Smoothness of the inversion map is redundant in the definition of Lie groups The question I want to ask is different from this one. 

Let $M$ be a smooth manifold which admits a group structure such that the multiplication map $m:G\times G\to G$ defined as $m(g, h)=gh$ for all $g, h\in G$ is a smooth map. Then $G$ is a Lie group.

Hint from professor Lee's book: The map $F\colon G\times G\to G\times G$ defined by $F(g,h) = (g,gh)$ is is a bijective local diffeomorphism.
By taking advantage of the result here(this is the problem 7-2 of Lee's book), I can prove that $dF_{(e,e)}$ is an isomorphism from $T_eG\oplus T_eG$ to itself sending $(X,Y)$ to $(X,X+Y)$, which implies that $F$ is a local diffeomorphism at $(e,e)$. But I don't know how to use this result to obtain that $F$ is local diffeomorphism everywhere. And I also find it hard to prove that $dF_{(g,h)}$ is an isomorphism for general element $(g,h) \in G\times G$.
Thanks in advance!

Let $L_g(h) = gh$ be left-multiplication by $g$, which is a diffeomorphism. Similarly, $R_g(h) = hg$ is a diffeomorphism. You know that $F$ is a diffeomorphism when restricted to some open neighborhood $U$ of $(e,e)$; then $$F|_{(g,h) \cdot U} = (L_g \times L_gR_h) \circ F|_U \circ (L_{g^{-1}} \times R_{h^{-1}}),$$ since the right hand side maps $$(x,y) \mapsto (g^{-1}x, yh^{-1}) \mapsto (g^{-1}x, g^{-1}xyh^{-1}) \mapsto (x, xy),$$ shows that $F$ is a local diffeomorphism on $(g,h) \cdot U$. So it's a bijective map that's a local diffeomorphism everywhere, which means it's a true diffeomorphism.
