equal
deleted
inserted
replaced
1191 shows "\<lbrakk>S \<subseteq> T; dim T \<le> dim S\<rbrakk> \<Longrightarrow> span S = span T" |
1191 shows "\<lbrakk>S \<subseteq> T; dim T \<le> dim S\<rbrakk> \<Longrightarrow> span S = span T" |
1192 by (simp add: span_mono subspace_dim_equal) |
1192 by (simp add: span_mono subspace_dim_equal) |
1193 |
1193 |
1194 lemma dim_psubset: |
1194 lemma dim_psubset: |
1195 "span S \<subset> span T \<Longrightarrow> dim S < dim T" |
1195 "span S \<subset> span T \<Longrightarrow> dim S < dim T" |
1196 by (metis (no_types, hide_lams) dim_span less_le not_le subspace_dim_equal subspace_span) |
1196 by (metis (no_types, opaque_lifting) dim_span less_le not_le subspace_dim_equal subspace_span) |
1197 |
1197 |
1198 lemma dim_eq_full: |
1198 lemma dim_eq_full: |
1199 shows "dim S = dimension \<longleftrightarrow> span S = UNIV" |
1199 shows "dim S = dimension \<longleftrightarrow> span S = UNIV" |
1200 by (metis dim_eq_span dim_subset_UNIV span_Basis span_span subset_UNIV |
1200 by (metis dim_eq_span dim_subset_UNIV span_Basis span_span subset_UNIV |
1201 dim_UNIV dim_span dimension_def) |
1201 dim_UNIV dim_span dimension_def) |