Relation Induced by Strict Positivity Property is Compatible with Multiplication

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

Let the relation $<$ be defined on $D$ as:

$\forall a, b \in D: a < b \iff \map P {-a + b}$

Then $<$ is compatible with $\times$ in the following sense:

$\forall x, y, z \in D: x < y, \map P z \implies \paren {z \times x} < \paren {z \times y}$
$\forall x, y, z \in D: x < y, \map P z \implies \paren {x \times z} < \paren {y \times z}$


If $x < y$ then $\map P {-x + y}$.


\(\ds \) \(\) \(\ds \map P {-x + y}, \, \map P z\)
\(\ds \) \(\leadsto\) \(\ds \map P {z \times \paren {-x + y} }\) Definition of Strict Positivity Property
\(\ds \) \(\leadsto\) \(\ds \map P {-z \times x + z \times y}\) Distributivity of $\times$ over $+$
\(\ds \) \(\leadsto\) \(\ds z \times x < z \times y\)

The other result follows from the fact that $\times$ is commutative in an integral domain.