Integer Multiplication is Well-Defined

Theorem
Integer multiplication is well-defined.

Proof
From the definition of integers, an integer is an element of the quotient structure $$\left({\frac {\mathbb{N} \times \mathbb{N}} {\boxminus}, \oplus_{\boxminus}}\right)$$ defined by $$\boxminus$$, where:


 * $$\boxminus$$ is the congruence relation defined on $$\mathbb{N} \times \mathbb{N}$$ by $$\left({x_1, y_1}\right) \boxminus \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$$;


 * $$\oplus$$ is the operation on $\mathbb{N} \times \mathbb{N}$ induced by $+$ on $\mathbb{N}$.

Let us denote the equivalence class of $$\left({a, b}\right) \in \mathbb{N}\times \mathbb{N}$$ by $$\left[\left[{a, b}\right]\right]_\boxminus$$.

We need to show that:

$$\left[\left[{a, b}\right]\right]_\boxminus = \left[\left[{p, q}\right]\right]_\boxminus \land \left[\left[{c, d}\right]\right]_\boxminus = \left[\left[{r, s}\right]\right]_\boxminus \Longrightarrow \left[\left[{a, b}\right]\right]_\boxminus \times \left[\left[{c, d}\right]\right]_\boxminus = \left[\left[{p, q}\right]\right]_\boxminus \times \left[\left[{r, s}\right]\right]_\boxminus$$

We have $$\left[\left[{a, b}\right]\right]_\boxminus = \left[\left[{p, q}\right]\right]_\boxminus \land \left[\left[{c, d}\right]\right]_\boxminus = \left[\left[{r, s}\right]\right]_\boxminus \iff a + q = b + p \land c + s = d + r$$ by the definition of $$\boxminus$$.

From the definition of integer multiplication, we have:

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

So, suppose that $$\left[\left[{a, b}\right]\right]_\boxminus = \left[\left[{p, q}\right]\right]_\boxminus$$ and $$\left[\left[{c, d}\right]\right]_\boxminus = \left[\left[{r, s}\right]\right]_\boxminus$$.

Both $$+$$ and $$\times$$ are commutative and associative on $$\mathbb{N}$$. Thus:

So we have $$ac + bd + ps + qr = ad + bc + pr + qs$$ and so, by the definition of $$\boxminus$$, we have:

$$\left[\left[{ac + bd, ad + bc}\right]\right]_\boxminus = \left[\left[{pr + qs, ps + qr}\right]\right]_\boxminus$$

So, by the definition of integer multiplication, this leads to:

$$\left[\left[{a, b}\right]\right]_\boxminus \times \left[\left[{c, d}\right]\right]_\boxminus = \left[\left[{p, q}\right]\right]_\boxminus \times \left[\left[{r, s}\right]\right]_\boxminus$$.

Thus integer multiplication has been shown to be well-defined.