Integer Divisor Results/Divisors of Negative Values

Theorem
Let $m, n \in \Z$, i.e. let $m, n$ be integers.
 * $m \mathrel \backslash n \iff -m \divides n \iff m \divides -n \iff -m \divides -n$

That is, $m$ divides $n$ $-m$ divides $n$  $m$ divides $-n$  $-m$ divides $-n$.

Proof
Let $m \mathrel \backslash n$.

From Integer Divides its Negative, we have $-m \divides m$.

From Divisor Relation on Positive Integers is Partial Ordering it follows that $-m \divides n$.

From Integer Divides its Negative, we have $n \divides -n$.

From Divisor Relation on Positive Integers is Partial Ordering it follows that $m \divides -n$.

From above, we have $-m \divides n$.

From Integer Divides its Negative, we have $n \divides -n$.

From Divisor Relation on Positive Integers is Partial Ordering it follows that $-m \divides -n$.

The reverse implications follow similarly.