src/HOL/Analysis/Path_Connected.thy
changeset 69064 5840724b1d71
parent 68913 55b12fde48d0
child 69144 f13b82281715
equal deleted inserted replaced
69045:8c240fdeffcb 69064:5840724b1d71
  1735   obtain y where x: "x = p y" and y: "0 \<le> y" "y \<le> 1"
  1735   obtain y where x: "x = p y" and y: "0 \<le> y" "y \<le> 1"
  1736     using x by (auto simp: path_image_def)
  1736     using x by (auto simp: path_image_def)
  1737   show ?thesis
  1737   show ?thesis
  1738     unfolding path_component_def 
  1738     unfolding path_component_def 
  1739   proof (intro exI conjI)
  1739   proof (intro exI conjI)
  1740     have "continuous_on {0..1} (p \<circ> (( *) y))"
  1740     have "continuous_on {0..1} (p \<circ> ((*) y))"
  1741       apply (rule continuous_intros)+
  1741       apply (rule continuous_intros)+
  1742       using p [unfolded path_def] y
  1742       using p [unfolded path_def] y
  1743       apply (auto simp: mult_le_one intro: continuous_on_subset [of _ p])
  1743       apply (auto simp: mult_le_one intro: continuous_on_subset [of _ p])
  1744       done
  1744       done
  1745     then show "path (\<lambda>u. p (y * u))"
  1745     then show "path (\<lambda>u. p (y * u))"