if $f$ is open and has constant rank, then $f$ is a submersion.

Question

Suppose $f:U\subseteq\mathbb{R}^m\to \mathbb{R}^{n}$ has constant rank. Show that $f$ is a submersion if and only if $f$ is open.

I know that a submersion is an open function.

Now if $r$ is the rank of $f$ By the theorem of constant rank for each $a\in U$ there are difeomorphisms $\alpha:V\subseteq\mathbb{R}^{r}\times\mathbb{R}^{p} \to U_{a}$ and $\beta: W_{f(a)}\to Z\subseteq \mathbb{R}^{r}\times \mathbb{R}^{q}$ such that $\beta\circ f\circ \alpha(x,y)=(x,0)$.

how can I use this to show that f is a submersion?. Thanks

Answer 1

$\beta \circ f \circ \alpha$ is open iff $q=0$