Divisor Relation is Antisymmetric/Corollary/Proof 2

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Corollary to Divisor Relation is Antisymmetric

Let $a, b \in \Z$.

If $a \divides b$ and $b \divides a$ then $a = b$ or $a = -b$.


Proof

Let $a \divides b$ and $b \divides a$.

Then by definition of divisor:

$\exists c, d \in \Z: a c = b, b d = a$

Thus:

\(\displaystyle a c d\) \(=\) \(\displaystyle a\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle c d\) \(=\) \(\displaystyle 1\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle d\) \(=\) \(\displaystyle \pm 1\) Divisors of One
\(\displaystyle \leadsto \ \ \) \(\displaystyle a\) \(=\) \(\displaystyle \pm b\) as $a = b d$

$\blacksquare$


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