use the definition for a ring to prove that Z7 is a ring under the operations

use the definition for a ring to prove that Z7 is a ring under the operations + and x as defined as follows: [a]7+[b]7 = [a+b]7 and [a]7 x [b]7 = [a x b]7 1. state each step of your proof 2. provide written justification for each step. But First I must use the six definitions of a ring to prove that z7 satisfies them. A ring is a set R equipped with two binary operations,1 here denoted by + and *, that have the following properties. 1. (a + b) + c = a + (b + c) for all a, b and c in R (addition is associative).
2. a + b = b + a for all a and b in R (addition is commutative). 3. There is an element 0 ∈ R such that 0 + x = x + 0 = x for all x ∈ R. 4. For each element x ∈ R, there is a unique element y ∈ R such that x + y = y + x = 0. (We denote y by −x.) 5. (a * b)* c = a * (b * c) for all a, b, and c in R (multiplication is associative). 6. (The distributive law) a * (b + c) = a * b + a * c and (b + c)* a = b * a + c * a for all a, b, and c in R. So I must have SIX proofs proving these, before I can answer a and b of the question