User:Dfeuer/OR3

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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.