How to prove this theorem using field axioms of real numbers?

Question

I need to prove that $0*a=a*0=0$ I'm guessing I have to use the existence of inverse axiom. But I'm not sure how the proof goes. Thanks a lot in advance!

Answer 1

$0*a = (0+0)*a = 0*a + 0*a$

Add the additive inverse of $0*a$ to both sides to get:

$0 = 0*a$

---

$a*0 = a*(0+0) = a*0 + a*0$

Add the additive inverse of $a*0$ to both sides to get:

$0 = a*0$