Sylvester domains and pro-$p$ groups

Let $G$ be a finitely generated torsion-free pro-$p$ group containing an open free-by-${\mathbb{Z}}_p$ pro-$p$ subgroup. We show that the completed group algebra of $G$ over ${\mathbb{F}}_p$ is a Sylvester domain. Moreover the inner rank of a matrix $A$ over this completed group algebra can be calculated by approximation by ranks corresponding to finite quotients of $G$, that is, if $G=G_1>G_2>\dots$ is a chain of normal open subgroups of $G$ with trivial intersection and $A_i$ is the matrix over ${\mathbb{F}}_p[G\mathrm{/}{G}_i]$ obtained from the matrix $A$ by applying the natural homomorphism induced from $G\to G\mathrm{/}{G}_i$, then the inner rank of $A$ equals ${\lim }_{i\to \mathrm{\infty }}\frac{{\mathrm{rk}}_{{\mathbb{F}}_p}(A_i)}{\mathrm{\mid }G:G_i\mathrm{\mid }}$. As a consequence, we obtain a particular case of the mod $p$ Lück approximation for abstract finitely generated subgroups of free-by-${\mathbb{Z}}_p$ pro-$p$ groups.