Determining whether a given mapping between tangent bundle $TU$ and $\phi(U)\times\mathbb{R}$ is also a diffeomorphism Let $M$ be an $n$-dimensional smooth manifold, $(U,\phi) = (U,x^1\dots,x^n)$ a coordinate chart and $c^1,\dots,c^n$ coefficients. Then at the point $p\in M$, any $v \in T_pU = T_pM$ can be written as $v = \sum_{i=1}^nc^i \frac{\partial}{\partial x^i}\bigg|_p$. Given Loring's (An Introduction to Manifolds, pp.130-131) homeomorphism $\psi:TU\to \phi(U)\times \mathbb{R}^n$, $\psi(v) = (x^1(p),\dots,x^n(p),c^1(v),\dots,c^n(v))$ and its inverse $\psi^{-1}(v) = (\phi(p),c^1,\dots,c^n) = \sum c^i \frac{\partial}{\partial x^i}\bigg|_p$, I'd like to know whether this mapping is also a diffeomorphism?
Bijectivity and smoothness of $\psi$ are evident (smootness is due to the smoothness of $\phi$ and the projection mappings $c^1,\dots,c^n$). And while it is easy to argue about the partial derivatives of $\psi^{-1}$ w.r.t. $c^1,\dots,c^n$, I've hit a brick while trying to argue about the partial derivative w.r.t. $\phi(p)$. Namely how can you argue that the evaluation mapping $p \mapsto \frac{\partial}{\partial x^i}\bigg|_p$ is smooth in this context, where the partial derivatives represent the basis vectors of $TU$ at the point $p$?
 A: To talk about smoothness or smoothness of the inverse of the map $\psi:TU\to \phi(U)\times \mathbb{R}^n$, you need a differentiable structure on $TU$. A few pages later, the differentiable structure on $TM$ is exactly defined in such a way, that such all these maps $\psi$ become coordinate charts for $TM$. Then $TU$ is an open submanifold of $TM$ and hence $\psi$ is a (global) coordinate chart for $TU$. Coordinate charts are allways diffeomorphisms. So yes, $\psi$ is a diffeomorphism, but this is true more or less directly by definition.
$\textbf {Edit}:$
In my version of the book (2. Edition) a vectorfield is defined to be smooth iff its smooth as a map $X:M\to TM$ (Def. 12.7). If $\phi$ and $\psi$ are as above then $\psi\circ X_{|U}:U\to\phi(U)\times\mathbb R^n$, which is of the form $(\phi,c)$ with $c:U\to\mathbb R^n$. From this it follows, that $X_{|U}$ is smooth iff its component functions $c_i$ are. In particular the locally defined vectorfields $p \mapsto \frac{\partial}{\partial x^i}\bigg|_p$ are smooth since there the component functions $c_j\equiv \delta_{ij}$ are constant.
