If $f$ is a $C^1$-diffeomorphism between open subsets of Banach spaces, is ${\rm D}f$ surjective at each point?

Question

Let $E_i$ be a $\mathbb R$-Banach space, $\Omega_i\subseteq E_i$ be open and $f:\Omega_1\to\Omega_2$ be bijective. If $f$ is Fréchet differentiable at $\omega_1\in\Omega_1$ and $f^{-1}$ is Fréchet differentiable at $\omega_2:=f(\omega_1)$, then $f^{-1}\circ f$ is Fréchet differentiable at $\omega_1$ and $$\operatorname{id}_{E_1}={\rm D}\operatorname{id}_{\Omega_1}(\omega_1)={\rm D}(f^{-1}\circ f)(\omega_1)={\rm D}f^{-1}(\omega_2)\circ{\rm D}f(\omega_1)\tag1.$$ Can we conclude that ${\rm D}f(\omega_1)$ is surjective?

Answer 1

Yes. If instead of differentiating $f^{-1}\circ f$ you differentiate $f\circ f^{-1}$, you get $$ \operatorname{id}_{E_2}={\rm D}\operatorname{id}_{\Omega_2}(\omega_2)={\rm D}(f\circ f^{-1})(\omega_2)={\rm D}f (\omega_1)\circ{\rm D}f^{-1}(\omega_2)\tag2. $$ So $Df(\omega_1)$ is invertible.