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.

Let $c \in \gcd \set {m, n} \Z$.

By definition of integral ideal:
 * $\gcd \set {m, n} \divides c$

By Set of Integer Combinations equals Set of Multiples of GCD:


 * $\exists x, y \in \Z: c = x m + y n$

But as $m \Z \subseteq r \Z$ and $n \Z \subseteq r \Z$:
 * $m \in r \Z$ and $n \in \r Z$

Thus by definition of integral ideal:
 * $x m + y n \in r \Z$

So:
 * $c \in \gcd \set {m, n} \Z \implies c \in r \Z$

and the result follows by definition of subset.