∀ {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (v : ℕ) (f : ℤ_[p] → ℕ) (f_spec : ∀ (x : ℤ_[p]), x - ↑(f x) ∈ Ideal.span {↑v})
  (f_congr : ∀ (x : ℤ_[p]) (a b : ℕ), x - ↑a ∈ Ideal.span {↑v} → x - ↑b ∈ Ideal.span {↑v} → ↑a = ↑b),
  PadicInt.toZModHom v f f_spec f_congr =
    { toFun := fun x => ↑(f x), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }