src/HOL/Real/HahnBanach/Linearform.thy
changeset 27611 2c01c0bdb385
parent 25762 c03e9d04b3e4
child 27612 d3eb431db035
     1.1 --- a/src/HOL/Real/HahnBanach/Linearform.thy	Tue Jul 15 16:02:10 2008 +0200
     1.2 +++ b/src/HOL/Real/HahnBanach/Linearform.thy	Tue Jul 15 16:50:09 2008 +0200
     1.3 @@ -20,9 +20,10 @@
     1.4  declare linearform.intro [intro?]
     1.5  
     1.6  lemma (in linearform) neg [iff]:
     1.7 -  includes vectorspace
     1.8 +  assumes "vectorspace V"
     1.9    shows "x \<in> V \<Longrightarrow> f (- x) = - f x"
    1.10  proof -
    1.11 +  interpret vectorspace [V] by fact
    1.12    assume x: "x \<in> V"
    1.13    hence "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1)
    1.14    also from x have "... = (- 1) * (f x)" by (rule mult)
    1.15 @@ -31,9 +32,10 @@
    1.16  qed
    1.17  
    1.18  lemma (in linearform) diff [iff]:
    1.19 -  includes vectorspace
    1.20 +  assumes "vectorspace V"
    1.21    shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y"
    1.22  proof -
    1.23 +  interpret vectorspace [V] by fact
    1.24    assume x: "x \<in> V" and y: "y \<in> V"
    1.25    hence "x - y = x + - y" by (rule diff_eq1)
    1.26    also have "f ... = f x + f (- y)" by (rule add) (simp_all add: x y)
    1.27 @@ -44,9 +46,10 @@
    1.28  text {* Every linear form yields @{text 0} for the @{text 0} vector. *}
    1.29  
    1.30  lemma (in linearform) zero [iff]:
    1.31 -  includes vectorspace
    1.32 +  assumes "vectorspace V"
    1.33    shows "f 0 = 0"
    1.34  proof -
    1.35 +  interpret vectorspace [V] by fact
    1.36    have "f 0 = f (0 - 0)" by simp
    1.37    also have "\<dots> = f 0 - f 0" using `vectorspace V` by (rule diff) simp_all
    1.38    also have "\<dots> = 0" by simp