Projection map is smooth I want to prove that $\pi: M \times N \rightarrow M$ is smooth where $M, N$ are smooth manifolds. Let $(U \times V, \phi \times \varphi)$ be a chart on $M \times N$, and $(W, \psi)$ be another chart on $M$, then $\psi \circ \pi\circ (\phi \times \varphi)^{-1}$ is smooth. I know that function $\psi \circ \pi\circ (\phi \times \varphi)^{-1} = \psi \circ \phi^{-1}$, which is smooth since $M$ is smooth, but those two have different domains, so how can they be equal?

The problem is that $\psi \circ \pi\circ (\phi \times \varphi)^{-1} = \psi \circ \phi^{-1}$ isn't valid, since as you've noticed the domain don't coincide. What is true is the following:
$$\psi \circ \pi\circ (\phi \times \varphi)^{-1}(x,y) = \psi \circ \pi(\phi^{-1}(x),\varphi^{-1}(y)) = \psi(\phi^{-1}(x)) = \psi \circ \phi^{-1}(x)$$
which is a smooth map from $\mathbb{R}^{m+n} \to \mathbb{R}^m$, as $\psi \circ \phi^{-1}$ is a smooth map itself.
