I have just started reading about Lie groups and by definition (at least the way it is defined in my class),
A group G is a Lie group if it is a smooth manifold and the map $G \times G \ni (g,h) \mapsto gh^{-1} \in G$ is smooth.
I am not comfortable with the notion of smoothness here. Moreover, if you want a specific problem where I am stuck, in my class notes, it is written that "this definition implies that the inverse map $G \ni h \mapsto h^{-1} \in G$ is smooth and this follows from the fact that $G \times G \ni (e,h) \mapsto eh^{-1} \in G$ is smooth ($e$ is the identity element of G).
Here's where exactly I am stuck in trying to prove the above statement. Consider the maps I have mentioned above excluding the one in definition. Call the first map $i$, the second map $f$ and define a third map $g$ which takes $G \ni h \mapsto (e,h) \in G \times G$. It can be seen that $i=f \circ g$ and if I can prove smoothness of $g$, then I am done.
To show smoothness of $g$ at a point $h$ on the smooth manifold $G$, I take a chart $(U,\phi)$ of $G$ such that $h \in U$. Then I take a chart $(U' \times V, \phi ' \times \psi)$ of $G \times G$ such that $g(U) \subset U' \times V$. Now to show smoothness, I have to show that the map $(\phi ' \times \psi) \circ g \circ \phi ^{-1}: \phi (U) \subset \Bbb{R}^n \to \Bbb{R}^m \times \Bbb{R}^p \supset \phi ' (U') \times \psi(V) $ is smooth. I am clueless now. How do I proceed?
Proof. Let $x_0\in X$ and consider a chart $(U,\phi)$ around $x_0$ in $X$. Now consider any chart $(V,\theta)$ around $y_0$ in $Y$. Then we have the chart $(U\times V, \phi\times\theta)$ around $(x_0,y_0)$ in $X\times Y$. With this we have
$$f(U)=U\times\{y_0\}\subseteq U\times V$$ $$(\phi\times\theta)\circ f\circ \phi^{-1}(v)=\big(v,\theta(y_0)\big)$$
i.e. the composition is the product of identity and constant map which is easily seen to be $C^n$.
Note that the domain of the composition is some open subset of $\mathbb{R}^s$ (i.e. $\phi(U)$) while the codomain is $\mathbb{R}^s\times\mathbb{R}^t$ where $s=\dim X$ and $t=\dim Y$. $\Box$