equal
deleted
inserted
replaced
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>)" |