How to determine logical form and / or type of proof? (beginner) I have some very confusion when analysing the logical form / structure of a proof.
In Basic Mathematics, there is a proof of the rule $0a = 0$ that is done by using the rules of addition and multiplication and the rule $1a = a$
It begins on one line:
$$0a + a = 0a + 1a = (0 +1)a = 1a = a$$
Is the first statement $0a + a= 0a +1a$? The reason he can  begin with this is because of the previously established rule that $1a = a$? Is this basically a biconditional, one object in and of itself? His second statement would be $0a + 1a = (0 + 1)a$ and he can justify it by distributivity? Sometimes these inline equations mess me up, I used to think of the leftmost side as continually 'transforming' after each equal sign. Is it actually a series of statements like I described above?
I have found as I've been learning more I've been questioning everything and possibly overthinking things, is this common when people are introduced to logic and proofs?
 A: Overthinking is a good thing when you start learning. The chained equations really do represent a sequence of deductions. Here is a very wordy way to spell that out:

To prove $0a = 0$ I will start with
$$ 1a =a \quad \text{    (one of
 the vector space axioms)}. 
$$ Then the axiom that says scalar product
distributes says $$ 0a + a = 0a + 1a = (0+1)a . $$ Then because $0+1 =
 1$ (which I would usually not bother making explicit) $$  (0+1)a = 1a
 = a $$ so $$ 0a + a = a. $$ Since we know there is a vector $-a$ to add to $a$ to get $0$ we can add it to both sides and use the
associativity of vector addition to conclude that $$ 0a + 0 = 0 $$ but
of course on the left side $$ 0a + 0 = 0a $$ so we're done.

As you learn to write proofs you will gradually come to know when a step needs words, when not.
I like to tell my students that the purpose of a proof isn't to convince me - I know the assertion is true - but to convince me that you have convinced yourself for good reason. How much you have to write to convey that evolves. In general, more words are better.
