Let $i_{q0} : M\rightarrow M\times N$, $i_{q0}(p) = (p, q0)$ be a mapping between smooth manifolds. I need some hints to show that it is $C^{\infty}$.
I have so far... Let $(U,\phi)$ and $(V,\psi)$ be charts about $p$ and $i_{q0}$, and let $r^{i}$ be the $ith$ coordinate function on Euclidean space. Then we need to show that $\frac{\partial (r^{i}\circ \psi \circ i_{q0} \circ \phi^{-1})}{\partial r^{j}}$ exists and is continuous at $\phi(p)$ and that we can keep taking partial derivatives.