Let $\phi \in C^{\infty}(\mathbb{R}^n)$. Consider $\phi(x,y)$ as a function on $\mathbb{R}^p \times \mathbb{R}^q$, where $p+q=n$.
For any $y \in \mathbb{R}^q$, consider the map $\phi_y:\mathbb{R}^p \rightarrow \mathbb{R} : x \mapsto \phi(x,y)$. Then do we have $\phi_y \in C^{\infty}(\mathbb{R}^p)$ for each $y \in \mathbb{R}^q$?
I'm wondering if there's a nice way to prove it, or if someone could point me towards a book where it's proved.
Here is a slick way to prove it. If you know that the composition of two smooth functions is smooth, then the function $\phi_y$ is the composition of $\phi$ and the inclusion map $\iota_y: \mathbb{R}^p\hookrightarrow \mathbb{R}^p \times \mathbb{R}^q$, $x\mapsto (x,y)$, which is clearly smooth.