinite : (a : Fin m → R) → Fin m →₀ R) fun (x : Fin m) ↦ 1,
                  (Equiv.symm
                        (DFinsupp.equivFunOnFintype (β := fun (i : ι) ↦
                          HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))) :
                      (a : (i : ι) → R ⧸ R ∙ p i ^ n i) → Π₀ (i : ι), R ⧸ R ∙ p i ^ n i)
                    fun (x : ι) ↦ (mkQ (R ∙ p x ^ n x) : (a : R) → R ⧸ R ∙ p x ^ n x) 1)).2 :
            (a : ι) → R ⧸ R ∙ p a ^ n a)
          i =
        (0.2 : (a : ι) → R ⧸ R ∙ p a ^ n a) i

Tactic:
rw [mem_ker, toSpanSingleton_apply, ← map_smul,
    e.symm.map_eq_zero_iff, Prod.ext_iff, Finsupp.ext_iff, DFinsupp.ext_iff] at h

Premises used: LinearEquiv.symm, Prod.ext_iff, LinearMap.toSpanSingleton_apply, LinearMap.mem_ker, DFinsupp.ext_iff, LinearEquiv.map_eq_zero_iff, Finsupp.ext_iff, map_smul
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
ι : Type u
w : Fintype ι
p : ι → R
irr : ∀ (i : ι), Irreducible (p i)
n : ι → ℕ
x : R
e : M ≃ₗ[R] (Fin 0 →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)
h :
  (∀ (a : Fin 0),
      Eq (α := R)
        (((HSMul.hSMul (β := (Fin 0 →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)) x
                  ((Equiv.symm Finsupp.equivFunOnFinite : (a : Fin 0 → R) → Fin 0 →₀ R) fun (x : Fin 0) ↦ 1,
                    (Equiv.symm
                          (DFinsupp.equivFunOnFintype (β := fun (i : ι) ↦
                            HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))) :
                        (a : (i : ι) → R ⧸ R ∙ p i ^ n i) → Π₀ (i : ι), R ⧸ R ∙ p i ^ n i)
                      fun (x : ι) ↦ (mkQ (R ∙ p x ^ n x) : (a : R) → R ⧸ R ∙ p x ^ n x) 1)).1 :
              (a : Fin 0) → R)
            a :
          R)
        ((0.1 : (a : Fin 0) → R) a : R)) ∧
    ∀ (i : ι),
      ((HSMul.hSMul (β := (Fin 0 →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)) x
                ((Equiv.symm Finsupp.equivFunOnFinite : (a : Fin 0 → R) → Fin 0 →₀ R) fun (x : Fin 0) ↦ 1,
                  (Equiv.symm
                        (DFinsupp.equivFunOnFintype (β := fun (i : ι) ↦
                          HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))) :
                      (a : (i : ι) → R ⧸ R ∙ p i ^ n i) → Π₀ (i : ι), R ⧸ R ∙ p i ^ n i)
                    fun (x : ι) ↦ (mkQ (R ∙ p x ^ n x) : (a : R) → R ⧸ R ∙ p x ^ n x) 1)).2 :
            (a : ι) → R ⧸ R ∙ p a ^ n a)
          i =
        (0.2 : (a : ι) → R ⧸ R ∙ p a ^ n a) i
zero :
  Membership.mem (γ := Submodule R R)
    (annihilator R (Fin 0 →₀ R) ⊓
      annihilator R (⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)))
    (HSMul.hSMul (β := R) x 1)
⊢ Membership.mem (γ := Submodule R R) (annihilator R M) x

Tactic:
rw [← mul_one x, ← smul_eq_mul, e.annihilator_eq, annihilator_prod]

Premises used: smul_eq_mul, mul_one, LinearEquiv.annihilator_eq, Module.annihilator_prod
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
ι : Type u
w : Fintype ι
p : ι → R
irr : ∀ (i : ι), Irreducible (p i)
n : ι → ℕ
x : R
e : M ≃ₗ[R] (Fin 0 →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)
h :
  (∀ (a : Fin 0),
      Eq (α := R)
        (((HSMul.hSMul (β := (Fin 0 →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)) x
                  ((Equiv.symm Finsupp.equivFunOnFinite : (a : Fin 0 → R) → Fin 0 →₀ R) fun (x : Fin 0) ↦ 1,
                    (Equiv.symm
                          (DFinsupp.equivFunOnFintype (β := fun (i : ι) ↦
                            HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))) :
                        (a : (i : ι) → R ⧸ R ∙ p i ^ n i) → Π₀ (i : ι), R ⧸ R ∙ p i ^ n i)
                      fun (x : ι) ↦ (mkQ (R ∙ p x ^ n x) : (a : R) → R ⧸ R ∙ p x ^ n x) 1)).1 :
              (a : Fin 0) → R)
            a :
          R)
        ((0.1 : (a : Fin 0) → R) a : R)) ∧
    ∀ (i : ι),
      ((HSMul.hSMul (β := (Fin 0 →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)) x
                ((Equiv.symm Finsupp.equivFunOnFinite : (a : Fin 0 → R) → Fin 0 →₀ R) fun (x : Fin 0) ↦ 1,
                  (Equiv.symm
                        (DFinsupp.equivFunOnFintype (β := fun (i : ι) ↦
                          HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))) :
                      (a : (i : ι) → R ⧸ R ∙ p i ^ n i) → Π₀ (i : ι), R ⧸ R ∙ p i ^ n i)
                    fun (x : ι) ↦ (mkQ (R ∙ p x ^ n x) : (a : R) → R ⧸ R ∙ p x ^ n x) 1)).2 :
            (a : ι) → R ⧸ R ∙ p a ^ n a)
          i =
        (0.2 : (a : ι) → R ⧸ R ∙ p a ^ n a) i
zero :
  ∀ (i : ι),
    Eq (α := HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))
      (Submodule.Quotient.mk (HSMul.hSMul (β := R) x 1)) 0
⊢ Membership.mem (γ := Submodule R R)
    (annihilator R (Fin 0 →₀ R) ⊓
      annihilator R (⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)))
    (HSMul.hSMul (β := R) x 1)

Tactic:
simp_rw [annihilator_eq_top_iff.mpr inferInstance, DirectSum, annihilator_dfinsupp,
      top_inf_eq, mem_iInf, Ideal.annihilator_quotient, ← Quotient.mk_eq_zero]

Premises used: Iff.mpr, Submodule.mem_iInf, inferInstance, top_inf_eq, Module.annihilator_eq_top_iff, Ideal.annihilator_quotient, Module.annihilator_dfinsupp, DirectSum, Submodule.Quotient.mk_eq_zero
-- Module: Mathlib.Algebra.Module.PID
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
ι : Type u
w : Fintype ι
p : ι → R
irr : ∀ (i : ι), Irreducible (p i)
n : ι → ℕ
x : R
e : M ≃ₗ[R] (Fin 0 →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)
h :
  (∀ (a : Fin 0),
      Eq (α := R)
        (((HSMul.hSMul (β := (Fin 0 →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)) x
                  ((Equiv.symm Finsupp.equivFunOnFinite : (a : Fin 0 → R) → Fin 0 →₀ R) fun (x : Fin 0) ↦ 1,
                    (Equiv.symm
                          (DFinsupp.equivFunOnFintype (β := fun (i : ι) ↦
                            HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))) :
                        (a : (i : ι) → R ⧸ R ∙ p i ^ n i) → Π₀ (i : ι), R ⧸ R ∙ p i ^ n i)
                      fun (x : ι) ↦ (mkQ (R ∙ p x ^ n x) : (a : R) → R ⧸ R ∙ p x ^ n x) 1)).1 :
              (a : Fin 0) → R)
            a :
          R)
        ((0.1 : (a : Fin 0) → R) a : R)) ∧
    ∀ (i : ι),
      ((HSMul.hSMul (β := (Fin 0 →₀ R) × ⨁ (i : ι), HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i)) x
                ((Equiv.symm Finsupp.equivFunOnFinite : (a : Fin 0 → R) → Fin 0 →₀ R) fun (x : Fin 0) ↦ 1,
                  (Equiv.symm
                        (DFinsupp.equivFunOnFintype (β := fun (i : ι) ↦
                          HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))) :
                      (a : (i : ι) → R ⧸ R ∙ p i ^ n i) → Π₀ (i : ι), R ⧸ R ∙ p i ^ n i)
                    fun (x : ι) ↦ (mkQ (R ∙ p x ^ n x) : (a : R) → R ⧸ R ∙ p x ^ n x) 1)).2 :
            (a : ι) → R ⧸ R ∙ p a ^ n a)
          i =
        (0.2 : (a : ι) → R ⧸ R ∙ p a ^ n a) i
⊢ ∀ (i : ι),
    Eq (α := HasQuotient.Quotient R (B := Submodule R R) (R ∙ p i ^ n i))
      (Submodule.Quotient.mk (HSMul.hSMul (β := R) x 1)) 0

Tactic:
h.2

Premises used: Submodule.Quotient.mk, And.right, Quot.map, Quotient.map'