Length of irreducible factorization is strictly multiplicatively monotone on unique factorization domain

Statement
Suppose $$R$$ is a fact about::unique factorization domain. Define $$N: R \setminus \{ 0 \} \to \mathbb{N}_0$$ as follows:


 * $$N(u) = 0$$ for any unit.
 * $$N(up_1p_2 \dots p_r) = r$$ where $$u$$ is a unit and $$p_i$$ are irreducible elements in $$R$$.

Note that this norm is well-defined because unique factorization guarantees that every non-unit can be written as a product of irreducibles and any two such expressions are equivalent up to ordering and associates.

Then, $$N$$ has the following properties:


 * $$N(ab) = N(a) + N(b)$$ for $$ab \ne 0$$.
 * $$N(a) = 0$$ if and only if $$a$$ is a unit.
 * $$N(ab) \ge N(a)$$ with equality occurring if and only if $$b$$ is a unit. (This last fact makes $$N$$ a fact about::strictly multiplicatively monotone norm).