Ordering on Natural Numbers is Compatible with Multiplication/Corollary

From ProofWiki
Jump to navigation Jump to search

Corollary to Ordering on Natural Numbers is Compatible with Multiplication

Let $a, b, c, d \in \N$ where $\N$ is the set of natural numbers.

Then:

$a > b, c > d \implies a c > b d$


Proof

\(\displaystyle a\) \(>\) \(\displaystyle b\)
\((1):\quad\) \(\displaystyle \implies \ \ \) \(\displaystyle a c\) \(>\) \(\displaystyle b c\) Ordering on Natural Numbers is Compatible with Multiplication


\(\displaystyle c\) \(>\) \(\displaystyle d\)
\((2):\quad\) \(\displaystyle \implies \ \ \) \(\displaystyle b c\) \(>\) \(\displaystyle b d\) Ordering on Natural Numbers is Compatible with Multiplication


Finally:

\(\displaystyle a c\) \(>\) \(\displaystyle b c\) from $(1)$
\(\displaystyle b c\) \(>\) \(\displaystyle b d\) from $(2)$
\(\displaystyle \implies \ \ \) \(\displaystyle a c\) \(>\) \(\displaystyle b d\) Ordering on Natural Numbers is Trichotomy

$\blacksquare$


Sources