Powerful subgroups of extensions of powerful p-groups and pro-p groups

Let H be a finitely generated pro-p group. We prove that, if H/Zₙ₋₁(H) is a powerful group, then the n-th terms of the lower p-series and the lower central series of H are powerfully embedded in H. As a consequence, we obtain that if H/Zₙ(H) is a p-adic analytic pro-p group for some positive integer n, then H is a p-adic analytic pro-p group. Furthermore, we extend these results to pro-p groups that are not necessarily finitely generated, provided some additional conditions are imposed.