I have tried a lot to prove this well-known result. The basic idea behind the proof is clear to me. But I'm stuck at showing either of the following conditions:
- The map $\ G \rightarrow TG \ $ given by $ \ g \mapsto (L_{g})_{*}(e)(v_{e}) \ $ is smooth.
$\ D\star(g,e)(0_{g},-):T_{e}(G) \rightarrow T_{g}(G)\ $ is vector space isomorphism.
Where G is the Lie group, e is the identity element and star is the multiplication operation. There is already an available solution in stackexchage but the solution is not correct completely. So it'll be very helpful if someone provides details. I'm completely stuck at showing smoothness of the map using charts. Please help.
Denote $L_g$ as the map $G\to G$ given by left multiplication by $G$. Consider the map $f: G \times T_eG \to TG$ where $f(g,v)=(g,dL_g(v))$. If we differentiate the product map, $P: G\times G \to G$ at $(g,e)$ from the splitting $T_gG\times T_eG \cong T_{(g,e)}(G\times G)$, we get $dP(g,e)=dL_g|_{T_eG}+Id|_{T_gG}$. Therefore by smoothness of the product map $f$ is smooth. $dL_g$ is the derivative of a diffeomorphism $((L_g)^{-1}=L_{g^-1})$ hence is a linear isomorphism, so $f$ is an isomorphism of vector bundles.