General Positivity Rule in Ordered Integral Domain/Corollary

Corollary to General Positivity Rule in Ordered Integral Domain

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

Let $\map P x$ where $x \in D$.

Then:

$\map P {n \cdot x}$ and $\map P {x^n}$

Proof

From the definition of power of an element:

$\ds n \cdot x = \sum_{i \mathop = 1}^n x$

$\ds x^n = \prod_{i \mathop = 1}^n x$

The result then follows directly from General Positivity Rule in Ordered Integral Domain.

$\blacksquare$