# Non-Zero Integers are Cancellable for Multiplication/Proof 1

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## Theorem

Every non-zero integer is cancellable for multiplication.

That is:

- $\forall x, y, z \in \Z, x \ne 0: x y = x z \iff y = z$

## Proof

Let $x y = x z$.

There are two cases to investigate: $x > 0$ and $x < 0$.

Let $x > 0$.

From Natural Numbers are Non-Negative Integers, $x \in \N_{> 0}$.

By the Extension Theorem for Distributive Operations and Ordering on Natural Numbers is Compatible with Multiplication, $x$ is cancellable for multiplication.

$\Box$

Let $x < 0$.

We know that the Integers form Integral Domain and are thus a ring.

Then $-x > 0$ and so:

\(\displaystyle \paren {-x} y\) | \(=\) | \(\displaystyle -\paren {x y}\) | Product with Ring Negative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\paren {x z}\) | $\struct {\Z, +}$ is a group: Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {-x} z\) | Product with Ring Negative | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle y\) | \(=\) | \(\displaystyle z\) | from above: case where $x > 0$ |

$\Box$

So whatever non-zero value $x$ takes, it is cancellable for multiplication.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 20$: Theorem $20.10$