Integer Addition is Commutative/Proof 2

Theorem

The operation of addition on the set of integers $\Z$ is commutative:

$\forall x, y \in \Z: x + y = y + x$

Proof

Let $x = \eqclass {a, b} {}$ and $y = \eqclass {c, d} {}$ for some $x, y \in \Z$.

Then:

\(\ds x + y\)

\(=\)

\(\ds \eqclass {a, b} {} + \eqclass {c, d} {}\)

Definition of Integer

\(\ds \)

\(=\)

\(\ds \eqclass {a + c, b + d} {}\)

Definition of Integer Addition

\(\ds \)

\(=\)

\(\ds \eqclass {c + a, d + b} {}\)

Natural Number Addition is Commutative

\(\ds \)

\(=\)

\(\ds \eqclass {c, d} {} + \eqclass {a, b} {}\)

Definition of Integer Addition

\(\ds \)

\(=\)

\(\ds y + x\)

Definition of Integer

$\blacksquare$