src/HOL/Analysis/Connected.thy
changeset 69597 ff784d5a5bfb
parent 69544 5aa5a8d6e5b5
child 69611 42cc3609fedf
equal deleted inserted replaced
69596:c8a2755bf220 69597:ff784d5a5bfb
  5151 
  5151 
  5152 text\<open>But is the premise really necessary? Need to generalise @{thm euclidean_dist_l2}\<close>
  5152 text\<open>But is the premise really necessary? Need to generalise @{thm euclidean_dist_l2}\<close>
  5153 lemma Euclidean_dist_upper: "i \<in> Basis \<Longrightarrow> dist (x \<bullet> i) (y \<bullet> i) \<le> dist x y"
  5153 lemma Euclidean_dist_upper: "i \<in> Basis \<Longrightarrow> dist (x \<bullet> i) (y \<bullet> i) \<le> dist x y"
  5154   by (metis (no_types) member_le_L2_set euclidean_dist_l2 finite_Basis)
  5154   by (metis (no_types) member_le_L2_set euclidean_dist_l2 finite_Basis)
  5155 
  5155 
  5156 text\<open>But is the premise @{term \<open>i \<in> Basis\<close>} really necessary?\<close>
  5156 text\<open>But is the premise \<^term>\<open>i \<in> Basis\<close> really necessary?\<close>
  5157 lemma open_preimage_inner:
  5157 lemma open_preimage_inner:
  5158   assumes "open S" "i \<in> Basis"
  5158   assumes "open S" "i \<in> Basis"
  5159     shows "open {x. x \<bullet> i \<in> S}"
  5159     shows "open {x. x \<bullet> i \<in> S}"
  5160 proof (rule openI, simp)
  5160 proof (rule openI, simp)
  5161   fix x
  5161   fix x