src/HOL/Real/HahnBanach/Linearform.thy
 changeset 13547 bf399f3bd7dc parent 13515 a6a7025fd7e8 child 14254 342634f38451
```     1.1 --- a/src/HOL/Real/HahnBanach/Linearform.thy	Thu Aug 29 11:15:36 2002 +0200
1.2 +++ b/src/HOL/Real/HahnBanach/Linearform.thy	Thu Aug 29 16:08:30 2002 +0200
1.3 @@ -16,11 +16,9 @@
1.4    assumes add [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y"
1.5      and mult [iff]: "x \<in> V \<Longrightarrow> f (a \<cdot> x) = a * f x"
1.6
1.7 -locale (open) vectorspace_linearform =
1.8 -  vectorspace + linearform
1.9 -
1.10 -lemma (in vectorspace_linearform) neg [iff]:
1.11 -  "x \<in> V \<Longrightarrow> f (- x) = - f x"
1.12 +lemma (in linearform) neg [iff]:
1.13 +  includes vectorspace
1.14 +  shows "x \<in> V \<Longrightarrow> f (- x) = - f x"
1.15  proof -
1.16    assume x: "x \<in> V"
1.17    hence "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1)
1.18 @@ -29,21 +27,22 @@
1.19    finally show ?thesis .
1.20  qed
1.21
1.22 -lemma (in vectorspace_linearform) diff [iff]:
1.23 -  "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y"
1.24 +lemma (in linearform) diff [iff]:
1.25 +  includes vectorspace
1.26 +  shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y"
1.27  proof -
1.28    assume x: "x \<in> V" and y: "y \<in> V"
1.29    hence "x - y = x + - y" by (rule diff_eq1)
1.30 -  also have "f ... = f x + f (- y)"
1.31 -    by (rule add) (simp_all add: x y)
1.32 -  also from y have "f (- y) = - f y" by (rule neg)
1.33 +  also have "f ... = f x + f (- y)" by (rule add) (simp_all add: x y)
1.34 +  also from _ y have "f (- y) = - f y" by (rule neg)
1.35    finally show ?thesis by simp
1.36  qed
1.37
1.38  text {* Every linear form yields @{text 0} for the @{text 0} vector. *}
1.39
1.40 -lemma (in vectorspace_linearform) linearform_zero [iff]:
1.41 -  "f 0 = 0"
1.42 +lemma (in linearform) zero [iff]:
1.43 +  includes vectorspace
1.44 +  shows "f 0 = 0"
1.45  proof -
1.46    have "f 0 = f (0 - 0)" by simp
1.47    also have "\<dots> = f 0 - f 0" by (rule diff) simp_all
```