Divisor Relation in Integral Domain is Transitive

Theorem
Let $\left({D, +, \circ}\right)$ be an integral domain.

Let $x, y, z \in D$.

Then:
 * $x \mathop \backslash y \land y \mathop \backslash z \implies x \mathop \backslash z$

Corollary
"Divides" is a transitive relation on $\Z$, the set of integers.

Proof
Let $x \mathop \backslash y \land y \mathop \backslash z$.

Then from the definition of divisor, we have:


 * $x \mathop \backslash y \iff \exists s \in D: y = s \circ x$
 * $y \mathop \backslash z \iff \exists t \in D: z = t \circ y$

Then:
 * $z = t \circ \left({s \circ x}\right) = \left({t \circ s}\right) \circ x$

Thus:
 * $\exists \left({t \circ s}\right) \in D: z = \left({t \circ s}\right) \circ x$

and the result follows.

Proof of Corollary
Follows directly from the fact that Integers form Integral Domain.

Alternatively:


 * $\forall x, y, z \in \Z: x \mathop \backslash y \land y \mathop \backslash z \implies x \mathop \backslash z$

This follows because: