Definition: A principal bundle $\pi:P \rightarrow M$ with structure group $G$ is a fiber bundle $P$ with a right action of the Lie group $G$ on the fibers, such that $$\pi(pg)= \pi(p), \quad p \in P , g \in G, $$ and such that the action of $G$ is free and transitive on the fibers.
Context: I want to prove the following statement:
The fibers of the bundle $P$ are diffeomorphic to $G$ itself.
What I've done so far: To show it, I've fixed a fiber $P_m:=\pi^{-1}(m)$, $ m \in M$ and chosen an element $p $ in this fiber, and then considered the map $$f_p : G \rightarrow P_m$$ given by $f_p(g)= pg$ for $g\in G$.
Using the fact that the action of $G$ on $P_m$ is free and transitive, I've proven that $f_p$ is bijective.
Question: How does one show that this map $f_p$ is differentiable?
Any help please!