Relation Induced by Strict Positivity Property is Compatible with Addition/Corollary

Corollary to Relation Induced by Strict Positivity Property is Compatible with Addition

Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property.

Let $\le$ be the relation defined on $D$ as:

$\le \ := \ < \cup \Delta_D$

where $\Delta_D$ is the diagonal relation.

Then $\le$ is compatible with $+$.

Proof

Let $a \le b$.

If $a \ne b$ then:

$a < b$

and Relation Induced by Strict Positivity Property is Compatible with Addition applies.

Otherwise $a = b$.

But $\struct {D, +}$ is the additive group of $\struct {D, +, \times}$ and the Cancellation Laws apply:

$a + c = b + c \iff a = b \iff c + a = c + b$

So $\le$ is seen to be compatible with $+$.

$\blacksquare$