Related to GP 1.3.9 - Is projection function smooth?

I start to think of this question when I attempt Ex 1.3.9 on Guillemin and Pollack's Differential Topology GP 1.3.9(b) Every manifold is locally expressible as a graph..

I am under the impression that this is true. For the proof of the following question, I need the fact that both the projection function and its inverse are smooth. Is this true?

Let $x_1, \dots, x_N$ be the standard coordinate functions on $\mathbb{R}^N$, and let $X$ be a $k$-dimensional submanifold of $\mathbb{R}^N$. Prove that every point $x \in X$ has a neighborhood on which the restrictions of some $k$-coordinate functions $x_{i_1}, \dots, x_{i_k}$ form a local coordinate system.

• What projection do you mean? – Mariano Suárez-Álvarez Jul 3 '13 at 5:56
• Hi @MarianoSuárez-Alvarez thanks. this kind: $$d \varphi: T_x(X) \mapsto \text{ span}(e_{i_1}, \dots, e_{i_k})$$ – WishingFish Jul 3 '13 at 15:53