User:Dfeuer/OR2

Theorem

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

Let $x, y \in R$.

Then the following equivalences hold:

$x \le y \iff 0_R \le y + (-x)$

$x \le y \iff 0_R \le (-x) + y$

$x < y \iff 0_R < y + (-x)$

$x < y \iff 0_R < (-x) + y$

Proof

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

Thus by User:Dfeuer/OG2, the stated equivalences hold.