Let $M$ be a manifold of dimension $n$ and let $\pi :E \rightarrow M$ be a (real) vector bundle of rank $r$, we want to associate a $GL_r-$ principal bundle to it in the following way:
Let $Fr(M)= \{(x,u) |\space x\in M \text{ and } u \text{ is a frame of }E_x:=\pi ^{-1}(x)\}$
To put a smooth structure we observe that we want any local frame of the vetore bundle to be a $\textbf{smooth}$ local section of the fiber bundle, so for each local frame on the open $U$ of the vector bundle (the local trivialty condition tells us that such frames exist) we define the chart:
$\Phi_{\pi^{-1}(U)}:\pi^{-1}(U)\rightarrow \mathbb{R}^{n+r^2}$
$(x,u)\mapsto (\phi(x),v_i^j)$
where $\phi$ is a chart on $U\subset M$, and $v_i^j$ are the coordinates of the frame $u$ with respect to the local frame of the fiber bundle.
Is my reasoning correct until now? If so my question is: how do i prove that this charts are compatible?