I am posting a picture, containing a theorem and its proof from these notes on Lie groups.
I'm quoting one of the lines:
$UH$ is open in $G$ (which easily follows from inverse function theorem applied to the map $f:U\times H\to G$).
What exactly is meant by this? Is the argument that $UH=f^{-1}(G)$, and hence open? Where does the inverse function theorem come in?
The differential of the map $f:U\times H\rightarrow G$ defined by $(u,h)\rightarrow uh$ is an isomorphism since $M$ is transversal to $H$, this justify the application of the local inverse theorem which implies that $f$ is a local diffeomorphism, hence $UH$ is open.