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Given smooth manifold $M$ how do you prove that the projection map $\pi : TM\to M$, $(p,v)\mapsto p$ is smooth?

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    Let me go back to @Pierre-Yves' comment: once you have chosen an atlas on $TM$ and another on $M$ such that in local coordinates $\pi:(p,v)\mapsto p$, then you are done, as obviously coordinate projections $\mathbb{R}^{2k} \to \mathbb{R}^k$ are smooth maps. So if you defined the smooth structure on $TM$ using the charts you gave above, there is pretty much nothing left to prove. What you need to think about is "why am I guaranteed to be able to find local coordinates such that $\pi$ is given as the canonical projection?"2012-03-26

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There are nice charts for $M$ and $TM$, given a chart $x$ for $M$: $ x:U\rightarrow V \text{ and the induced } Tx:TU\rightarrow U \times \mathbb R^n $ One way to show that a map is smooth, is to express it in charts, which typically is written $y\circ f \circ x^{-1}$ or something like that. And it is enough to show smoothness for one pair of charts around $p$ and $f(p)$ for every point $p$. For $\pi$ and the nice charts this is just $ x \circ \pi \circ (Tx)^{-1} $ which is the projection $U \times \mathbb R^n \rightarrow U$