User:Dfeuer/OR1

Theorem

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

Let $x, y, z \in R$.

Then the following equivalences hold:

$x \le y \iff x + z \le y + z$

$x \le y \iff z + x \le z + y$

$x < y \iff x + z < y + z$

$x < y \iff z + x < z + y$

Proof

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

Thus by User:Dfeuer/OG1, the stated equivalences hold.