Suppose $E$ is a smooth vector bundle over $M$. Show that the projection map $\pi :E \to M$ is a surjective smooth submersion.
Since $E$ is a smooth vector bundle for each $p \in M$ there exists a neighborhood $U$ of $p$ such that $\varphi: \pi^{-1}(U) \to U \times \Bbb R^k$ is a diffeomorphism and that $\pi_1 \circ \varphi = \pi$.
In order for $\pi$ to be a smooth submersion the differential $d\pi_p : T_p E \to T_{\pi(p)} M$ must be surjective.
Since we have the characterization $\pi_1 \circ \varphi = \pi$ we get that $$d\pi_{p} = d(\pi_1 \circ \varphi)_p = d\pi_{1_{\varphi(p)}} \circ d\varphi_{p}.$$
Now I need to show that $d\pi_{1_{\varphi(p)}}$ and $d\varphi_{p}$ surjective maps in order to conclude the proof.
Since $\varphi$ is a diffeomorphism we have that $\varphi^{-1}$ is a smooth bijection. So defining $d\varphi^{-1}_{q} : T_{q}(U \times \Bbb R^k) \to T_p \pi^{-1}(U)$ we have that $$d\varphi_p \circ d\varphi^{-1}_{q} = d(\varphi \circ \varphi^{-1})_{q} = d(id)_{q}$$
so $d\varphi^{-1}_{\varphi(p)}$ is an inverse for $d\varphi_p$ and so it's bijective and in particular surjective.
Similarly for $d\pi_{1_p}:T_p(U \times \Bbb R^k) \to T_{\pi_1(p)}U$ define $d\iota_{U_q} : T_qU \to T_{\iota_U(q)}(U \times \Bbb R^k)$ where $\iota_U:U \to U \times \Bbb R^k$ is the right inverse of $\pi_1$. Then we get that $$d\pi_{1_p} \circ d\iota_{U_q} = d(\pi_1 \circ \iota_U)_q=d(id)_q$$ which by same reasoning makes $d\pi_1$ surjective.
Is there some problems with the proofs for the surjectivities for these differentials? I'm not very confident about them.