After reading an article by Peter Norvig about Beal's Conjecture I became pretty interested in digging into it. In his article here he goes into the use of a technique to optimize checking by this:
If $A^x + B^y = C^z$ and $p$ is some word-sized prime then $A^x \pmod p + B^y \pmod p = C^z \pmod p$.
I am aware of a few different properties of modular arithmetic with primes that are useful, but I do not see how this one is derived. I have a feeling that it involves the Chinese Remainder Theorem and some fiddling, but I couldn't find a proof of this equality anywhere and it's really starting to bother me.
Could someone tell me how Norvig arrived at this conclusion? I would really appreciate it. I've never seen such an equality before. Thank you!