Let $P$ be a smooth manifold, $G$ a Lie group, $\alpha:P\times G\to P$ a smooth action and $p:P\to P/G$ a smooth principal $G$-bundle. Then, we have the sequence
$$ G \xrightarrow{\alpha_a} P \xrightarrow{\pi} P/G$$
I'm trying to prove that the induced sequence in the tangent spaces is exact:
$$ 0\rightarrow T_eG \xrightarrow{T_e\alpha_a} T_aP \xrightarrow{T_a\pi} T_{[a]}P/G\rightarrow 0$$
In particular, the problem is proving that $T_a\pi$ is surjective and that $kerT_a\pi\subseteq ImT_e\alpha_a$, so any help would be greatly appreciated.
I proved the injectivity of $T_e\alpha_a$ using that $\alpha_a$ is injective (since the action is free), and then it's jacobian matrix is injective, but suggestions are also welcome.
Cheers