Natural Numbers with Divisor Operation is Isomorphic to Subgroups of Integer Multiples under Inclusion

Theorem
Let $\N_{>0}$ denote the set of strictly positive natural numbers.

For $n \in \N_{>0}$, let $n \Z$ denote the set of integer multiples of $n$.

Let $\struct {\Z, +}$ denote the additive group of integers.

Let $\mathscr G$ be the set of all subgroups of $\struct {\Z, +}$.

Consider the algebraic structure $\struct {\N_{>0}, \divides}$, where $\divides$ denotes the divisor operator:
 * $a \divides b$ denotes that $a$ is a divisor of $b$

Let $\phi: \struct {\N_{>0}, \divides} \to \struct {\mathscr G, \supseteq}$ be the mapping defined as:
 * $\forall n \in \N_{>0}: \map \phi n = n \Z$

Then $\phi$ is an order isomorphism.

Proof
We note that from Subgroups of Additive Group of Integers, the subgroups of $\struct {\Z, +}$ are precisely the sets of integer multiples $n \Z$, for $n \in \N_{>0}$.

For each $n \in \N_{>0}$, there is a unique $n \Z \in \mathscr G$.

Hence $\phi$ is a bijection.

It remains to be demonstrated that $\phi$ is order-preserving in both directions.

Thus:

Then:

Hence the result.