-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (n : ℕ), ∀ᶠ (i : ι) in cofinite, Eq (α := R) ((f i : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
this_1 : UniformSpace R := ⊥
this : IsUniformInducing.{u_2, u_2} (α := ArithmeticFunction R) (β := (a : ℕ) → R) (DFunLike.coe (β := fun (x : ℕ) ↦ R))
suffices suffices₁ suffices₂ : Multipliable f
suffices₃ suffices₄ :
  ∀ (n : ℕ),
    ∃ (a : Finset ι),
      ∀ (k l : Finset ι),
        a ≤ k → a ≤ l → Eq.{u_2 + 1} (α := R) ((∏ b ∈ k, f b : (a : ℕ) → R) n : R) ((∏ b ∈ l, f b : (a : ℕ) → R) n : R)
suffices₅ :
  ∀ (n : ℕ)
    (hf : ∀ k ∈ Set.Iic n, ∀ᶠ (x : ι) in cofinite, Eq (α := R) ((f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)),
    ∃ (a : Finset ι),
      ∀ (k l : Finset ι),
        a ≤ k → a ≤ l → Eq.{u_2 + 1} (α := R) ((∏ b ∈ k, f b : (a : ℕ) → R) n : R) ((∏ b ∈ l, f b : (a : ℕ) → R) n : R)
suffices₆ :
  ∀ (n : ℕ)
    (hf :
      ∃ (v : Set ι),
        Membership.mem (γ := Filter ι) cofinite v ∧
          ∀ y ∈ v, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)),
    ∃ (a : Finset ι),
      ∀ (k l : Finset ι),
        a ≤ k → a ≤ l → Eq.{u_2 + 1} (α := R) ((∏ b ∈ k, f b : (a : ℕ) → R) n : R) ((∏ b ∈ l, f b : (a : ℕ) → R) n : R)
suffices₇ suffices₈ :
  ∀ (n : ℕ) (s : Set ι) (hs : Membership.mem (γ := Filter ι) cofinite s)
    (hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)),
    ∃ (a : Finset ι),
      ∀ (k l : Finset ι),
        a ≤ k → a ≤ l → Eq.{u_2 + 1} (α := R) ((∏ b ∈ k, f b : (a : ℕ) → R) n : R) ((∏ b ∈ l, f b : (a : ℕ) → R) n : R)
suffices₉ :
  ∀ (n : ℕ) (s : Set ι) (hs : Membership.mem (γ := Filter ι) cofinite s)
    (hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)),
    let t := ⋯.toFinset (s := sᶜ);
    ∀ (u v : Finset ι) (hu : t ≤ u) (hv : t ≤ v),
      Eq.{u_2 + 1} (α := R) ((∏ b ∈ u, f b : (a : ℕ) → R) n : R) ((∏ b ∈ v, f b : (a : ℕ) → R) n : R)
suffices₁₀ :
  ∀ (n : ℕ) (s : Set ι) (hs : Membership.mem (γ := Filter ι) cofinite s)
    (hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)),
    let t := ⋯.toFinset (s := sᶜ);
    ∀ (u v : Finset ι) (hu : t ≤ u) (hv : t ≤ v),
      Eq (α := R) ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ u \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
        ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ v \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
suffices₁₁ :
  ∀ (n : ℕ) (s : Set ι) (hs : Membership.mem (γ := Filter ι) cofinite s)
    (hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)),
    let t := ⋯.toFinset (s := sᶜ);
    ∀ (u v : Finset ι) (hv : t ≤ v) (hu : ∀ i ∈ u \ t, i ∈ s),
      Eq (α := R) ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ u \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
        ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ v \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
suffices₁₂ :
  ∀ (n : ℕ) (s : Set ι) (hs : Membership.mem (γ := Filter ι) cofinite s)
    (hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)),
    let t := ⋯.toFinset (s := sᶜ);
    ∀ (u v : Finset ι) (hu : ∀ i ∈ u \ t, i ∈ s) (hv : ∀ i ∈ v \ t, i ∈ s),
      Eq (α := R) ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ u \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
        ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ v \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
suffices₁₃ :
  ∀ (n : ℕ) (s : Set ι) (hs : Membership.mem (γ := Filter ι) cofinite s)
    (hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)),
    let t := ⋯.toFinset (s := sᶜ);
    ∀ (u v : Finset ι) (hu : ∀ i ∈ u \ t, i ∈ s) (hv : ∀ i ∈ v \ t, i ∈ s),
      ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) k : R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k : R)
suffices₁₄ suffices₁₅ :
  ∀ (n : ℕ) (s : Set ι) (hs : Membership.mem (γ := Filter ι) cofinite s)
    (hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)),
    let t := ⋯.toFinset (s := sᶜ);
    ∀ (u v : Finset ι) (hu : ∀ i ∈ u \ t, i ∈ s) (hv : ∀ i ∈ v \ t, i ∈ s) (w : Finset ι) (hw : ∀ i ∈ w, i ∈ s),
      ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ w, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
suffices₁₆ :
  ∀ (n : ℕ) (s : Set ι) (hs : Membership.mem (γ := Filter ι) cofinite s)
    (hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)),
    let t := ⋯.toFinset (s := sᶜ);
    ∀ (u v : Finset ι) (hu : ∀ i ∈ u \ t, i ∈ s) (hv : ∀ i ∈ v \ t, i ∈ s)
      (hw : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → i ∈ s),
      ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ ∅, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
insert insert₁ :
  ∀ (n : ℕ) (s : Set ι) (hs : Membership.mem (γ := Filter ι) cofinite s)
    (hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)),
    let t := ⋯.toFinset (s := sᶜ);
    ∀ (u v : Finset ι) (hu : ∀ i ∈ u \ t, i ∈ s) (hv : ∀ i ∈ v \ t, i ∈ s) (a : ι) (s_1 : Finset ι),
      a ∉ s_1 →
        (∀ (hw : ∀ i ∈ s_1, i ∈ s),
            ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ s_1, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)) →
          ∀ (hw : ∀ i ∈ Insert.insert a s_1, i ∈ s),
            ∀ k ≤ n,
              Eq.{u_2 + 1} (α := R) ((∏ x ∈ Insert.insert a s_1, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
⊢ ∀ (n : ℕ),
    ∀ᶠ (s : Finset ι) in atTop, Eq (α := R) ((∏ i ∈ s, f i : (a : ℕ) → R) n : R) ((eulerProduct f : (a : ℕ) → R) n : R)

Tactic:
classical
  suffices Multipliable f from tendsto_iff.mp this.hasProd
  simp_rw [multipliable_iff_cauchySeq_finset, CauchySeq, ← this.cauchy_map_iff,
    Filter.map_map, cauchy_map_iff', Pi.uniformity, DiscreteUniformity.eq_principal_setRelId,
    tendsto_iInf, tendsto_comap_iff, tendsto_principal, Function.comp_apply, prod_atTop_atTop_eq,
    eventually_atTop_prod_self, SetRel.mem_id]
  intro n
  replace hf : ∀ k ∈ Set.Iic n, ∀ᶠ (x : ι) in cofinite, (f x) k = (1 : ArithmeticFunction R) k :=
    fun k hk ↦ hf k
  rw [← eventually_all_finite (Set.finite_Iic n), eventually_iff_exists_mem] at hf
  obtain ⟨s, hs, hs'⟩ := hf
  let t := (mem_cofinite.mp hs).toFinset
  refine ⟨t, fun u v hu hv ↦ ?_⟩
  rw [← Finset.prod_sdiff hu, ← Finset.prod_sdiff hv]
  replace hu : ∀ i ∈ u \ t, i ∈ s := by
    intro i hi
    rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
    exact hi.2
  replace hv : ∀ i ∈ v \ t, i ∈ s := by
    intro i hi
    rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
    exact hi.2
  suffices ∀ k ≤ n, (∏ x ∈ u \ t, f x) k = (∏ x ∈ v \ t, f x) k by
    rw [mul_apply, mul_apply]
    refine Finset.sum_congr rfl fun k hk ↦ ?_
    rw [this k.1 (Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hk))]
  suffices ∀ w, (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = (1 : ArithmeticFunction R) k by
    intro k hk
    rw [this (u \ t) hu k hk, this (v \ t) hv k hk]
  intro w hw
  induction w using Finset.induction_on
  case empty => simp
  case insert i w hi hw' =>
    intro k hk
    rw [← one_mul (1 : ArithmeticFunction R), Finset.prod_insert hi, mul_apply, mul_apply]
    refine Finset.sum_congr rfl fun j hj ↦ ?_
    have h1 := hs' i (hw i (Finset.mem_insert_self i w)) j.1
      ((Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hj)).trans hk)
    have h2 := hw' (fun i hi ↦ hw i (Finset.mem_insert_of_mem hi)) j.2
      ((Nat.divisor_le (Nat.snd_mem_divisors_of_mem_antidiagonal hj)).trans hk)
    rw [h1, h2]

Premises used: CauchySeq, Filter.map_map, Insert.insert, Filter.tendsto_comap_iff, Filter.tendsto_principal, multipliable_iff_cauchySeq_finset, DiscreteUniformity.eq_principal_setRelId, SetRel.mem_id, cauchy_map_iff', Filter.tendsto_iInf, Filter.prod_atTop_atTop_eq, Function.comp_apply, ArithmeticFunction.mul_apply, Filter.eventually_atTop_prod_self, OfNat.ofNat, Finset.sum_congr, one_mul, rfl, Filter.eventually_iff_exists_mem, IsUniformInducing.cauchy_map_iff, Pi.uniformity