src/HOL/Algebra/FiniteProduct.thy
changeset 67341 df79ef3b3a41
parent 67091 1393c2340eec
child 67613 ce654b0e6d69
equal deleted inserted replaced
67340:150d40a25622 67341:df79ef3b3a41
   500   with \<open>\<not> finite A\<close> show ?case by simp
   500   with \<open>\<not> finite A\<close> show ?case by simp
   501 qed (auto simp add: Pi_def)
   501 qed (auto simp add: Pi_def)
   502 
   502 
   503 lemma finprod_const:
   503 lemma finprod_const:
   504   assumes a [simp]: "a : carrier G"
   504   assumes a [simp]: "a : carrier G"
   505     shows "finprod G (%x. a) A = a (^) card A"
   505     shows "finprod G (%x. a) A = a [^] card A"
   506 proof (induct A rule: infinite_finite_induct)
   506 proof (induct A rule: infinite_finite_induct)
   507   case (insert b A)
   507   case (insert b A)
   508   show ?case 
   508   show ?case 
   509   proof (subst finprod_insert[OF insert(1-2)])
   509   proof (subst finprod_insert[OF insert(1-2)])
   510     show "a \<otimes> (\<Otimes>x\<in>A. a) = a (^) card (insert b A)"
   510     show "a \<otimes> (\<Otimes>x\<in>A. a) = a [^] card (insert b A)"
   511       by (insert insert, auto, subst m_comm, auto)
   511       by (insert insert, auto, subst m_comm, auto)
   512   qed auto
   512   qed auto
   513 qed auto
   513 qed auto
   514 
   514 
   515 (* The following lemma was contributed by Jesus Aransay. *)
   515 (* The following lemma was contributed by Jesus Aransay. *)