∀ (p : ℕ) [hp : Fact (Nat.Prime p)] (n : ℕ) {x : ℤ_[p]} (hx : x ∈ Ideal.span {↑p}),
  PadicInt.Coe.ringHom ⟨PadicInt.dpow'✝ p n ↑x, ⋯⟩ =
    (DividedPowers.RatAlgebra.dividedPowers ⊤).dpow n (PadicInt.Coe.ringHom x)