"0" is defined as the "additive identity". That is, x+ 0= x for any number x.
We also have the "distributive law" for arithmetic: a(b+ c)= ab+ ac for any numbers, a, b, and c.
In particular a(b+ 0)= ab+ a0. But since b+ 0= b, a(b+ 0)= ab so ab= ab+ a0. Subtracting ab from both sides 0= a0.
