Let $E$ and $F$ be two vector bundles over a manifold $M$. Consider the Whitney sum $$E\oplus F\longrightarrow M.$$

We have projections $\pi\_E:E\oplus F\longrightarrow E$ and $\pi\_F:E\oplus F\longrightarrow F$. Is it true that $$\textrm{pr}\_E^\*\wedge \textrm{pr}\_F^\*: \Omega^p(E)\otimes \Omega^q(F)\longrightarrow \Omega^{p+q}(E\oplus F),$$ is an isomorphism of $C^\infty(M)$-modules? The map $\textrm{pr}\_E^\*\wedge \textrm{pr}\_F^\*$ is of course given by:

$$\omega\_E\otimes \omega\_F\longmapsto \textrm{pr}\_E^\*\omega\_E\wedge \textrm{pr}\_F^\*\omega\_F,$$ where $\textrm{pr}\_E^\*:\Omega^p(E)\longrightarrow \Omega^p(E\oplus F)$ and $\textrm{pr}\_F^\*:\Omega^q(F)\longrightarrow \Omega^q(E\oplus F)$ are the induce maps on forms. 

Thanks.

Obs: I just messed everything up. I know that $$\Omega^k(E\oplus F)\simeq \bigoplus\_{p+q=k}\Omega^p(E)\otimes \Omega^q(F),$$ so I'll rewrite the question later.