User:Dfeuer/OR3

Theorem
Let $\left({R, +, \circ, \le}\right)$ be an ordered ring with zero $0_R$.

Let $x \in R$.

Then the following equivalences hold:


 * $x \le 0 \iff 0_R \le (-x)$
 * $x \le 0 \iff 0_R \le (-x)$
 * $x < 0 \iff 0_R < (-x)$
 * $x < 0 \iff 0_R < (-x)$

Proof
By the definition of an ordered ring, $\left({R, +, \le}\right)$ is an ordered group.

Thus by User:Dfeuer/Group Inverse Reverses Ordering in Ordered Group, the stated equivalences hold.