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

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

This follows because:

$$ $$ $$ $$ $$ $$

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

$$ $$ $$

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


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

Hence the result.