src/HOL/Vector_Spaces.thy
changeset 73932 fd21b4a93043
parent 72302 d7d90ed4c74e
equal deleted inserted replaced
73866:66bff50bc5f1 73932:fd21b4a93043
  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)