Pullback by a Smooth Map Commutes with the Exterior Derivative

Let $n,m\in$ $\mathbb{N}$ and $k\in\mathbb{N}\cup\{0\}$. Let $U\subseteq\mathbb{R}^n$ and $V\subseteq\mathbb{R}^m$ be open subsets of Euclidean space, let $F:U\to V$ be a smooth map, and let $\omega$ be a $C^1$ differential $k$-form on $V$. Then the pullback $F^{*}\omega$ is a $C^1$ differential $k$-form on $U$, and

$$d(F^{*}\omega)=F^{*}(d\omega),$$

where on the left $d$ denotes the exterior derivative on $U$ applied to $F^{*}\omega$, and on the right $d\omega$ is the exterior derivative of $\omega$ on $V$, whose pullback under $F$ is again taken in the sense of Pullback of a Differential Form by a C^1 Map.