src/HOL/Algebra/Subrings.thy
changeset 73932 fd21b4a93043
parent 70160 8e9100dcde52
equal deleted inserted replaced
73866:66bff50bc5f1 73932:fd21b4a93043
   355     using subfield_m_inv[OF assms A(1)] by auto
   355     using subfield_m_inv[OF assms A(1)] by auto
   356   hence "inv k \<otimes> (k \<otimes> a) \<in> K"
   356   hence "inv k \<otimes> (k \<otimes> a) \<in> K"
   357     using k' A(3) subring_props(6) by auto
   357     using k' A(3) subring_props(6) by auto
   358   thus "a \<in> K"
   358   thus "a \<in> K"
   359     using m_assoc[of "inv k" k a] A(2) inv_k subring_props(1)
   359     using m_assoc[of "inv k" k a] A(2) inv_k subring_props(1)
   360     by (metis (no_types, hide_lams) A(1) Diff_iff l_one subsetCE)
   360     by (metis (no_types, opaque_lifting) A(1) Diff_iff l_one subsetCE)
   361 qed
   361 qed
   362 
   362 
   363 lemma (in ring) subfield_iff:
   363 lemma (in ring) subfield_iff:
   364   shows "\<lbrakk> field (R \<lparr> carrier := K \<rparr>); K \<subseteq> carrier R \<rbrakk> \<Longrightarrow> subfield K R"
   364   shows "\<lbrakk> field (R \<lparr> carrier := K \<rparr>); K \<subseteq> carrier R \<rbrakk> \<Longrightarrow> subfield K R"
   365     and "subfield K R \<Longrightarrow> field (R \<lparr> carrier := K \<rparr>)"
   365     and "subfield K R \<Longrightarrow> field (R \<lparr> carrier := K \<rparr>)"