Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative

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Theorem

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

Let $N$ be the (strict) negativity property on $D$:

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


Then for all $a \in D$:

$\map N a, \map P b \implies \map N {a \times b}$


Proof

\(\displaystyle \map N a, \map P b\) \(\leadsto\) \(\displaystyle \map P {-a}, \map P b\) Definition of Strict Negativity Property
\(\displaystyle \) \(\leadsto\) \(\displaystyle \map P {\paren {-a} \times b}\) Strict Positivity Property: $(P \, 2)$
\(\displaystyle \) \(\leadsto\) \(\displaystyle \map P {-\paren {a \times b} }\) Product with Ring Negative
\(\displaystyle \) \(\leadsto\) \(\displaystyle \map N {a \times b}\) Definition of Strict Negativity Property

$\blacksquare$


Sources