I think you're step 2 is incorrect, if you're working with ring axioms, you can say in general for a number (where R is a ring), there exists a neutral element (and for the addition: 0) wherefore: , that means in this case (for the addition): .

I would prove it this way:

(this is an axiom of the existence of a neutral element in a commutative group in a ring)

(putting 0 to the other side)

(in a ring the addition is commutative)

Because of the existence of the neutral element for the addition, like I stated earlier: therefore: