Proof Verification: A differentiable aplication $\psi : M\to N$ is differentiable if and only if: $\psi^{*}f\in C^{\infty}(M)$

Question

Show that a differentiable aplication $\psi$ over $M$ to a differentiable variety $N$ is differentiable if and only if: $$\psi^{*}f\in C^{\infty}(M)$$ For: $f\in C^{\infty}(N)$

Where $\psi^{*}f$ is the pull back of $f$ over $\psi$ and $C^{\infty}(N)$ is the set of all differentiable functions of $N$ over $\mathbb R$

What i have done: Knowing that:

1. Let $(U,\phi)$ and $(V,\chi)$ be two n-charts over $M$. And $g:M\to \mathbb R$ When the n-charts are $C^{\infty}$-related, the following identities: $$g\circ \phi^{-1}=(g\circ \chi^{-1})\circ(\chi \circ \phi^{-1})$$ $$g\circ \chi^{-1}=(g\circ \phi^{-1})\circ(\phi \circ \chi^{-1})$$ Then $g\circ \phi^{-1}$ is differentiable if and only if $g\circ \chi^{-1}$ is.
2. The pull back si defined as: $$\psi^{*}f=f\circ \psi$$

Answer 1

Proof Let $\varphi : M\to N$ and $f\in C^{\infty}(N)$:

The relation:

$$(\psi^{*}f)\circ \phi^{-1}=(f\circ \chi^{-1})\circ(\chi \circ \psi \circ \phi^{-1})$$ It follows: $$\psi^{*}f\in C^{\infty}(M)$$