Real Number Ordering is Compatible with Multiplication/Negative Factor

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Theorem

$\forall a, b, c \in \R: a < b \land c < 0 \implies a c > b c$


where $\R$ is the set of real numbers.


Proof

From Real Numbers form Ordered Integral Domain, $\struct {\R, +, \times, \le}$ forms an ordered integral domain.

Thus:

\(\displaystyle a\) \(<\) \(\displaystyle b\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle b - a\) \(>\) \(\displaystyle 0\) Definition of Positivity Property
\(\displaystyle \leadsto \ \ \) \(\displaystyle c \times \paren {b - a}\) \(<\) \(\displaystyle 0\) Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative
\(\displaystyle \leadsto \ \ \) \(\displaystyle b \times c - a \times c\) \(<\) \(\displaystyle 0\) Ring Axioms: Product is Distributive over Addition
\(\displaystyle \leadsto \ \ \) \(\displaystyle a \times c\) \(>\) \(\displaystyle b \times c\) Definition of Positivity Property

$\blacksquare$


Sources