Identity Function is Completely Multiplicative

Theorem

Let $I_{\Z_{>0}}: \Z_{>0} \to \Z_{>0}$ be the identity function:

$\forall n \in \Z_{>0}: I_{\Z_{>0}} \left({n}\right) = n$

Then $I_{\Z_{>0}}$ is completely multiplicative.


The validity of the material on this page is questionable.
In particular: Completely multiplicativity is defined for fields but $\Z_{>0}$ is not a field
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Proof

$\forall m, n \in \Z_{>0}: I_{\Z_{>0}} \left({m n}\right) = m n = I_{\Z_{>0}} \left({m}\right) I_{\Z_{>0}} \left({n}\right)$

$\blacksquare$

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