Background and definitions: For a vector bundle $E$ let $F(E)$ denote its frame bundle. For a principal bundle $P$, and representation $(\rho,GL(n,\mathbb{R}))$ let $E(P,\mathbb R^n)$ denote the associated vector bundle that is $E(P,\mathbb R^n)=P\times \mathbb{R}/\sim$ where $(p,v)\sim(pg,\rho(g^{-1}v))$.
Problem Show that $F(E(P,\mathbb R^n)\cong P$.
Attempt: The fibers of $F(E(P,\mathbb R^n)$ consist of frames of the vector bundle $E(P,GL(n,\mathbb R))$ but the elements in $P$ are just points on the manifold $P$ so I am not sure how to reconcile these two ideas.
Update: We want to show that $F(E(P,\mathbb R^n)_x\cong P_x$. Since $P$ is a $GL(n,\mathbb{R})$ bundle $P_x\cong GL(n,\mathbb{R})$. I think that $F(E(P,\mathbb R^n)_x$ are frames of $E(P,\mathbb R^n)_x=\{[p,v]: p\in P_x \text{ and } v \in \mathbb R^n \}$. I think we need to use the $GL(n,\mathbb R)$ invariance of $[p,v]$ to show that any frame is uniquely defined by $p$, but I don't see how to do this.