Sorry that I sounded dismissive. It's a nice proof, and it wasn't obvious to me.

I am uncomfortable with using Pearson correlation to mean correlation. Consider y=sin(x), dy/dx = cos(x). These are "uncorrelated" according to Pearson correlation, but given one, there are at most 2 possibilties for the other. So knowing one gives you almost complete info about the other. So calling them "independent" seems wrong.