Integer Divisor Results/Divisors of Negative Values

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

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

Proof
Let $m \backslash n$.

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

From Divides is Partial Ordering on Positive Integers it follows that $-m \mathop \backslash n$.

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

From Divides is Partial Ordering on Positive Integers it follows that $m \mathop \backslash -n$.

From above, we have $-m \mathop \backslash n$.

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

From Divides is Partial Ordering on Positive Integers it follows that $-m \mathop \backslash -n$.

The reverse implications follow similarly.