Let $\pi:P\rightarrow M$ be a $G$-principal bundle and $\rho:G\rightarrow GL(V)$ a representation.
The space of $k$ forms on $M$ with values in $P\times_G V$ (denote as $\Omega^k(M;P\times_G V)$ can be identified with with the space of horizontal, right invariant $k$-forms on $M$ (denote as $\Omega^k_G(P;V))$.
Ie, there is an isomorphism:
$\Omega^k_G(P;V)\cong \Omega^k(M;P\times_G V)$.
Could you please give me the proof of this result or suggest me a reference where I can find the proof in details (since I found the proof in some lecture notes but it was not clear).
Thanks for your help!