:: SETWISEO semantic presentation
theorem Th1: :: SETWISEO:1
canceled;
theorem Th2: :: SETWISEO:2
canceled;
theorem Th3: :: SETWISEO:3
theorem Th4: :: SETWISEO:4
theorem Th5: :: SETWISEO:5
for
X,
Y,
x being
set holds
( not
X c= Y \/ {x} or
x in X or
X c= Y )
theorem Th6: :: SETWISEO:6
for
x,
X,
y being
set holds
(
x in X \/ {y} iff (
x in X or
x = y ) )
theorem Th7: :: SETWISEO:7
canceled;
theorem Th8: :: SETWISEO:8
for
X,
x,
Y being
set holds
(
X \/ {x} c= Y iff (
x in Y &
X c= Y ) )
theorem Th9: :: SETWISEO:9
canceled;
theorem Th10: :: SETWISEO:10
canceled;
theorem Th11: :: SETWISEO:11
theorem Th12: :: SETWISEO:12
theorem Th13: :: SETWISEO:13
theorem Th14: :: SETWISEO:14
theorem Th15: :: SETWISEO:15
theorem Th16: :: SETWISEO:16
Lemma32:
for X, Y being non empty set
for f being Function of X,Y
for A being Element of Fin X holds f .: A is Element of Fin Y
by FINSUB_1:def 5;
theorem Th17: :: SETWISEO:17
canceled;
theorem Th18: :: SETWISEO:18
theorem Th19: :: SETWISEO:19
:: deftheorem Def1 defines {}. SETWISEO:def 1 :
:: deftheorem Def2 defines having_a_unity SETWISEO:def 2 :
theorem Th20: :: SETWISEO:20
canceled;
theorem Th21: :: SETWISEO:21
canceled;
theorem Th22: :: SETWISEO:22
theorem Th23: :: SETWISEO:23
definition
let X be non
empty set ,
Y be non
empty set ;
let F be
BinOp of
Y;
let B be
Element of
Fin X;
let f be
Function of
X,
Y;
assume that E14:
(
B <> {} or
F has_a_unity )
and E15:
F is
commutative
and E16:
F is
associative
;
func c3 $$ c4,
c5 -> Element of
a2 means :
Def3:
:: SETWISEO:def 3
ex
G being
Function of
Fin X,
Y st
(
it = G . B & ( for
e being
Element of
Y st
e is_a_unity_wrt F holds
G . {} = e ) & ( for
x being
Element of
X holds
G . {x} = f . x ) & ( for
B' being
Element of
Fin X st
B' c= B &
B' <> {} holds
for
x being
Element of
X st
x in B \ B' holds
G . (B' \/ {x}) = F . (G . B'),
(f . x) ) );
existence
ex b1 being Element of Y ex G being Function of Fin X,Y st
( b1 = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B' being Element of Fin X st B' c= B & B' <> {} holds
for x being Element of X st x in B \ B' holds
G . (B' \/ {x}) = F . (G . B'),(f . x) ) )
uniqueness
for b1, b2 being Element of Y st ex G being Function of Fin X,Y st
( b1 = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B' being Element of Fin X st B' c= B & B' <> {} holds
for x being Element of X st x in B \ B' holds
G . (B' \/ {x}) = F . (G . B'),(f . x) ) ) & ex G being Function of Fin X,Y st
( b2 = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B' being Element of Fin X st B' c= B & B' <> {} holds
for x being Element of X st x in B \ B' holds
G . (B' \/ {x}) = F . (G . B'),(f . x) ) ) holds
b1 = b2
end;
:: deftheorem Def3 defines $$ SETWISEO:def 3 :
theorem Th24: :: SETWISEO:24
canceled;
theorem Th25: :: SETWISEO:25
theorem Th26: :: SETWISEO:26
theorem Th27: :: SETWISEO:27
theorem Th28: :: SETWISEO:28
theorem Th29: :: SETWISEO:29
theorem Th30: :: SETWISEO:30
theorem Th31: :: SETWISEO:31
theorem Th32: :: SETWISEO:32
theorem Th33: :: SETWISEO:33
theorem Th34: :: SETWISEO:34
theorem Th35: :: SETWISEO:35
theorem Th36: :: SETWISEO:36
theorem Th37: :: SETWISEO:37
theorem Th38: :: SETWISEO:38
theorem Th39: :: SETWISEO:39
theorem Th40: :: SETWISEO:40
theorem Th41: :: SETWISEO:41
theorem Th42: :: SETWISEO:42
theorem Th43: :: SETWISEO:43
theorem Th44: :: SETWISEO:44
theorem Th45: :: SETWISEO:45
:: deftheorem Def4 defines FinUnion SETWISEO:def 4 :
theorem Th46: :: SETWISEO:46
canceled;
theorem Th47: :: SETWISEO:47
canceled;
theorem Th48: :: SETWISEO:48
canceled;
theorem Th49: :: SETWISEO:49
theorem Th50: :: SETWISEO:50
theorem Th51: :: SETWISEO:51
theorem Th52: :: SETWISEO:52
theorem Th53: :: SETWISEO:53
theorem Th54: :: SETWISEO:54
theorem Th55: :: SETWISEO:55
:: deftheorem Def5 defines FinUnion SETWISEO:def 5 :
theorem Th56: :: SETWISEO:56
theorem Th57: :: SETWISEO:57
theorem Th58: :: SETWISEO:58
theorem Th59: :: SETWISEO:59
theorem Th60: :: SETWISEO:60
theorem Th61: :: SETWISEO:61
theorem Th62: :: SETWISEO:62
theorem Th63: :: SETWISEO:63
theorem Th64: :: SETWISEO:64
theorem Th65: :: SETWISEO:65
:: deftheorem Def6 defines singleton SETWISEO:def 6 :
theorem Th66: :: SETWISEO:66
canceled;
theorem Th67: :: SETWISEO:67
theorem Th68: :: SETWISEO:68
theorem Th69: :: SETWISEO:69
Lemma163:
for D being non empty set
for X, P being set
for f being Function of X,D holds f .: P c= D
;
theorem Th70: :: SETWISEO:70
theorem Th71: :: SETWISEO:71
theorem Th72: :: SETWISEO:72