src/HOL/Multivariate_Analysis/Linear_Algebra.thy
changeset 54489 03ff4d1e6784
parent 54413 88a036a95967
child 54703 499f92dc6e45
equal deleted inserted replaced
54488:b60f1fab408c 54489:03ff4d1e6784
   335 
   335 
   336 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
   336 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
   337   by (simp add: bilinear_def linear_iff)
   337   by (simp add: bilinear_def linear_iff)
   338 
   338 
   339 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
   339 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
   340   by (simp only: scaleR_minus1_left [symmetric] bilinear_lmul)
   340   by (drule bilinear_lmul [of _ "- 1"]) simp
   341 
   341 
   342 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
   342 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
   343   by (simp only: scaleR_minus1_left [symmetric] bilinear_rmul)
   343   by (drule bilinear_rmul [of _ _ "- 1"]) simp
   344 
   344 
   345 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
   345 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
   346   using add_imp_eq[of x y 0] by auto
   346   using add_imp_eq[of x y 0] by auto
   347 
   347 
   348 lemma bilinear_lzero:
   348 lemma bilinear_lzero: