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Products of lists of prime elements. #

This file contains some theorems relating prime and products of lists.

theorem prime.dvd_prod_iff {M : Type u_1} [comm_monoid_with_zero M] {p : M} {L : list M} (pp : prime p) :
p (a : M) (H : a L), p a

Prime p divides the product of a list L iff it divides some a ∈ L

theorem prime.not_dvd_prod {M : Type u_1} [comm_monoid_with_zero M] {p : M} {L : list M} (pp : prime p) (hL : (a : M), a L ¬p a) :
theorem mem_list_primes_of_dvd_prod {M : Type u_1} [cancel_comm_monoid_with_zero M] [unique Mˣ] {p : M} (hp : prime p) {L : list M} (hL : (q : M), q L prime q) (hpL : p :
p L
theorem perm_of_prod_eq_prod {M : Type u_1} [cancel_comm_monoid_with_zero M] [unique Mˣ] {l₁ l₂ : list M} :
l₁.prod = l₂.prod ( (p : M), p l₁ prime p) ( (p : M), p l₂ prime p) l₁ ~ l₂