Let M be a smooth manifold and $E_1$ and $E_2$ be two smooth manifolds which are also vector bundles with projections \begin{align*} &\pi_1:E_1\to M\\ &\pi_2:E_2\to M \end{align*} and local trivilisations which are diffeomorphisms \begin{align*} &\varphi_{1}:\pi_1^{-1}(U)\to U\times \mathbb{R}^{k_1}\\ &\varphi_{2}:\pi_2^{-1}(U)\to U\times \mathbb{R}^{k_2} \end{align*} respectively.
Take the tensor product of these two vector bundles and there is the projection \begin{equation*} \pi: E_1\otimes E_2 \to M \end{equation*} and local trivilisations: \begin{equation*} \varphi:\pi^{-1}(U)\to U\times (\mathbb{R}^{k_1}\otimes \mathbb{R^{k_2}}) \end{equation*} which can be defined by: \begin{equation*} \varphi(v\otimes w)=(\pi(v\otimes w), \varphi_1(v)\otimes \varphi_2(w)). \end{equation*}
My issue is in showing that $\varphi$ is a diffeomorphism so that $E$ is a smooth vector bundle.
My attempt so far has been to put $U$ and $\mathbb{R}^{k_1}\otimes \mathbb{R}^{k_2}$ into local coordinates and then use $\varphi$ to have local coordinates on $\varphi^{-1}(U)$. Given this we have that $\varphi$ is a diffeomorphism.
But this doesn't use the fact that $\varphi_1$ and $\varphi_2$ are diffeomorphisms. Is this still ok?