Exact sequence of tangent spaces of principal $G$-bundles

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

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I like this question. What have you tried to prove that $T_a\pi$ is surjective? – Sanath K. Devalapurkar Jul 2 '14 at 0:00
I think I have it: Consider a local trivialization $(\varphi_\alpha,U_\alpha)$ of $P/G$ such that $p\circ\varphi_\alpha = proj_1$. If $[a]\in U_\alpha$, and we consider any curve $\gamma$ through $[a]$ contained in $U_\alpha$ then, by choosing any smooth section $s:U_\alpha\to p^{-1}(U_\alpha)$, ... – hjhjhj57 Jul 2 '14 at 0:31
... we can define another curve in $P$ given by $s\circ\gamma$. Finally $T_a\pi[s\circ\gamma] = [\pi\circ s \circ\gamma] = [\gamma]$. (I'm using germs of curves as tangent vectors). Is the proof right? – hjhjhj57 Jul 2 '14 at 0:35
Seems so! ${}{}{}{}{}{}{}{}{}{}{}$ – Sanath K. Devalapurkar Jul 2 '14 at 0:46