A detail in the proof that left invariant vector fields are smooth

Question

I am trying to understand a proof of the following proposition:

Let $X$ be a left invariant vector field on the Lie Group $G$. Then, $X$ is smooth.

The proof goes as follows:

"Let $X$ be left invariant. It suffices to prove that$^1$ $$ Xf: G\rightarrow \mathbb{R} \\ x\mapsto f_*(x)(X(x)) $$ is smooth for any function $f$.

$$ (Xf)(x) = f_*(x)(X(x)) \\ = f_*(x)(L_{x,*}(e)(X(e)))\\ = (f\circ L_x)_*(e)(X(e))) $$ which is clearly smooth."

Question:

$^1$: I thought that the definition of $Xf$ was such that $Xf(x):= X_xf$. And this is not the same as $f_*(x)(X(x))$.

Answer 1

I think the problem here is the canonical isomorphism $\mathbb{R}\simeq T_p\mathbb{R}$ for all $p\in\mathbb{R}$. Notice that if $f:G\to\mathbb{R}$, then $f_*|_x : T_x G\to T_{f(x)}\mathbb{R}\simeq \mathbb{R}$, and that is why you can identify $f_*|_x(X|_x)\in T_{f(x)}\mathbb{R}$ with $X|_x f\in \mathbb{R}$.