:: HILBERT3 semantic presentation
theorem Th1: :: HILBERT3:1
theorem Th2: :: HILBERT3:2
theorem Th3: :: HILBERT3:3
theorem Th4: :: HILBERT3:4
theorem Th5: :: HILBERT3:5
theorem Th6: :: HILBERT3:6
theorem Th7: :: HILBERT3:7
theorem Th8: :: HILBERT3:8
theorem Th9: :: HILBERT3:9
theorem Th10: :: HILBERT3:10
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( b
1 <> b
2 & b
1,b
2 --> b
3,b
4 = b
1,b
2 --> b
5,b
6 implies ( b
3 = b
5 & b
4 = b
6 ) )
theorem Th11: :: HILBERT3:11
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( b
1 <> b
2 & b
3 in b
5 & b
4 in b
6 implies b
1,b
2 --> b
3,b
4 in product (b1,b2 --> b5,b6) )
theorem Th12: :: HILBERT3:12
theorem Th13: :: HILBERT3:13
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> b
2 implies
(b1,b2 --> b3,b4) * (b1,b2 --> b2,b1) = b
1,b
2 --> b
4,b
3 )
theorem Th14: :: HILBERT3:14
theorem Th15: :: HILBERT3:15
theorem Th16: :: HILBERT3:16
theorem Th17: :: HILBERT3:17
theorem Th18: :: HILBERT3:18
theorem Th19: :: HILBERT3:19
theorem Th20: :: HILBERT3:20
theorem Th21: :: HILBERT3:21
theorem Th22: :: HILBERT3:22
canceled;
theorem Th23: :: HILBERT3:23
theorem Th24: :: HILBERT3:24
theorem Th25: :: HILBERT3:25
definition
let c
1, c
2 be non
empty set ;
let c
3 be
Permutation of c
1;
let c
4 be
Function of c
2,c
2;
func c
3 => c
4 -> Function of
Funcs a
1,a
2,
Funcs a
1,a
2 means :
Def1:
:: HILBERT3:def 1
for b
1 being
Function of a
1,a
2 holds a
5 . b
1 = (a4 * b1) * (a3 " );
existence
ex b1 being Function of Funcs c1,c2, Funcs c1,c2 st
for b2 being Function of c1,c2 holds b1 . b2 = (c4 * b2) * (c3 " )
uniqueness
for b1, b2 being Function of Funcs c1,c2, Funcs c1,c2 holds
( ( for b3 being Function of c1,c2 holds b1 . b3 = (c4 * b3) * (c3 " ) ) & ( for b3 being Function of c1,c2 holds b2 . b3 = (c4 * b3) * (c3 " ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines => HILBERT3:def 1 :
theorem Th26: :: HILBERT3:26
theorem Th27: :: HILBERT3:27
theorem Th28: :: HILBERT3:28
definition
let c
1 be
SetValuation;
func SetVal c
1 -> ManySortedSet of
HP-WFF means :
Def2:
:: HILBERT3:def 2
( a
2 . VERUM = 1 & ( for b
1 being
Nat holds a
2 . (prop b1) = a
1 . b
1 ) & ( for b
1, b
2 being
Element of
HP-WFF holds
( a
2 . (b1 '&' b2) = [:(a2 . b1),(a2 . b2):] & a
2 . (b1 => b2) = Funcs (a2 . b1),
(a2 . b2) ) ) );
existence
ex b1 being ManySortedSet of HP-WFF st
( b1 . VERUM = 1 & ( for b2 being Nat holds b1 . (prop b2) = c1 . b2 ) & ( for b2, b3 being Element of HP-WFF holds
( b1 . (b2 '&' b3) = [:(b1 . b2),(b1 . b3):] & b1 . (b2 => b3) = Funcs (b1 . b2),(b1 . b3) ) ) )
uniqueness
for b1, b2 being ManySortedSet of HP-WFF holds
( b1 . VERUM = 1 & ( for b3 being Nat holds b1 . (prop b3) = c1 . b3 ) & ( for b3, b4 being Element of HP-WFF holds
( b1 . (b3 '&' b4) = [:(b1 . b3),(b1 . b4):] & b1 . (b3 => b4) = Funcs (b1 . b3),(b1 . b4) ) ) & b2 . VERUM = 1 & ( for b3 being Nat holds b2 . (prop b3) = c1 . b3 ) & ( for b3, b4 being Element of HP-WFF holds
( b2 . (b3 '&' b4) = [:(b2 . b3),(b2 . b4):] & b2 . (b3 => b4) = Funcs (b2 . b3),(b2 . b4) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines SetVal HILBERT3:def 2 :
:: deftheorem Def3 defines SetVal HILBERT3:def 3 :
theorem Th29: :: HILBERT3:29
theorem Th30: :: HILBERT3:30
theorem Th31: :: HILBERT3:31
theorem Th32: :: HILBERT3:32
registration
let c
1 be
SetValuation;
let c
2, c
3, c
4 be
Element of
HP-WFF ;
cluster Function-yielding M5(
SetVal a
1,
(a2 => a3),
SetVal a
1,
(a2 => a4));
existence
ex b1 being Function of SetVal c1,(c2 => c3), SetVal c1,(c2 => c4) st b1 is Function-yielding
cluster Function-yielding Element of
SetVal a
1,
(a2 => (a3 => a4));
existence
ex b1 being Element of SetVal c1,(c2 => (c3 => c4)) st b1 is Function-yielding
end;
:: deftheorem Def4 defines Permutation HILBERT3:def 4 :
definition
let c
1 be
SetValuation;
let c
2 be
Permutation of c
1;
func Perm c
2 -> ManySortedFunction of
SetVal a
1,
SetVal a
1 means :
Def5:
:: HILBERT3:def 5
( a
3 . VERUM = id 1 & ( for b
1 being
Nat holds a
3 . (prop b1) = a
2 . b
1 ) & ( for b
1, b
2 being
Element of
HP-WFF holds
ex b
3 being
Permutation of
SetVal a
1,b
1ex b
4 being
Permutation of
SetVal a
1,b
2 st
( b
3 = a
3 . b
1 & b
4 = a
3 . b
2 & a
3 . (b1 '&' b2) = [:b3,b4:] & a
3 . (b1 => b2) = b
3 => b
4 ) ) );
existence
ex b1 being ManySortedFunction of SetVal c1, SetVal c1 st
( b1 . VERUM = id 1 & ( for b2 being Nat holds b1 . (prop b2) = c2 . b2 ) & ( for b2, b3 being Element of HP-WFF holds
ex b4 being Permutation of SetVal c1,b2ex b5 being Permutation of SetVal c1,b3 st
( b4 = b1 . b2 & b5 = b1 . b3 & b1 . (b2 '&' b3) = [:b4,b5:] & b1 . (b2 => b3) = b4 => b5 ) ) )
uniqueness
for b1, b2 being ManySortedFunction of SetVal c1, SetVal c1 holds
( b1 . VERUM = id 1 & ( for b3 being Nat holds b1 . (prop b3) = c2 . b3 ) & ( for b3, b4 being Element of HP-WFF holds
ex b5 being Permutation of SetVal c1,b3ex b6 being Permutation of SetVal c1,b4 st
( b5 = b1 . b3 & b6 = b1 . b4 & b1 . (b3 '&' b4) = [:b5,b6:] & b1 . (b3 => b4) = b5 => b6 ) ) & b2 . VERUM = id 1 & ( for b3 being Nat holds b2 . (prop b3) = c2 . b3 ) & ( for b3, b4 being Element of HP-WFF holds
ex b5 being Permutation of SetVal c1,b3ex b6 being Permutation of SetVal c1,b4 st
( b5 = b2 . b3 & b6 = b2 . b4 & b2 . (b3 '&' b4) = [:b5,b6:] & b2 . (b3 => b4) = b5 => b6 ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines Perm HILBERT3:def 5 :
:: deftheorem Def6 defines Perm HILBERT3:def 6 :
theorem Th33: :: HILBERT3:33
theorem Th34: :: HILBERT3:34
theorem Th35: :: HILBERT3:35
theorem Th36: :: HILBERT3:36
theorem Th37: :: HILBERT3:37
theorem Th38: :: HILBERT3:38
theorem Th39: :: HILBERT3:39
theorem Th40: :: HILBERT3:40
:: deftheorem Def7 defines canonical HILBERT3:def 7 :
theorem Th41: :: HILBERT3:41
theorem Th42: :: HILBERT3:42
theorem Th43: :: HILBERT3:43
theorem Th44: :: HILBERT3:44
theorem Th45: :: HILBERT3:45
theorem Th46: :: HILBERT3:46
theorem Th47: :: HILBERT3:47
:: deftheorem Def8 defines pseudo-canonical HILBERT3:def 8 :
theorem Th48: :: HILBERT3:48
theorem Th49: :: HILBERT3:49
theorem Th50: :: HILBERT3:50
theorem Th51: :: HILBERT3:51
theorem Th52: :: HILBERT3:52
theorem Th53: :: HILBERT3:53
theorem Th54: :: HILBERT3:54
theorem Th55: :: HILBERT3:55