Divisor Relation on Positive Integers is Partial Ordering

Theorem
"Divides" is a partial ordering of $\Z^*_+$.

Proof
Checking in turn each of the critera for an ordering:

Reflexivity

 * $\forall n \in \Z: n \backslash n$ from Integer Divisor Results‎.

Transitivity

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

This follows because:

Antisymmetry
We have $\forall a, b \in \Z: a \backslash b \land b \backslash a \implies \left\vert{a}\right\vert = \left\vert{b}\right\vert$ which follows from Integer Absolute Value Greater than Divisors:

If we restrict ourselves to the domain of positive integers, we can see:


 * $\forall a, b \in \Z^*_+: a \backslash b \land b \backslash a \implies a = b$

Hence the result.