I am trying to understand some basic stuff about Manifolds and Lie Groups.
Suppose $H$ is a Lie group which is also a subgroup of a Lie Group $G$. Suppose the inclusion map is smooth. Then $H$ is a Lie Subgroup of $G$.
I am not really sure if this result is true. I am trying to give the following argument.
Since the inclusion map is smooth, we can take the differential. The differential map must be an inclusion map, hence injective. Thus the original map is an immersion. Thus $H$ is an immersed submanifold of $G$. Hence its a Lie subgroup.
Is it correct?