Relation Induced by Strict Positivity Property is Compatible with Addition/Corollary
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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 $+$.
$\blacksquare$