Tangent bundle of a Lie Group is trivial

Hello,

Let G be a Lie Group and g his Lie Algebra, that is the tangent space at $\displaystyle e \in G$ I have to show that the map $\displaystyle \phi$: G x g -> TG, $\displaystyle \phi(h,X)=dL_h[X]$, (whereas

$\displaystyle dL_h[X](f)=X(f \circ L_h)$ with L_h left multiplication )

is a diffeomorphism and a linear isomorphism on each fibre.

I'm really hopeless! I don't know how i can show this property's of $\displaystyle \phi$.

Do you know some books or links, where i can read the proof? Or can you please explain it to me?

Regards