Equivalence between submersion and a particular coordinate system

Question

I'm proving the next:

Let $\psi:M^{m}\rightarrow N^{n}$ be a smooth map and let $p\in M.$ The following are equivalent:

i) The differential map $d\psi_{p}$ is onto,

ii) If $y^{1},\ldots,y^{n}$ is a coordinate system for $N$ at $\psi(p),$ there is a coordinate system fon $p$ in $M$ of the form $(y^{1}\circ\psi,\ldots,y^{n}\circ\psi,x^{n+1},\ldots,x^{m}).$

I've proved ii) implies i) proving jacobian matrix of $d\psi_{p}$ has rank $n.$

I'm stuck in the other direction. Is there a theorem or a easy way to prove this?

Any kind of help is thanked in advanced.

Answer 1

You have to apply the inverse function theorem, and there is a bit of work needed to bring things into the right form. As a hint, I would suggest that you start with that case that $M$ is an open neighbourhood of $0$ in $\mathbb R^m$, $p=0$ and the kernel of $d\psi_p$ is spanned by the standard basis vectors $e_{n+1},\dots,e_m$. Then reduce the general case to this special case by constructing appropriate local coordinates on $M$ around $p$.