Let $M, N$ be smooth manifolds. Show that $T(M \times N)$ is diffeomorphic to $TM \times TN$.
Here is what I have so far. Define $F: T(M \times N) \to TM \times TN$ as $$F((p,q), v)= ((p, d\pi_{1_{(p,q)}}(v)), (q, d\pi_{2_{(p,q)}}(v))).$$ The map $F$ is bijective since $\alpha:T_{(p,q)}(M\times N) \to T_pM \oplus T_qN, \alpha(v)=(d\pi_{1_{(p,q)}}(v), d\pi_{2_{(p,q)}}(v))$ is an isomorphism.
Now we need to show that $F$ is smooth. Since the tangent bundles can be themselves considered as manifolds we let $(p,q) \in M\times N$ and pick charts $(U, \varphi) = (U, (x^1, \dots ,x^m))$ for $M$ and $(V, \psi) = (V, (y^1, \dots, y^n))$ for $N$. Also let $\pi : T(M \times N) \to M \times N$ be the natural projection.
Consider now smooth maps $\widetilde{\varphi \times \psi} : \pi^{-1}(U \times V) \to \Bbb R^{2(n+m)}$ defined by $$\widetilde{\varphi \times \psi} \left( v^i \frac{\partial}{\partial x^i} \bigg|_{(p,q)} \right)=((x^1\times y^1)(p,q), \dots, (x^{n+m}\times y^{n+m})(p,q), v^1, \dots, v^{n+m}),$$
and $\tilde{\varphi} \times \tilde{\psi}: \pi^{-1}(U) \times \pi^{-1}(V) \to \Bbb R^{2(n+m)}$ defined by $$\tilde{\varphi} \times \tilde{\psi}\left( v^i \frac{\partial}{\partial x^i} \bigg|_{p}, w^i\frac{\partial}{\partial y^i} \bigg|_{q} \right)=(x^1(p), \dots,x^n(p), y^1(q), \dots, y^m(q), v^1, \dots v^n, w^1, \dots, w^m).$$
Now the coordinate representation for $F$ would be given by $$(\tilde{\varphi} \times \tilde{\psi}) \circ F \circ (\widetilde{\varphi \times \psi})^{-1}$$ but I cannot show that this is smooth.
What can I do to show the smoothness of this composition? Also is the definitions for $\tilde{\varphi} \times \tilde{\psi}$ and $\widetilde{\varphi \times \psi}$ correct? The former one seems good, but the latter one ($\widetilde{\varphi \times \psi}$) feels a bit sketchy.
