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

Corollary to Natural Numbers with Divisor Operation is Isomorphic to Subgroups of Integer Multiples under Inclusion
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 $\struct {\Z, +}$.

Let $\struct {\mathscr G, \times_\PP, \supseteq}$ be the ordered structure such that the operation $\times_\PP$ is defined as:
 * $\forall n \Z, m \Z \in \mathscr G: n \Z \times_\PP m \Z := \paren {n m} \Z$

Consider the ordered semigroup $\struct {\N_{>0}, \times, \divides}$, where:
 * $\divides$ denotes the divisor operator:
 * $a \divides b$ denotes that $a$ is a divisor of $b$
 * $\times$ denotes integer multiplication.

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

Then $\phi$ is an ordered semigroup isomorphism.

Proof
Recall that from Non-Zero Natural Numbers under Multiplication with Divisibility forms Ordered Semigroup, $\struct {\N_{>0}, \times, \divides}$ is indeed an ordered semigroup.

From Natural Numbers with Divisor Operation is Isomorphic to Subgroups of Integer Multiples under Inclusion, we have that $\phi: \struct {\N_{>0}, \divides} \to \struct {\mathscr G, \supseteq}$ is an order isomorphism.

We need to ascertain that $\struct {\mathscr G, \times_\PP, \supseteq}$ is an ordered semigroup.

Then we need to establish that $\phi: \struct {\N_{>0}, \times} \to \struct {\mathscr G, \times_\PP}$ is a semigroup isomorphism.