∀ {R : Type u_1} {A : Type u_2} [inst : CommSemiring R] [inst_1 : CommSemiring A] [inst_2 : Algebra R A] {p : ℕ}
  [inst_3 : ExpChar A p] {x : A}, (∃ n, x ^ p ^ n ∈ (algebraMap R A).rangeS) = (x ∈ Subalgebra.perfectClosure R A p)