Let $φ : G → H$ be a continous morphism of linear Lie groups with H connected.
Prove that $φ$ is surjective iff $dφ$ surjective
Using the expression : $exp\circ dφ = φ \circ exp$ we can see that
$φ$ is surjective $\Rightarrow φ \circ exp = exp\circ dφ$ is surjective but how to prove $dφ$ is surjective?
I have no idea on how to tackle the other implication.
Thanks to comments I see now that
$dφ$ is surjective $\Rightarrow dφ(Lie(g))=Lie(h) \Rightarrow exp\circ dφ(Lie(g))=\exp\circ Lie(h) \Rightarrow φ\circ exp(Lie(g))=H$ (because H is connected) $ \Rightarrow φ$ is surjective.