I am going through Lie groups by Rossman. I'm currently in section 2.3, and I'm a bit lost as to what is going on and why certain statements aren't trivial. I don't have a specific point of confusion but if it's ok I'm going to summarize what I THINK is going on, and please correct me / elaborate if there is something I'm missing.
Proposition 1. A map $f$ from an open subset of $\mathbb{R}^p$ into a linear group G is of class $C^k$ iff it is of class $C^k$ as a map into the matrix space M.
Now, from my understanding, the proof is basically this: Consider $g$ being the lie algebra of $G$. A map f from $\mathbb{R}^p$ into $G$ can be thought of as a map from bases $g_1...g_p$ representing $e_1...e_p$ as a vector space into G and and M can just be thought of as exp($g^*(a_1e_1...a_pe_p)$) where g* takes $e_1...e_p$ to $g_1...g_p$. In other words: $g$, $G$, and $\mathbb{R}^p$ are just talking different vector spaces and the maps between them are all analytic.
The actual proof goes: $C^k$ implies $f(x)$ = $f(x_0)*$exp(X) for all $x$ near $x_o$. Implies exp(X) = $f(x_o)^{-1}*f(x)$ implies X = log($f(x_o)^{-1}*f(x))$ which supposedly finishes the proof if we know that the RHS is in the lie algebra since log is analytic. By finishing the proof, are supposed to consider X here as some element in M aka $\mathbb{R}^p$?
I'm confused about how $\mathbb{R}^p$ is different from M and why this is not a completely trivial proof. Is the story something like: we have these different things going on. We are mapping from $\mathbb{R}^p$ to $G$. This ends up being exactly like a map from $g$ to $G$. This is invertible around things in $G$, which gets us to $g$, which gets us to M, which is basically just where we started ($\mathbb{R}^p$)?
There is then a corollary that for G, the map from $a$ to $a^{-1}$ is analytic. Is the idea here that we can think of this as a map from G to G, which is basically (by taking log) a map from $g$ to $g$, which is basically the negation map from $\mathbb{R}^p$ to $\mathbb{R}^p$?