Congruence (Number Theory) is Congruence Relation

Theorem
Congruence modulo $m$ is a congruence relation on $\struct {\Z, +}$.

Proof
Suppose $a \equiv b \bmod m$ and $c \equiv d \bmod m$.

Then by the definition of congruence there exists $k, k' \in \Z$ such that:


 * $\paren {a - b} = k m$


 * $\paren {c - d} = k' m$

Hence:


 * $\paren {a - b} + \paren {c - d} = k m + k' m$

Using the properties of the integers:


 * $\paren {a + c} - \paren {b + d} = m \paren {k + k'}$

Hence $\paren {a + c} \equiv \paren {b + d} \bmod m$ and congruence modulo $m$ is a congruence relation.