equal
deleted
inserted
replaced
566 from fold_image_related[where h="(\<lambda>m. a(m) mod n)" and g=a, OF th1 th3 fS, OF th4] show ?thesis unfolding setprod_def by (simp add: fS) |
566 from fold_image_related[where h="(\<lambda>m. a(m) mod n)" and g=a, OF th1 th3 fS, OF th4] show ?thesis unfolding setprod_def by (simp add: fS) |
567 qed |
567 qed |
568 |
568 |
569 lemma nproduct_cmul: |
569 lemma nproduct_cmul: |
570 assumes fS:"finite S" |
570 assumes fS:"finite S" |
571 shows "setprod (\<lambda>m. (c::'a::{comm_monoid_mult,recpower})* a(m)) S = c ^ (card S) * setprod a S" |
571 shows "setprod (\<lambda>m. (c::'a::{comm_monoid_mult})* a(m)) S = c ^ (card S) * setprod a S" |
572 unfolding setprod_timesf setprod_constant[OF fS, of c] .. |
572 unfolding setprod_timesf setprod_constant[OF fS, of c] .. |
573 |
573 |
574 lemma coprime_nproduct: |
574 lemma coprime_nproduct: |
575 assumes fS: "finite S" and Sn: "\<forall>x\<in>S. coprime n (a x)" |
575 assumes fS: "finite S" and Sn: "\<forall>x\<in>S. coprime n (a x)" |
576 shows "coprime n (setprod a S)" |
576 shows "coprime n (setprod a S)" |