Integer Multiplication is Closed

Theorem
The set of integers is closed under multiplication.

Proof
Integer multiplication is defined as:

$$\forall a, b, c, d \in \mathbb{N}: \left[\left[{a, b}\right]\right]_\boxminus \times \left[\left[{c, d}\right]\right]_\boxminus = \left[\left[{ac + bd, ad + bc}\right]\right]_\boxminus$$.

where $$\left[\left[{a, b}\right]\right]_\boxminus$$ is the equivalence class as defined in the definition of integers.

$$\forall a, b, c, d \in \mathbb{N}: \left[\left[{a, b}\right]\right]_\boxminus \in \mathbb{Z}, \left[\left[{c, d}\right]\right]_\boxminus \in \mathbb{Z}$$.

Also, $$\forall a, b, c, d \in \mathbb{N}: \left[\left[{a, b}\right]\right]_\boxminus \times \left[\left[{c, d}\right]\right]_\boxminus = \left[\left[{ac + bd, ad + bc}\right]\right]_\boxminus$$.

But $$ac + bd \in \mathbb{N}, ad + bc \in \mathbb{N}$$.

So $$\forall a, b, c, d \in \mathbb{N}: \left[\left[{ac + bd, ad + bc}\right]\right]_\boxminus \in \mathbb{Z}$$.

Therefore integer multiplication is closed.