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.