I want to prove that unit cricle is a Lie grouop. $i.e$, want to show that the multiplication and inverse function is smooth.
The multiplication and inverse is given by \begin{align} &m(e^{i\theta}, e^{i\phi}) = e^{i(\theta+\phi)} \\ &i(e^{i\theta}) = e^{-i\theta} \end{align}
i tried to find some proof for this but many textbook just state $\mathbb{S}^1$ is Lie group, and some textbook states multiplication and inverse as above
My naive guess is is introducing $f:\mathbb{R}^1 \rightarrow \mathbb{S}^1$ such that $f(x) = e^{ix}$, and since $f$ is smooth, i think $m$ and $i$ are also smooth.
Is there something wrong with my interpretation?
Or is there any other ways to show that multiplication and inverse map is smooth?