My question pertains to Ted Shifrin's Multivariable Mathematics Section 8-2.3 Pullbacks.
We are to use the following pieces, given in the definition, to build the pullback $\mathbf{g}^{*}\omega\in\mathcal{A}^{k}\left(U\right)$ of $\omega$ by $\mathbf{g}.$ The symbol $\mathbb{I}$ is a multi-index. The last four lines are definitive, giving the pullback of a function; the pullback of a basis 1-form; the pullback of a wedge product; and the pullback of a sum, respectively:
\begin{align*} U\subset & \mathbb{R}^{m}\\ \mathbf{g}: & U\to\mathbb{R}^{n}\\ f: & \mathbb{R}^{n}\to\mathbb{R}\\ \mathbf{g}\left(\mathbf{u}\right)= & \mathbf{x}\\ \mathbf{g}^{*}f= & f\circ\mathbf{g}\\ \mathbf{g}^{*}dx_{i}= & dg_{i}=\sum_{j=1}^{m}\frac{\partial g_{i}}{\partial u_{j}}du_{j}\\ \mathbf{g}^{*}\left(dx_{i_{1}}\wedge\dots\wedge dx_{i_{k}}\right)= & dg_{i_{1}}\wedge\dots\wedge dg_{i_{k}}=d\mathbf{g}_{\mathbb{I}}\\ \mathbf{g}^{*}\left(\sum_{\mathbb{I}}f_{\mathbb{I}}d\mathbf{x}_{\mathbb{I}}\right)= & \sum_{\mathbb{I}}\left(f_{\mathbb{I}}\circ\mathbf{g}\right)d\mathbf{g}_{\mathbb{I}} \end{align*}
Before the final equation he states "Last, we take the pullback of the sum to be the sum of the pullbacks."
It may seem pedantic, but this is mathematics. The last equation appears to define both the pullback of a sum, as well as the pullback of the product of a scalar function with a basis k-form. I'm left wondering if this is something I should derive from the previous parts of the definition, or it is simply an unverbalized part of the final part of the definition. So which is it?
To my way of thinking, that product "distribution" rule is the most significant part of the definition.