In my Math book (Elementary Analysis by Kenneth Ross), it says to prove Theorem 3.1 (ii), that $a * 0 = 0$ for all a. However, the solution in the book is:
A3: $a + 0 = a$ for all a
DL: $a(b + c) = ab + ac$ for all a,b,c.
(i) $a + c = b + c$ implies $a = b$
We use A3 and DL to obtain $a * 0 = a * (0 + 0) = a * 0 + a * 0$, so $0 + a * 0 = a * 0 + a * 0$. By (i) we conclude $0 = a * 0$.
I understand how to prove (i). However, what confuses me is the starting point for this next proof. How do you know and where do you start in a mathematical proof? I've seen other proofs proving this theorem on here at StackExchange, but I want to understand how my book is proving it. One way I've seen that makes sense to me is starting with the distributive law:
$a(b + c) = ab + ac$
Let b and c = 0
$$ \begin{align*} a(0 + 0) &= a * 0 + a * 0 \\ a * 0 &= a * 0 + a * 0 \\ (a * 0) + (-a * 0) &= (a * 0 + a * 0) + (-a * 0) \\ 0 &= a * 0 + (a * 0 + (-a * 0)) \\ 0 &= a * 0 + 0 \\ 0 &= a * 0 \\ \end{align*} $$
This makes sense. However, in the above expression provided by the book, they start from $a * 0$ and get to $a * 0 + a * 0$. This makes sense. But, I don't understand where they're getting $0 + a * 0$ from. Also, is there a general guideline or tip to help one know where to start with a proof? Do we start with the equation/expression itself that we want to prove and manipulate it with the known axioms? Or do we start with an axiom and try to work our way back to the proposition/theorem?
EDIT: Is there any recommended books to help me learn proofs as a total beginner? It feels like every math book in college is written for people who already understand how proofs work and such.
When I read this I saw $a+b=b+c$ then $a=c$ as applied to the problem to mean $a=0$, $b=a*0$, and $c=a*0$. then in the $a+b=b+c$, $a=c$ would imply $(a=0)=(c=a*0)$, so $0=a*0$