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Let M be an n-manifold with coordinate charts [tex](U_\alpha, \phi_\alpha)[/tex]. Therefore [tex](\pi^{-1}(U_\alpha), \tilde{\phi_\alpha})[/tex] are charts for TM where [tex]\pi[/tex] is the projection map and [tex]\tilde{\phi_\alpha}(p, v^i \frac{\partial}{\partial x^i}\vert_p) = (\phi(p),v^1, \ldots, v^n)[/tex]. I claim that the map F from TM to M x R^n given by [tex]F(p, v^i \frac{\partial}{\partial x^i}\vert_p) = (p, v^1, \ldots, v^n)[/tex] is a diffeomorphism. Clearly F is bijective so it is sufficient to check that F is a local diffeomorphism. Thus let [tex](p, v^i \frac{\partial}{\partial x^i}\vert_p)[/tex] be an arbitrary point in TM. [tex]p \in U_\alpha[/tex] for some [tex]\alpha[/tex] so [tex]\pi^{-1}(U_\alpha)[/tex] is an open set (indeed a chart) of TM containing [tex](p, v^i \frac{\partial}{\partial x^i}\vert_p)[/tex] and [tex]F(\pi^{-1}(U_\alpha)) = U_\alpha \times R^n[/tex] is a chart of M x R^n. But the restriction of F to [tex]\pi^{-1}(U_\alpha)[/tex] is [tex](\phi_\alpha^{-1} \times Id_{R^n}) \circ \tilde{\phi_\alpha}[/tex], which is a diffeomorphism (being a composition of diffeomorphisms).

Where did I go wrong?