Let $G$ be a Lie group, let $X\in T_eG$, and let $\alpha_X:\mathbb{R}\to G$ be the maximal curve of $X$ with starting point $e$. In his notes, my teacher proves that $F:\mathbb{R}\times T_eG\to G,(t,X)\to\alpha_X(t)$ is smooth, as follows: firstly, note that $\alpha_X(t)=\phi_X(t,e)$, where $\phi_X:\mathbb{R}\times G\to G$ is the flow of $X^L$. He says that it is a well-known (local) result that the map $(X,t,x)\mapsto\phi_X(t,x)$ is smooth, from which it of course follows that $F$ is smooth. What is this well-known (local) result, i.e. why is that map smooth? He also mentions that $X^L$ depends linearly hence smoothly on $X$; this much is clear. But how does this translates to smoothness of $(X,t,x)\to\phi_X(t,x)$?
Addendum: this result is used to establish that $\exp:T_eG\to G,X\to\alpha_X(1)$ is smooth.