# Thread: Proving Theorems with Axioms

1. ## Proving Theorems with Axioms

Apologies for the lack of a better way to state the problem I'm having right now.

I'm trying to prove 2 theorems given by my professor:

1. ∀a∈ ℝ (Real numbers), $\displaystyle (-1)*a = -a$

2. If a, b ∈ (real numbers) and a =/= 0 and b =/= 0, then a*b =/=. If a, b ∈ (real numbers) and a*b = 0 and a=/= 0, then b = 0

For the first one, I understand that I need to use the axiom which states: 1*a = a, but I don't know how to continue.

The second one, honestly, confuses me and I don't know where to start.

2. For the first, use the distributive property: 0=a*0=a(1-1).

For the second, suppose ab=0, then multiply on the left by 1/a.

3. Originally Posted by Tinyboss
For the first, use the distributive property: 0=a*0=a(1-1).

For the second, suppose ab=0, then multiply on the left by 1/a.
Thank you for the help. I have a question on what you stated.

How do you approach the step " 0 = a*0 = a(1 -1) ? I'm lost there.

4. Use the distributive property to continue: 0 = a(1-1) = a(1+(-1)) = a*1 + a*(-1) = a + a*(-1).