Join of Sets of Integer Multiples is Set of Integer Multiples of GCD

Theorem
Let $m, n \in \Z$.

Let $m \Z$ denote the set of integer multiples of $m$

Let $r \in \Z$ such that:
 * $m \Z \subseteq r \Z$

and:
 * $n \Z \subseteq r \Z$

Then:
 * $\gcd \set {m, n} \Z \subseteq r \Z$

where $\gcd$ denotes greatest common divisor.

Proof
From Set of Integer Multiples is Integral Ideal, each of $m \Z$, $n \Z$, $r \Z$ and $\gcd \set {m, n} \Z$ are integral ideals.

By definition of Integral Ideal is Ideal of Ring, each is a subgroup of the additive group of integers $\struct {\Z, +}$.

Let $S$ be the join of $m \Z$ and $n \Z$:
 * $S = m \Z \vee n \Z$

Then by Join of Subgroups is Group Generated by Union:
 * $S = \gen {m \Z \cup n \Z}$

where $\gen {m \Z \cup n \Z}$ denotes the subgroup generated by $m \Z \cup n \Z$.

By definition of generator, $S$ is the smallest subgroup of $\Z$ containing $m \Z \vee n \Z$

Thus $S \subseteq r \Z$.

The result follows from Subgroup of Additive Group of Integers Generated by Two Integers.