I need a little help understanding this proof. I'm new to this and I'm a little slow =/ I've added questions in italics next to the two places I'm unsure about.
Let F be a field. There is an element 0 F such that a + 0 = a for all a F. Assume that z F is such that a + z = a for all a F. Show that z = 0.
Suppose z F satisfies a + z = a for all a F.
Then in particular 0 + z = 0. so we're simply using a + z = a from the previous line and putting a = 0 here to get 0 + z = 0?
By the commutative law for addition 0 + z = z + 0.
As a + 0 = a for all a F by definition of 0 we moreover have z + 0 = z. I don't see how z + 0 = z can follow from a + 0 = a...I thought z + 0 = z was simply one of the axioms of a field (the neutral element for addition). I.e., I don't see how we have used 'a + 0 = a' at all in this proof.
Combining everything we get 0 = 0 + z = z + 0 = z as required.