Suppose $G$ is a smooth manifold and also a topological group. Also suppose that left multiplication $L_g : G \rightarrow G$ is smooth for any $g \in G$. Finally suppose that the multiplication map is smooth in a neighborhood of the identity $e$. By this i mean there exists open $U$ containing $e$ such that $\mu : U \times U \rightarrow M$ is smooth.
Can we conclude that $\mu$ makes G into a Lie group (with the given smooth structure)?
I know there are some strong theorems we could use here. Like the answer to one of hilbert's problems. I was hoping there was an elementary way to see it. I've tried writing the multiplication map near two different points as a composition of these maps, but I cant get things to work out. Thanks for your time.