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.

$\blacksquare$


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