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given two differential manifolds $M_1$ and $M_2$. I have to show that the projection $\pi: M_1 \times M_2 \to M_1$ is smooth. By definition, I then need to show that for a point $(a,b)\in M_1\times M_2$ with charts $(\psi,\phi): U\times \tilde U\ni (a,b) \to V\times \tilde V$ and open sets $U\subset M_1, \tilde U \subset M_2, V\subset \mathbb{R}^n, \tilde{V}\subset \mathbb{R}^k$, we have that \begin{align} \phi^\prime \circ \pi \circ (\phi^{-1},\psi^{-1}) \end{align} is smooth.

I know that $\phi^\prime, \phi\text{ and }\psi$ are smooth but I only know that $\pi$ is linear. Is it sufficent to conclude that this map is smooth?

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One has $(\phi(a),\psi(b))\mapsto (a,b)\mapsto a \mapsto \phi'(a)$. This can be written a little more compactly as the map $(x,y)\mapsto \phi'\circ \phi^{-1}(x)$, which is clearly smooth.

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