I have a question about the proof of Rudin Theorem 10.20
Theorem 10.20
(a)If $\omega, \lambda $ are $k-$ and $m-$ forms, respectively of class $C^{1}$ in $E$, then $$d(\omega \land \lambda)=(d\omega)\land \lambda + (-1)^{k}\omega \land d\lambda$$
(b) If $\omega$ is of class $C^{1}$, then $d^{2} \omega=0$
I have two questions regarding the proof
First, $\omega=\sum b_{I}(x)dx_{I}$ and $\lambda=\sum c_{J}(x)dx_{J}$, but the proof said (a) follows if (a) is proved for the special case $\omega=fdx_{I}$ and $\lambda=gdx_{J}$, Why?
Second, the proof said $d(dx_{I})=0$ since $d\omega=\sum db_{I} \land dx_{I}$, I think if we let $\omega=dx_{I}$, by this definition, we get $d(dx_{I})=0\land dx_{I}$ , is $0\land dx_{I}=0$? if so, how to prove it?
Thanks in advance!