Non-Zero Integers are Cancellable for Multiplication/Proof 2

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 $y, z \in \Z: y \ne z$.

\(\ds y\)

\(\ne\)

\(\ds z\)

\(\ds \leadstoandfrom \ \ \)

\(\ds y - z\)

\(\ne\)

\(\ds 0\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x \paren {y - z}\)

\(\ne\)

\(\ds 0\)

Ring of Integers has no Zero Divisors‎

\(\ds \leadstoandfrom \ \ \)

\(\ds x y - x z\)

\(\ne\)

\(\ds 0\)

Integer Multiplication Distributes over Subtraction

The result follows by transposition.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers