src/HOL/Analysis/Homeomorphism.thy
changeset 63918 6bf55e6e0b75
parent 63627 6ddb43c6b711
child 63945 444eafb6e864
equal deleted inserted replaced
63917:40d1c5e7f407 63918:6bf55e6e0b75
  1005   have *: "(\<Sum>a \<in> Basis. a \<bullet> basf b * (x \<bullet> basg a)) = x \<bullet> b" if "b \<in> Basis" for x b
  1005   have *: "(\<Sum>a \<in> Basis. a \<bullet> basf b * (x \<bullet> basg a)) = x \<bullet> b" if "b \<in> Basis" for x b
  1006     using surbf that by auto
  1006     using surbf that by auto
  1007   have gf[simp]: "g (f x) = x" for x
  1007   have gf[simp]: "g (f x) = x" for x
  1008     apply (rule euclidean_eqI)
  1008     apply (rule euclidean_eqI)
  1009     apply (simp add: f_def g_def inner_setsum_left scaleR_setsum_left algebra_simps)
  1009     apply (simp add: f_def g_def inner_setsum_left scaleR_setsum_left algebra_simps)
  1010     apply (simp add: Groups_Big.setsum_right_distrib [symmetric] *)
  1010     apply (simp add: Groups_Big.setsum_distrib_left [symmetric] *)
  1011     done
  1011     done
  1012   then have "inj f" by (metis injI)
  1012   then have "inj f" by (metis injI)
  1013   have gfU: "g ` f ` U = U"
  1013   have gfU: "g ` f ` U = U"
  1014     by (rule set_eqI) (auto simp: image_def)
  1014     by (rule set_eqI) (auto simp: image_def)
  1015   have "S homeomorphic U"
  1015   have "S homeomorphic U"