Integer Multiplication is Closed

Theorem
The set of integers is closed under multiplication.

Proof
Integer multiplication is defined as:

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

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

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

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

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

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

Therefore integer multiplication is closed.