Let's use mod 5 arithmetic. You have 5 elements in the ring -- 0,1,2,3,4
The additive inverses are as follows:
1 + 4 = 0
2 + 3 = 0
So what happens if you multiply 2 * -3 (2 times the additive inverse of 3).
The additive inverse of 3 is just 2, so the answer is 2 * -3 = 2 * 2 = 4.
The other way to calculate it is to find the additive inverse of the product:
2 * -3 = -(2 * 3) = -(1) which is the additive inverse of 1: 4 again.
Here is your argument elaborated step by step.
STEP 1: First we want to show that ab is the additive inverse of (-a)b. This is Theorem 3 of the post.
STEP 2: Next we want to show that (-a)(-b) is the additive inverse of (-a)b. This follows similarly to the proof of Theorem 3: (-a)(-b) + (-a)(b) = (-a)(-b + b) = (-a)(0) and (-a)(0) = 0 by Theorem 2 of the post.
But nothing in the ring axioms directly says that the above results mean ab and (-a)(-b) must be equal. How do we know for sure that ab and (-a)(-b) are not two distinct additive inverses of (-a)b?
THEOREM 5: We now prove the uniqueness of additive inverse of an element from the ring axioms. Let b and c both be additive inverses of a. Therefore b = b + 0 = b + (a + c) = (b + a) + c = 0 + c = c.
Now from Steps 1 and 2, and Theorem 5, it follows that ab = (-a)(-b).
So what did we save in terms of intermediate theorems? Nothing! We no longer need Theorem 1 (inverse of inverse) of the post. But now we introduced Theorem 5 (uniqueness of additive inverse). We have exactly the same number of intermediate theorems with your approach.
JadeNB•7mo ago