:: CFUNCT_1 semantic presentation

definition

mode Complex is Element of COMPLEX ;

end;

definition

let C be non empty set ;
let f1, f2 be PartFunc of C, COMPLEX ;
deffunc H₁( set ) -> Element of COMPLEX = (f1 /. $1) * ((f2 /. $1) " );
func f1 / f2 -> PartFunc of C, COMPLEX means :Def1: :: CFUNCT_1:def 1
( dom it = (dom f1) /\ ((dom f2) \ (f2 " {0})) & ( for c being Element of C st c in dom it holds
it /. c = (f1 /. c) * ((f2 /. c) " ) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines / CFUNCT_1:def 1 :

definition

let C be non empty set ;
let f be PartFunc of C, COMPLEX ;
deffunc H₁( set ) -> Element of COMPLEX = (f /. $1) " ;
func f ^ -> PartFunc of C, COMPLEX means :Def2: :: CFUNCT_1:def 2
( dom it = (dom f) \ (f " {0}) & ( for c being Element of C st c in dom it holds
it /. c = (f /. c) " ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines ^ CFUNCT_1:def 2 :

theorem :: CFUNCT_1:1

canceled;

theorem :: CFUNCT_1:2

canceled;

theorem Th3: :: CFUNCT_1:3

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds
( dom (f1 + f2) = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom (f1 + f2) holds
(f1 + f2) /. c = (f1 /. c) + (f2 /. c) ) )

proof end;

theorem Th4: :: CFUNCT_1:4

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds
( dom (f1 - f2) = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom (f1 - f2) holds
(f1 - f2) /. c = (f1 /. c) - (f2 /. c) ) )

proof end;

theorem Th5: :: CFUNCT_1:5

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds
( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom (f1 (#) f2) holds
(f1 (#) f2) /. c = (f1 /. c) * (f2 /. c) ) )

proof end;

theorem :: CFUNCT_1:6

canceled;

theorem Th7: :: CFUNCT_1:7

for C being non empty set
for f being PartFunc of C, COMPLEX
for r being Element of COMPLEX holds
( dom (r (#) f) = dom f & ( for c being Element of C st c in dom (r (#) f) holds
(r (#) f) /. c = r * (f /. c) ) )

proof end;

theorem :: CFUNCT_1:8

canceled;

theorem Th9: :: CFUNCT_1:9

for C being non empty set
for f being PartFunc of C, COMPLEX holds
( dom (- f) = dom f & ( for c being Element of C st c in dom (- f) holds
(- f) /. c = - (f /. c) ) )

proof end;

Lm1: for x, Y being set
for C being non empty set
for f being PartFunc of C, COMPLEX holds
( x in f " Y iff ( x in dom f & f /. x in Y ) )
by PARTFUN2:44;

theorem :: CFUNCT_1:10

canceled;

theorem :: CFUNCT_1:11

canceled;

theorem :: CFUNCT_1:12

canceled;

theorem :: CFUNCT_1:13

canceled;

theorem :: CFUNCT_1:14

canceled;

theorem Th15: :: CFUNCT_1:15

for C being non empty set
for g being PartFunc of C, COMPLEX holds
( dom (g ^ ) c= dom g & (dom g) /\ ((dom g) \ (g " {0})) = (dom g) \ (g " {0}) )

proof end;

theorem Th16: :: CFUNCT_1:16

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0}) = ((dom f1) \ (f1 " {0})) /\ ((dom f2) \ (f2 " {0}))

proof end;

theorem Th17: :: CFUNCT_1:17

for C being non empty set
for c being Element of C
for f being PartFunc of C, COMPLEX st c in dom (f ^ ) holds
f /. c <> 0

proof end;

theorem Th18: :: CFUNCT_1:18

for C being non empty set
for f being PartFunc of C, COMPLEX holds (f ^ ) " {0} = {}

proof end;

theorem Th19: :: CFUNCT_1:19

for C being non empty set
for f being PartFunc of C, COMPLEX holds
( |.f.| " {0} = f " {0} & (- f) " {0} = f " {0} )

proof end;

theorem Th20: :: CFUNCT_1:20

for C being non empty set
for f being PartFunc of C, COMPLEX holds dom ((f ^ ) ^ ) = dom (f | (dom (f ^ )))

proof end;

theorem Th21: :: CFUNCT_1:21

for C being non empty set
for f being PartFunc of C, COMPLEX
for r being Element of COMPLEX st r <> 0 holds
(r (#) f) " {0} = f " {0}

proof end;

theorem :: CFUNCT_1:22

for C being non empty set
for f1, f2, f3 being PartFunc of C, COMPLEX holds (f1 + f2) + f3 = f1 + (f2 + f3)

proof end;

theorem Th23: :: CFUNCT_1:23

for C being non empty set
for f1, f2, f3 being PartFunc of C, COMPLEX holds (f1 (#) f2) (#) f3 = f1 (#) (f2 (#) f3)

proof end;

theorem Th24: :: CFUNCT_1:24

for C being non empty set
for f1, f2, f3 being PartFunc of C, COMPLEX holds (f1 + f2) (#) f3 = (f1 (#) f3) + (f2 (#) f3)

proof end;

theorem :: CFUNCT_1:25

for C being non empty set
for f3, f1, f2 being PartFunc of C, COMPLEX holds f3 (#) (f1 + f2) = (f3 (#) f1) + (f3 (#) f2) by Th24;

theorem Th26: :: CFUNCT_1:26

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX
for r being Element of COMPLEX holds r (#) (f1 (#) f2) = (r (#) f1) (#) f2

proof end;

theorem Th27: :: CFUNCT_1:27

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX
for r being Element of COMPLEX holds r (#) (f1 (#) f2) = f1 (#) (r (#) f2)

proof end;

theorem Th28: :: CFUNCT_1:28

for C being non empty set
for f1, f2, f3 being PartFunc of C, COMPLEX holds (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3)

proof end;

theorem :: CFUNCT_1:29

for C being non empty set
for f3, f1, f2 being PartFunc of C, COMPLEX holds (f3 (#) f1) - (f3 (#) f2) = f3 (#) (f1 - f2) by Th28;

theorem :: CFUNCT_1:30

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX
for r being Element of COMPLEX holds r (#) (f1 + f2) = (r (#) f1) + (r (#) f2)

proof end;

theorem Th31: :: CFUNCT_1:31

for C being non empty set
for f being PartFunc of C, COMPLEX
for r, q being Element of COMPLEX holds (r * q) (#) f = r (#) (q (#) f)

proof end;

theorem :: CFUNCT_1:32

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX
for r being Element of COMPLEX holds r (#) (f1 - f2) = (r (#) f1) - (r (#) f2)

proof end;

theorem :: CFUNCT_1:33

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds f1 - f2 = (- 1r ) (#) (f2 - f1)

proof end;

theorem :: CFUNCT_1:34

for C being non empty set
for f1, f2, f3 being PartFunc of C, COMPLEX holds f1 - (f2 + f3) = (f1 - f2) - f3

proof end;

theorem Th35: :: CFUNCT_1:35

for C being non empty set
for f being PartFunc of C, COMPLEX holds 1r (#) f = f

proof end;

theorem :: CFUNCT_1:36

for C being non empty set
for f1, f2, f3 being PartFunc of C, COMPLEX holds f1 - (f2 - f3) = (f1 - f2) + f3

proof end;

theorem :: CFUNCT_1:37

for C being non empty set
for f1, f2, f3 being PartFunc of C, COMPLEX holds f1 + (f2 - f3) = (f1 + f2) - f3

proof end;

theorem Th38: :: CFUNCT_1:38

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds |.(f1 (#) f2).| = |.f1.| (#) |.f2.|

proof end;

theorem Th39: :: CFUNCT_1:39

for C being non empty set
for f being PartFunc of C, COMPLEX
for r being Element of COMPLEX holds |.(r (#) f).| = |.r.| (#) |.f.|

proof end;

theorem Th40: :: CFUNCT_1:40

for C being non empty set
for f being PartFunc of C, COMPLEX holds - f = (- 1r ) (#) f

proof end;

theorem Th41: :: CFUNCT_1:41

for C being non empty set
for f being PartFunc of C, COMPLEX holds - (- f) = f

proof end;

theorem :: CFUNCT_1:42

canceled;

theorem :: CFUNCT_1:43

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds f1 - (- f2) = f1 + f2

proof end;

theorem Th44: :: CFUNCT_1:44

for C being non empty set
for f being PartFunc of C, COMPLEX holds (f ^ ) ^ = f | (dom (f ^ ))

proof end;

theorem Th45: :: CFUNCT_1:45

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds (f1 (#) f2) ^ = (f1 ^ ) (#) (f2 ^ )

proof end;

theorem Th46: :: CFUNCT_1:46

for C being non empty set
for f being PartFunc of C, COMPLEX
for r being Element of COMPLEX st r <> 0 holds
(r (#) f) ^ = (r " ) (#) (f ^ )

proof end;

Lm2: (- 1r ) " = - 1r
by COMPLEX1:def 7;

theorem :: CFUNCT_1:47

canceled;

theorem :: CFUNCT_1:48

canceled;

theorem Th49: :: CFUNCT_1:49

for C being non empty set
for f being PartFunc of C, COMPLEX holds (- f) ^ = (- 1r ) (#) (f ^ )

proof end;

theorem Th50: :: CFUNCT_1:50

for C being non empty set
for f being PartFunc of C, COMPLEX holds |.f.| ^ = |.(f ^ ).|

proof end;

theorem Th51: :: CFUNCT_1:51

for C being non empty set
for f, g being PartFunc of C, COMPLEX holds f / g = f (#) (g ^ )

proof end;

theorem Th52: :: CFUNCT_1:52

for C being non empty set
for g, f being PartFunc of C, COMPLEX
for r being Element of COMPLEX holds r (#) (g / f) = (r (#) g) / f

proof end;

theorem :: CFUNCT_1:53

for C being non empty set
for f, g being PartFunc of C, COMPLEX holds (f / g) (#) g = f | (dom (g ^ ))

proof end;

theorem Th54: :: CFUNCT_1:54

for C being non empty set
for f, g, f1, g1 being PartFunc of C, COMPLEX holds (f / g) (#) (f1 / g1) = (f (#) f1) / (g (#) g1)

proof end;

theorem Th55: :: CFUNCT_1:55

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds (f1 / f2) ^ = (f2 | (dom (f2 ^ ))) / f1

proof end;

theorem Th56: :: CFUNCT_1:56

for C being non empty set
for g, f1, f2 being PartFunc of C, COMPLEX holds g (#) (f1 / f2) = (g (#) f1) / f2

proof end;

theorem :: CFUNCT_1:57

for C being non empty set
for g, f1, f2 being PartFunc of C, COMPLEX holds g / (f1 / f2) = (g (#) (f2 | (dom (f2 ^ )))) / f1

proof end;

theorem :: CFUNCT_1:58

for C being non empty set
for f, g being PartFunc of C, COMPLEX holds
( - (f / g) = (- f) / g & f / (- g) = - (f / g) )

proof end;

theorem :: CFUNCT_1:59

for C being non empty set
for f1, f, f2 being PartFunc of C, COMPLEX holds
( (f1 / f) + (f2 / f) = (f1 + f2) / f & (f1 / f) - (f2 / f) = (f1 - f2) / f )

proof end;

theorem Th60: :: CFUNCT_1:60

for C being non empty set
for f1, f, g1, g being PartFunc of C, COMPLEX holds (f1 / f) + (g1 / g) = ((f1 (#) g) + (g1 (#) f)) / (f (#) g)

proof end;

theorem :: CFUNCT_1:61

for C being non empty set
for f, g, f1, g1 being PartFunc of C, COMPLEX holds (f / g) / (f1 / g1) = (f (#) (g1 | (dom (g1 ^ )))) / (g (#) f1)

proof end;

theorem :: CFUNCT_1:62

for C being non empty set
for f1, f, g1, g being PartFunc of C, COMPLEX holds (f1 / f) - (g1 / g) = ((f1 (#) g) - (g1 (#) f)) / (f (#) g)

proof end;

theorem :: CFUNCT_1:63

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds |.(f1 / f2).| = |.f1.| / |.f2.|

proof end;

theorem Th64: :: CFUNCT_1:64

for X being set
for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds
( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )

proof end;

theorem Th65: :: CFUNCT_1:65

for X being set
for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds
( (f1 (#) f2) | X = (f1 | X) (#) (f2 | X) & (f1 (#) f2) | X = (f1 | X) (#) f2 & (f1 (#) f2) | X = f1 (#) (f2 | X) )

proof end;

theorem Th66: :: CFUNCT_1:66

for X being set
for C being non empty set
for f being PartFunc of C, COMPLEX holds
( (- f) | X = - (f | X) & (f ^ ) | X = (f | X) ^ & |.f.| | X = |.(f | X).| )

proof end;

theorem :: CFUNCT_1:67

for X being set
for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds
( (f1 - f2) | X = (f1 | X) - (f2 | X) & (f1 - f2) | X = (f1 | X) - f2 & (f1 - f2) | X = f1 - (f2 | X) )

proof end;

theorem :: CFUNCT_1:68

for X being set
for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds
( (f1 / f2) | X = (f1 | X) / (f2 | X) & (f1 / f2) | X = (f1 | X) / f2 & (f1 / f2) | X = f1 / (f2 | X) )

proof end;

theorem :: CFUNCT_1:69

for X being set
for C being non empty set
for f being PartFunc of C, COMPLEX
for r being Element of COMPLEX holds (r (#) f) | X = r (#) (f | X)

proof end;

theorem Th70: :: CFUNCT_1:70

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds
( ( f1 is total & f2 is total implies f1 + f2 is total ) & ( f1 + f2 is total implies ( f1 is total & f2 is total ) ) & ( f1 is total & f2 is total implies f1 - f2 is total ) & ( f1 - f2 is total implies ( f1 is total & f2 is total ) ) & ( f1 is total & f2 is total implies f1 (#) f2 is total ) & ( f1 (#) f2 is total implies ( f1 is total & f2 is total ) ) )

proof end;

theorem Th71: :: CFUNCT_1:71

for C being non empty set
for f being PartFunc of C, COMPLEX
for r being Element of COMPLEX holds
( f is total iff r (#) f is total )

proof end;

theorem Th72: :: CFUNCT_1:72

for C being non empty set
for f being PartFunc of C, COMPLEX holds
( f is total iff - f is total )

proof end;

theorem Th73: :: CFUNCT_1:73

for C being non empty set
for f being PartFunc of C, COMPLEX holds
( f is total iff |.f.| is total )

proof end;

theorem Th74: :: CFUNCT_1:74

for C being non empty set
for f being PartFunc of C, COMPLEX holds
( f ^ is total iff ( f " {0} = {} & f is total ) )

proof end;

theorem Th75: :: CFUNCT_1:75

for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX holds
( ( f1 is total & f2 " {0} = {} & f2 is total ) iff f1 / f2 is total )

proof end;

theorem :: CFUNCT_1:76

for C being non empty set
for c being Element of C
for f1, f2 being PartFunc of C, COMPLEX st f1 is total & f2 is total holds
( (f1 + f2) /. c = (f1 /. c) + (f2 /. c) & (f1 - f2) /. c = (f1 /. c) - (f2 /. c) & (f1 (#) f2) /. c = (f1 /. c) * (f2 /. c) )

proof end;

theorem :: CFUNCT_1:77

for C being non empty set
for c being Element of C
for f being PartFunc of C, COMPLEX
for r being Element of COMPLEX st f is total holds
(r (#) f) /. c = r * (f /. c)

proof end;

theorem :: CFUNCT_1:78

for C being non empty set
for c being Element of C
for f being PartFunc of C, COMPLEX st f is total holds
( (- f) /. c = - (f /. c) & |.f.| . c = |.(f /. c).| )

proof end;

theorem :: CFUNCT_1:79

for C being non empty set
for c being Element of C
for f being PartFunc of C, COMPLEX st f ^ is total holds
(f ^ ) /. c = (f /. c) "

proof end;

theorem :: CFUNCT_1:80

for C being non empty set
for c being Element of C
for f1, f2 being PartFunc of C, COMPLEX st f1 is total & f2 ^ is total holds
(f1 / f2) /. c = (f1 /. c) * ((f2 /. c) " )

proof end;

definition

let C be non empty set ;
let f be PartFunc of C, COMPLEX ;
let Y be set ;
pred f is_bounded_on Y means :Def3: :: CFUNCT_1:def 3
|.f.| is_bounded_on Y;

end;

:: deftheorem Def3 defines is_bounded_on CFUNCT_1:def 3 :

theorem Th81: :: CFUNCT_1:81

for Y being set
for C being non empty set
for f being PartFunc of C, COMPLEX holds
( f is_bounded_on Y iff ex p being real number st
for c being Element of C st c in Y /\ (dom f) holds
|.(f /. c).| <= p )

proof end;

theorem :: CFUNCT_1:82

for Y, X being set
for C being non empty set
for f being PartFunc of C, COMPLEX st Y c= X & f is_bounded_on X holds
f is_bounded_on Y

proof end;

theorem :: CFUNCT_1:83

for X being set
for C being non empty set
for f being PartFunc of C, COMPLEX st X misses dom f holds
f is_bounded_on X

proof end;

theorem Th84: :: CFUNCT_1:84

for Y being set
for C being non empty set
for f being PartFunc of C, COMPLEX
for r being Element of COMPLEX st f is_bounded_on Y holds
r (#) f is_bounded_on Y

proof end;

theorem :: CFUNCT_1:85

for X being set
for C being non empty set
for f being PartFunc of C, COMPLEX holds |.f.| is_bounded_below_on X

proof end;

theorem Th86: :: CFUNCT_1:86

for Y being set
for C being non empty set
for f being PartFunc of C, COMPLEX st f is_bounded_on Y holds
( |.f.| is_bounded_on Y & - f is_bounded_on Y )

proof end;

theorem Th87: :: CFUNCT_1:87

for X, Y being set
for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX st f1 is_bounded_on X & f2 is_bounded_on Y holds
f1 + f2 is_bounded_on X /\ Y

proof end;

theorem Th88: :: CFUNCT_1:88

for X, Y being set
for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX st f1 is_bounded_on X & f2 is_bounded_on Y holds
( f1 (#) f2 is_bounded_on X /\ Y & f1 - f2 is_bounded_on X /\ Y )

proof end;

theorem :: CFUNCT_1:89

for X, Y being set
for C being non empty set
for f being PartFunc of C, COMPLEX st f is_bounded_on X & f is_bounded_on Y holds
f is_bounded_on X \/ Y

proof end;

theorem :: CFUNCT_1:90

for X, Y being set
for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX st f1 is_constant_on X & f2 is_constant_on Y holds
( f1 + f2 is_constant_on X /\ Y & f1 - f2 is_constant_on X /\ Y & f1 (#) f2 is_constant_on X /\ Y )

proof end;

theorem Th91: :: CFUNCT_1:91

for Y being set
for C being non empty set
for f being PartFunc of C, COMPLEX
for q being Element of COMPLEX st f is_constant_on Y holds
q (#) f is_constant_on Y

proof end;

theorem Th92: :: CFUNCT_1:92

for Y being set
for C being non empty set
for f being PartFunc of C, COMPLEX st f is_constant_on Y holds
( |.f.| is_constant_on Y & - f is_constant_on Y )

proof end;

theorem Th93: :: CFUNCT_1:93

for Y being set
for C being non empty set
for f being PartFunc of C, COMPLEX st f is_constant_on Y holds
f is_bounded_on Y

proof end;

theorem :: CFUNCT_1:94

for Y being set
for C being non empty set
for f being PartFunc of C, COMPLEX st f is_constant_on Y holds
( ( for r being Element of COMPLEX holds r (#) f is_bounded_on Y ) & - f is_bounded_on Y & |.f.| is_bounded_on Y )

proof end;

theorem Th95: :: CFUNCT_1:95

for X, Y being set
for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX st f1 is_bounded_on X & f2 is_constant_on Y holds
f1 + f2 is_bounded_on X /\ Y

proof end;

theorem :: CFUNCT_1:96

for X, Y being set
for C being non empty set
for f1, f2 being PartFunc of C, COMPLEX st f1 is_bounded_on X & f2 is_constant_on Y holds
( f1 - f2 is_bounded_on X /\ Y & f2 - f1 is_bounded_on X /\ Y & f1 (#) f2 is_bounded_on X /\ Y )

proof end;