Properties of Strict Negativity

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

Let $\le$ be the total ordering induced by $P$, and let $<$ be its strict total ordering counterpart.

Let $N$ be the negativity property on $D$:


 * $\forall a \in D: \map N a \iff \map P {-a}$

Then the following properties apply for all $a, b \in D$:
 * $(1): \quad \map N a \iff a < 0$
 * $(2): \quad \map P a \iff \map N {-a}$
 * $(3): \quad \map N a, \map N b \implies \map N {a + b}$
 * $(4): \quad \map N a, \map N b \implies \map P {a \times b}$
 * $(5): \quad \map N a, \map P b \implies \map N {a \times b}$