But is it possible to adapt all lemmas accordingly?  I could imagine
that some statements use the fact that the support of multiplicity are
the prime numbers.
Not really. Some facts will still require the primality assumption, e.g.

lemma prime_multiplicity_mult_distrib:
  assumes "is_prime_elem p" "x ≠ 0" "y ≠ 0"
  shows   "multiplicity p (x * y) = multiplicity p x + multiplicity p y"


Cheers,

Manuel

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