Loring Tu's in his An Introduction to Manifolds discusses the smoothness of a projection map in Example 6.17, and the smoothness of a map to a Cartesian product in Exercise 6.18 (Second Edition, page no. 65). They are given below.
Both the proofs are the straightforward application of the definition of a smooth map between smooth manifolds and eventually uses the following facts.
The projection map $p_{\mathbb{R}^m}: \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m$ is smooth. And the restriction of $p_{\mathbb{R}^m}$ to an open subset of $\mathbb{R}^m \times \mathbb{R}^n$ is also smooth. (cf. the solution of Example 6.17.)
The function $h:\mathbb{R}^n \to \mathbb{R}^{m_1 + m_2}$, $h(x) = (h_1(x), h_2(x))$, is smooth iff $h_1:\mathbb{R}^n \to \mathbb{R}^{m_1}$ and $h_2:\mathbb{R}^n \to \mathbb{R}^{m_2}$ are smooth functions. And the restriction of $h$ to an open subset of $\mathbb{R}^n$ is also smooth. (It is utilized in the proof of the Exercise 6.18.)
My Questions
It is not clear to me how we prove the smoothness of the functions $p_{\mathbb{R}^m}$, $h$, and their restrictions to an open subset of their domains. Do we use the definition of smoothness of a function in terms of infinite differentiability in these cases to prove their smoothness as in all cases the domains are Euclidean vector spaces? Any explanation would be highly appreciated.