src/HOL/Analysis/Inner_Product.thy
changeset 68611 4bc4b5c0ccfc
parent 68499 d4312962161a
child 68617 75129a73aca3
equal deleted inserted replaced
68609:9963bb965a0e 68611:4bc4b5c0ccfc
   151         by (simp add: inner_add inner_commute)
   151         by (simp add: inner_add inner_commute)
   152       show "0 \<le> norm x + norm y"
   152       show "0 \<le> norm x + norm y"
   153         unfolding norm_eq_sqrt_inner by simp
   153         unfolding norm_eq_sqrt_inner by simp
   154     qed
   154     qed
   155   have "sqrt (a\<^sup>2 * inner x x) = \<bar>a\<bar> * sqrt (inner x x)"
   155   have "sqrt (a\<^sup>2 * inner x x) = \<bar>a\<bar> * sqrt (inner x x)"
   156     by (simp add: real_sqrt_mult_distrib)
   156     by (simp add: real_sqrt_mult)
   157   then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
   157   then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
   158     unfolding norm_eq_sqrt_inner
   158     unfolding norm_eq_sqrt_inner
   159     by (simp add: power2_eq_square mult.assoc)
   159     by (simp add: power2_eq_square mult.assoc)
   160 qed
   160 qed
   161 
   161