Can the following theorem from calculus:
Theorem: Let $F:\Omega\rightarrow \mathbb{R}^m$ be a continuously differentiable map, where $\Omega\subseteq \mathbb{R}^n$ is a domain(Connected and open). If $\det (DF)\neq 0$ for all $x\in \Omega$, then $F$ is an open map.
This is a classic theorem in calculus, but I was wondering whether it can be deduced given that we know the Banach-Schauder theorem?(Open mapping between Banach spaces)
Since differentiablity is essentially a linear approximation, and at each point the linear approximation is invertible and open, can such an argument be made? I am aware that in this case there is a family of linear open surjective operators, but it seems to me that by a continuity of the differentials and a compactness argument , one can consider the worst linear approximation and show that open balls are sent to open balls.
I am however somewhat struggling in writing it down coherently, and would appreciate feedback.