:: Algebra of Vector Functions
:: by Hiroshi Yamazaki and Yasunari Shidama
::
:: Received October 27, 1992
:: Copyright (c) 1992 Association of Mizar Users
definition
let C be non
empty set ;
let V be
RealNormSpace;
let f1,
f2 be
PartFunc of
C,the
carrier of
V;
deffunc H1(
Element of
C)
-> Element of the
carrier of
V =
(f1 /. $1) + (f2 /. $1);
deffunc H2(
Element of
C)
-> Element of the
carrier of
V =
(f1 /. $1) - (f2 /. $1);
defpred S1[
set ]
means $1
in (dom f1) /\ (dom f2);
set X =
(dom f1) /\ (dom f2);
func f1 + f2 -> PartFunc of
C,the
carrier of
V means :
Def1:
:: VFUNCT_1:def 1
(
dom it = (dom f1) /\ (dom f2) & ( for
c being
Element of
C st
c in dom it holds
it /. c = (f1 /. c) + (f2 /. c) ) );
existence
ex b1 being PartFunc of C,the carrier of V st
( dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 /. c = (f1 /. c) + (f2 /. c) ) )
uniqueness
for b1, b2 being PartFunc of C,the carrier of V st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 /. c = (f1 /. c) + (f2 /. c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b2 holds
b2 /. c = (f1 /. c) + (f2 /. c) ) holds
b1 = b2
func f1 - f2 -> PartFunc of
C,the
carrier of
V means :
Def2:
:: VFUNCT_1:def 2
(
dom it = (dom f1) /\ (dom f2) & ( for
c being
Element of
C st
c in dom it holds
it /. c = (f1 /. c) - (f2 /. c) ) );
existence
ex b1 being PartFunc of C,the carrier of V st
( dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 /. c = (f1 /. c) - (f2 /. c) ) )
uniqueness
for b1, b2 being PartFunc of C,the carrier of V st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 /. c = (f1 /. c) - (f2 /. c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b2 holds
b2 /. c = (f1 /. c) - (f2 /. c) ) holds
b1 = b2
end;
:: deftheorem Def1 defines + VFUNCT_1:def 1 :
:: deftheorem Def2 defines - VFUNCT_1:def 2 :
:: deftheorem Def3 defines (#) VFUNCT_1:def 3 :
:: deftheorem Def4 defines (#) VFUNCT_1:def 4 :
:: deftheorem Def5 defines ||. VFUNCT_1:def 5 :
:: deftheorem Def6 defines - VFUNCT_1:def 6 :
theorem :: VFUNCT_1:1
canceled;
theorem :: VFUNCT_1:2
canceled;
theorem :: VFUNCT_1:3
canceled;
theorem :: VFUNCT_1:4
canceled;
theorem :: VFUNCT_1:5
canceled;
theorem :: VFUNCT_1:6
canceled;
theorem :: VFUNCT_1:7
theorem :: VFUNCT_1:8
theorem :: VFUNCT_1:9
theorem :: VFUNCT_1:10
theorem :: VFUNCT_1:11
theorem :: VFUNCT_1:12
theorem :: VFUNCT_1:13
theorem :: VFUNCT_1:14
theorem :: VFUNCT_1:15
theorem :: VFUNCT_1:16
theorem :: VFUNCT_1:17
theorem :: VFUNCT_1:18
theorem :: VFUNCT_1:19
theorem Th20: :: VFUNCT_1:20
theorem :: VFUNCT_1:21
theorem :: VFUNCT_1:22
theorem :: VFUNCT_1:23
theorem Th24: :: VFUNCT_1:24
theorem :: VFUNCT_1:25
theorem :: VFUNCT_1:26
theorem :: VFUNCT_1:27
theorem :: VFUNCT_1:28
theorem Th29: :: VFUNCT_1:29
theorem Th30: :: VFUNCT_1:30
theorem Th31: :: VFUNCT_1:31
theorem :: VFUNCT_1:32
theorem Th33: :: VFUNCT_1:33
theorem :: VFUNCT_1:34
theorem Th35: :: VFUNCT_1:35
theorem :: VFUNCT_1:36
theorem :: VFUNCT_1:37
theorem Th38: :: VFUNCT_1:38
theorem Th39: :: VFUNCT_1:39
theorem Th40: :: VFUNCT_1:40
theorem Th41: :: VFUNCT_1:41
theorem Th42: :: VFUNCT_1:42
theorem :: VFUNCT_1:43
theorem :: VFUNCT_1:44
theorem :: VFUNCT_1:45
theorem :: VFUNCT_1:46
:: deftheorem Def7 defines is_bounded_on VFUNCT_1:def 7 :
theorem :: VFUNCT_1:47
canceled;
theorem :: VFUNCT_1:48
theorem :: VFUNCT_1:49
theorem :: VFUNCT_1:50
theorem Th51: :: VFUNCT_1:51
theorem Th52: :: VFUNCT_1:52
theorem Th53: :: VFUNCT_1:53
theorem :: VFUNCT_1:54
theorem :: VFUNCT_1:55
theorem :: VFUNCT_1:56
theorem :: VFUNCT_1:57
theorem :: VFUNCT_1:58
theorem Th59: :: VFUNCT_1:59
theorem Th60: :: VFUNCT_1:60
theorem Th61: :: VFUNCT_1:61
theorem :: VFUNCT_1:62
theorem :: VFUNCT_1:63
theorem :: VFUNCT_1:64