:: Algebra of Vector Functions
:: by Hiroshi Yamazaki and Yasunari Shidama
::
:: Received October 27, 1992
:: Copyright (c) 1992-2012 Association of Mizar Users


begin

::
::OPERATIONS ON PARTIAL FUNCTIONS FROM A DOMAIN TO THE SET OF REAL NUMBERS
::
definition
let C be non empty set ;
let V be non empty addLoopStr ;
let f1, f2 be PartFunc of C,V;
deffunc H1( set ) ->  Element of the U1 of V = (f1 /. $1) + (f2 /. $1);
deffunc H2( set ) ->  Element of the U1 of V = (f1 /. $1) - (f2 /. $1);
defpred S1[ set ] means $1 in (dom f1) /\ (dom f2);
set X = (dom f1) /\ (dom f2);
funcf1 + f2 ->  PartFunc of C,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,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) ) )
proofend;
uniqueness
 for b1, b2 being PartFunc of C,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
proofend;
funcf1 - f2 ->  PartFunc of C,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,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) ) )
proofend;
uniqueness
 for b1, b2 being PartFunc of C,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
proofend;
end;

:: deftheorem Def1 defines + VFUNCT_1:def_1_:_
 for C being non empty set
for V being non empty addLoopStr
for f1, f2, b5 being PartFunc of C,V holds
( b5 = f1 + f2 iff ( dom b5 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b5 holds
b5 /. c = (f1 /. c) + (f2 /. c) ) ) );

:: deftheorem Def2 defines - VFUNCT_1:def_2_:_
 for C being non empty set
for V being non empty addLoopStr
for f1, f2, b5 being PartFunc of C,V holds
( b5 = f1 - f2 iff ( dom b5 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b5 holds
b5 /. c = (f1 /. c) - (f2 /. c) ) ) );

registration
let C be non empty set ;
let V be non empty addLoopStr ;
let f1, f2 be Function of C,V;
clusterf1 + f2 ->  total ;
coherence
 f1 + f2 is total
proofend;
clusterf1 - f2 ->  total ;
coherence
 f1 - f2 is total
proofend;
end;

definition
let C be non empty set ;
let V be non empty RLSStruct ;
let f1 be PartFunc of C,REAL;
let f2 be PartFunc of C,V;
deffunc H1( Element of C) ->  Element of the U1 of V = (f1 . $1) * (f2 /. $1);
defpred S1[ set ] means $1 in (dom f1) /\ (dom f2);
set X = (dom f1) /\ (dom f2);
funcf1 (#) f2 ->  PartFunc of C,V means :Def3: :: VFUNCT_1:def 3
( 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,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) ) )
proofend;
uniqueness
 for b1, b2 being PartFunc of C,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
proofend;
end;

:: deftheorem Def3 defines (#) VFUNCT_1:def_3_:_
 for C being non empty set
for V being non empty RLSStruct
for f1 being PartFunc of C,REAL
for f2, b5 being PartFunc of C,V holds
( b5 = f1 (#) f2 iff ( dom b5 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b5 holds
b5 /. c = (f1 . c) * (f2 /. c) ) ) );

registration
let C be non empty set ;
let V be non empty RLSStruct ;
let f1 be Function of C,REAL;
let f2 be Function of C,V;
clusterf1 (#) f2 ->  total ;
coherence
 f1 (#) f2 is total
proofend;
end;

definition
let C be non empty set ;
let V be non empty RLSStruct ;
let f be PartFunc of C,V;
let r be Real;
deffunc H1( Element of C) ->  Element of the U1 of V = r * (f /. $1);
defpred S1[ set ] means $1 in dom f;
funcr (#) f ->  PartFunc of C,V means :Def4: :: VFUNCT_1:def 4
( dom it = dom f & ( for c being Element of C st c in dom it holds
it /. c = r * (f /. c) ) );
existence
 ex b1 being PartFunc of C,V st
( dom b1 = dom f & ( for c being Element of C st c in dom b1 holds
b1 /. c = r * (f /. c) ) )
proofend;
uniqueness
 for b1, b2 being PartFunc of C,V st dom b1 = dom f & ( for c being Element of C st c in dom b1 holds
b1 /. c = r * (f /. c) ) & dom b2 = dom f & ( for c being Element of C st c in dom b2 holds
b2 /. c = r * (f /. c) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def4 defines (#) VFUNCT_1:def_4_:_
 for C being non empty set
for V being non empty RLSStruct
for f being PartFunc of C,V
for r being Real
for b5 being PartFunc of C,V holds
( b5 = r (#) f iff ( dom b5 = dom f & ( for c being Element of C st c in dom b5 holds
b5 /. c = r * (f /. c) ) ) );

registration
let C be non empty set ;
let V be non empty RLSStruct ;
let f be Function of C,V;
let r be Real;
clusterr (#) f ->  total ;
coherence
 r (#) f is total
proofend;
end;

definition
let C be non empty set ;
let V be non empty addLoopStr ;
let f be PartFunc of C,V;
deffunc H1( Element of C) ->  Element of the U1 of V = - (f /. $1);
defpred S1[ set ] means $1 in dom f;
func - f ->  PartFunc of C,V means :Def5: :: VFUNCT_1:def 5
( dom it = dom f & ( for c being Element of C st c in dom it holds
it /. c = - (f /. c) ) );
existence
 ex b1 being PartFunc of C,V st
( dom b1 = dom f & ( for c being Element of C st c in dom b1 holds
b1 /. c = - (f /. c) ) )
proofend;
uniqueness
 for b1, b2 being PartFunc of C,V st dom b1 = dom f & ( for c being Element of C st c in dom b1 holds
b1 /. c = - (f /. c) ) & dom b2 = dom f & ( for c being Element of C st c in dom b2 holds
b2 /. c = - (f /. c) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def5 defines - VFUNCT_1:def_5_:_
 for C being non empty set
for V being non empty addLoopStr
for f, b4 being PartFunc of C,V holds
( b4 = - f iff ( dom b4 = dom f & ( for c being Element of C st c in dom b4 holds
b4 /. c = - (f /. c) ) ) );

registration
let C be non empty set ;
let V be non empty addLoopStr ;
let f be Function of C,V;
cluster - f ->  total ;
coherence
 - f is total
proofend;
end;

theorem :: VFUNCT_1:1
for C being non empty set
for V being RealNormSpace
for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL holds (dom (f1 (#) f2)) \ ((f1 (#) f2) " {(0. V)}) = ((dom f1) \ (f1 " {0})) /\ ((dom f2) \ (f2 " {(0. V)}))
proofend;

theorem :: VFUNCT_1:2
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V holds
( ||.f.|| " {0} = f " {(0. V)} & (- f) " {(0. V)} = f " {(0. V)} )
proofend;

theorem :: VFUNCT_1:3
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V
for r being Real st r <> 0 holds
(r (#) f) " {(0. V)} = f " {(0. V)}
proofend;

::
:: BASIC PROPERTIES OF OPERATIONS
::
theorem Th4: :: VFUNCT_1:4
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V holds f1 + f2 = f2 + f1
proofend;

definition
let C be non empty set ;
let V be RealNormSpace;
let f1, f2 be PartFunc of C,V;
:: original:+
redefine funcf1 + f2 ->  PartFunc of C,V;
commutativity
 for f1, f2 being PartFunc of C,V holds f1 + f2 = f2 + f1 by Th4;
end;

theorem :: VFUNCT_1:5
for C being non empty set
for V being RealNormSpace
for f1, f2, f3 being PartFunc of C,V holds (f1 + f2) + f3 = f1 + (f2 + f3)
proofend;

theorem :: VFUNCT_1:6
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,REAL
for f3 being PartFunc of C,V holds (f1 (#) f2) (#) f3 = f1 (#) (f2 (#) f3)
proofend;

theorem :: VFUNCT_1:7
for C being non empty set
for V being RealNormSpace
for f3 being PartFunc of C,V
for f1, f2 being PartFunc of C,REAL holds (f1 + f2) (#) f3 = (f1 (#) f3) + (f2 (#) f3)
proofend;

theorem :: VFUNCT_1:8
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V
for f3 being PartFunc of C,REAL holds f3 (#) (f1 + f2) = (f3 (#) f1) + (f3 (#) f2)
proofend;

theorem :: VFUNCT_1:9
for C being non empty set
for V being RealNormSpace
for f2 being PartFunc of C,V
for r being Real
for f1 being PartFunc of C,REAL holds r (#) (f1 (#) f2) = (r (#) f1) (#) f2
proofend;

theorem :: VFUNCT_1:10
for C being non empty set
for V being RealNormSpace
for f2 being PartFunc of C,V
for r being Real
for f1 being PartFunc of C,REAL holds r (#) (f1 (#) f2) = f1 (#) (r (#) f2)
proofend;

theorem :: VFUNCT_1:11
for C being non empty set
for V being RealNormSpace
for f3 being PartFunc of C,V
for f1, f2 being PartFunc of C,REAL holds (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3)
proofend;

theorem :: VFUNCT_1:12
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V
for f3 being PartFunc of C,REAL holds (f3 (#) f1) - (f3 (#) f2) = f3 (#) (f1 - f2)
proofend;

theorem :: VFUNCT_1:13
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V
for r being Real holds r (#) (f1 + f2) = (r (#) f1) + (r (#) f2)
proofend;

theorem Th14: :: VFUNCT_1:14
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V
for r, p being Real holds (r * p) (#) f = r (#) (p (#) f)
proofend;

theorem :: VFUNCT_1:15
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V
for r being Real holds r (#) (f1 - f2) = (r (#) f1) - (r (#) f2)
proofend;

theorem :: VFUNCT_1:16
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V holds f1 - f2 = (- 1) (#) (f2 - f1)
proofend;

theorem :: VFUNCT_1:17
for C being non empty set
for V being RealNormSpace
for f1, f2, f3 being PartFunc of C,V holds f1 - (f2 + f3) = (f1 - f2) - f3
proofend;

theorem Th18: :: VFUNCT_1:18
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V holds 1 (#) f = f
proofend;

theorem :: VFUNCT_1:19
for C being non empty set
for V being RealNormSpace
for f1, f2, f3 being PartFunc of C,V holds f1 - (f2 - f3) = (f1 - f2) + f3
proofend;

theorem :: VFUNCT_1:20
for C being non empty set
for V being RealNormSpace
for f1, f2, f3 being PartFunc of C,V holds f1 + (f2 - f3) = (f1 + f2) - f3
proofend;

theorem :: VFUNCT_1:21
for C being non empty set
for V being RealNormSpace
for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL holds ||.(f1 (#) f2).|| = (abs f1) (#) ||.f2.||
proofend;

theorem :: VFUNCT_1:22
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V
for r being Real holds ||.(r (#) f).|| = (abs r) (#) ||.f.||
proofend;

theorem Th23: :: VFUNCT_1:23
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V holds - f = (- 1) (#) f
proofend;

theorem Th24: :: VFUNCT_1:24
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V holds - (- f) = f
proofend;

theorem Th25: :: VFUNCT_1:25
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V holds f1 - f2 = f1 + (- f2)
proofend;

theorem :: VFUNCT_1:26
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V holds f1 - (- f2) = f1 + f2
proofend;

theorem Th27: :: VFUNCT_1:27
for X being set
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V holds
( (f1 + f2) | X = (f1 | X) + (f2 | X) &amp; (f1 + f2) | X = (f1 | X) + f2 &amp; (f1 + f2) | X = f1 + (f2 | X) )
proofend;

theorem :: VFUNCT_1:28
for X being set
for C being non empty set
for V being RealNormSpace
for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL holds
( (f1 (#) f2) | X = (f1 | X) (#) (f2 | X) &amp; (f1 (#) f2) | X = (f1 | X) (#) f2 &amp; (f1 (#) f2) | X = f1 (#) (f2 | X) )
proofend;

theorem Th29: :: VFUNCT_1:29
for X being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V holds
( (- f) | X = - (f | X) &amp; ||.f.|| | X = ||.(f | X).|| )
proofend;

theorem :: VFUNCT_1:30
for X being set
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V holds
( (f1 - f2) | X = (f1 | X) - (f2 | X) &amp; (f1 - f2) | X = (f1 | X) - f2 &amp; (f1 - f2) | X = f1 - (f2 | X) )
proofend;

theorem :: VFUNCT_1:31
for X being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V
for r being Real holds (r (#) f) | X = r (#) (f | X)
proofend;

::
:: TOTAL PARTIAL FUNCTIONS FROM A DOMAIN TO REAL
::
theorem :: VFUNCT_1:32
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V holds
( ( f1 is total &amp; f2 is total implies f1 + f2 is total ) &amp; ( f1 + f2 is total implies ( f1 is total &amp; f2 is total ) ) &amp; ( f1 is total &amp; f2 is total implies f1 - f2 is total ) &amp; ( f1 - f2 is total implies ( f1 is total &amp; f2 is total ) ) )
proofend;

theorem :: VFUNCT_1:33
for C being non empty set
for V being RealNormSpace
for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL holds
( ( f1 is total &amp; f2 is total ) iff f1 (#) f2 is total )
proofend;

theorem Th34: :: VFUNCT_1:34
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V
for r being Real holds
( f is total iff r (#) f is total )
proofend;

theorem :: VFUNCT_1:35
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V holds
( f is total iff - f is total )
proofend;

theorem Th36: :: VFUNCT_1:36
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V holds
( f is total iff ||.f.|| is total )
proofend;

theorem :: VFUNCT_1:37
for C being non empty set
for c being Element of C
for V being RealNormSpace
for f1, f2 being PartFunc of C,V st f1 is total &amp; f2 is total holds
( (f1 + f2) /. c = (f1 /. c) + (f2 /. c) &amp; (f1 - f2) /. c = (f1 /. c) - (f2 /. c) )
proofend;

theorem :: VFUNCT_1:38
for C being non empty set
for c being Element of C
for V being RealNormSpace
for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL st f1 is total &amp; f2 is total holds
(f1 (#) f2) /. c = (f1 . c) * (f2 /. c)
proofend;

theorem :: VFUNCT_1:39
for C being non empty set
for c being Element of C
for V being RealNormSpace
for f being PartFunc of C,V
for r being Real st f is total holds
(r (#) f) /. c = r * (f /. c)
proofend;

theorem :: VFUNCT_1:40
for C being non empty set
for c being Element of C
for V being RealNormSpace
for f being PartFunc of C,V st f is total holds
( (- f) /. c = - (f /. c) &amp; ||.f.|| . c = ||.(f /. c).|| )
proofend;

::
:: BOUNDED AND CONSTANT PARTIAL FUNCTIONS FROM A DOMAIN TO A NORMED REAL SPACE
::
definition
let C be non empty set ;
let V be non empty NORMSTR ;
let f be PartFunc of C,V;
let Y be set ;
predf is_bounded_on Y means :Def6: :: VFUNCT_1:def 6
 ex r being Real st
for c being Element of C st c in Y /\ (dom f) holds
||.(f /. c).|| <= r;
end;

:: deftheorem Def6 defines is_bounded_on VFUNCT_1:def_6_:_
 for C being non empty set
for V being non empty NORMSTR
for f being PartFunc of C,V
for Y being set holds
( f is_bounded_on Y iff ex r being Real st
for c being Element of C st c in Y /\ (dom f) holds
||.(f /. c).|| <= r );

theorem :: VFUNCT_1:41
for Y, X being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V st Y c= X &amp; f is_bounded_on X holds
f is_bounded_on Y
proofend;

theorem :: VFUNCT_1:42
for X being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V st X misses dom f holds
f is_bounded_on X
proofend;

theorem :: VFUNCT_1:43
for Y being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V holds 0 (#) f is_bounded_on Y
proofend;

theorem Th44: :: VFUNCT_1:44
for Y being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V
for r being Real st f is_bounded_on Y holds
r (#) f is_bounded_on Y
proofend;

theorem Th45: :: VFUNCT_1:45
for Y being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V st f is_bounded_on Y holds
( ||.f.|| | Y is bounded &amp; - f is_bounded_on Y )
proofend;

theorem Th46: :: VFUNCT_1:46
for X, Y being set
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V st f1 is_bounded_on X &amp; f2 is_bounded_on Y holds
f1 + f2 is_bounded_on X /\ Y
proofend;

theorem :: VFUNCT_1:47
for X, Y being set
for C being non empty set
for V being RealNormSpace
for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL st f1 | X is bounded &amp; f2 is_bounded_on Y holds
f1 (#) f2 is_bounded_on X /\ Y
proofend;

theorem :: VFUNCT_1:48
for X, Y being set
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V st f1 is_bounded_on X &amp; f2 is_bounded_on Y holds
f1 - f2 is_bounded_on X /\ Y
proofend;

theorem :: VFUNCT_1:49
for X, Y being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V st f is_bounded_on X &amp; f is_bounded_on Y holds
f is_bounded_on X \/ Y
proofend;

theorem :: VFUNCT_1:50
for X, Y being set
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V st f1 | X is V8() &amp; f2 | Y is V8() holds
( (f1 + f2) | (X /\ Y) is V8() &amp; (f1 - f2) | (X /\ Y) is V8() )
proofend;

theorem :: VFUNCT_1:51
for X, Y being set
for C being non empty set
for V being RealNormSpace
for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL st f1 | X is V8() &amp; f2 | Y is V8() holds
(f1 (#) f2) | (X /\ Y) is V8()
proofend;

theorem Th52: :: VFUNCT_1:52
for Y being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V
for p being Real st f | Y is V8() holds
(p (#) f) | Y is V8()
proofend;

theorem Th53: :: VFUNCT_1:53
for Y being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V st f | Y is V8() holds
( ||.f.|| | Y is V8() &amp; (- f) | Y is V8() )
proofend;

theorem Th54: :: VFUNCT_1:54
for Y being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V st f | Y is V8() holds
f is_bounded_on Y
proofend;

theorem :: VFUNCT_1:55
for Y being set
for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V st f | Y is V8() holds
( ( for r being Real holds r (#) f is_bounded_on Y ) &amp; - f is_bounded_on Y &amp; ||.f.|| | Y is bounded )
proofend;

theorem :: VFUNCT_1:56
for X, Y being set
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V st f1 is_bounded_on X &amp; f2 | Y is V8() holds
f1 + f2 is_bounded_on X /\ Y
proofend;

theorem :: VFUNCT_1:57
for X, Y being set
for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V st f1 is_bounded_on X &amp; f2 | Y is V8() holds
( f1 - f2 is_bounded_on X /\ Y &amp; f2 - f1 is_bounded_on X /\ Y )
proofend;