:: CARD_FIN semantic presentation
Lemma1:
for b1, b2 being set holds
( b1 in b2 implies (b2 \ {b1}) \/ {b1} = b2 )
by ZFMISC_1:140;
Lemma2:
for b1, b2 being set holds
( not b1 in b2 implies (b2 \/ {b1}) \ {b1} = b2 )
by ZFMISC_1:141;
theorem Th1: :: CARD_FIN:1
theorem Th2: :: CARD_FIN:2
theorem Th3: :: CARD_FIN:3
theorem Th4: :: CARD_FIN:4
theorem Th5: :: CARD_FIN:5
theorem Th6: :: CARD_FIN:6
for b
1, b
2 being
Nat holds
(b1 ! ) / ((b1 -' b2) ! ) is
Nat
theorem Th7: :: CARD_FIN:7
theorem Th8: :: CARD_FIN:8
definition
let c
1 be
finite set ;
let c
2 be
Nat;
let c
3, c
4 be
set ;
func Choose c
1,c
2,c
3,c
4 -> Subset of
(Funcs a1,{a3,a4}) means :
Def1:
:: CARD_FIN:def 1
for b
1 being
set holds
( b
1 in a
5 iff ex b
2 being
Function of a
1,
{a3,a4} st
( b
2 = b
1 &
Card (b2 " {a3}) = a
2 ) );
existence
ex b1 being Subset of (Funcs c1,{c3,c4}) st
for b2 being set holds
( b2 in b1 iff ex b3 being Function of c1,{c3,c4} st
( b3 = b2 & Card (b3 " {c3}) = c2 ) )
uniqueness
for b1, b2 being Subset of (Funcs c1,{c3,c4}) holds
( ( for b3 being set holds
( b3 in b1 iff ex b4 being Function of c1,{c3,c4} st
( b4 = b3 & Card (b4 " {c3}) = c2 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4 being Function of c1,{c3,c4} st
( b4 = b3 & Card (b4 " {c3}) = c2 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines Choose CARD_FIN:def 1 :
theorem Th9: :: CARD_FIN:9
theorem Th10: :: CARD_FIN:10
theorem Th11: :: CARD_FIN:11
theorem Th12: :: CARD_FIN:12
theorem Th13: :: CARD_FIN:13
theorem Th14: :: CARD_FIN:14
theorem Th15: :: CARD_FIN:15
theorem Th16: :: CARD_FIN:16
Lemma18:
for b1, b2, b3 being set
for b4 being finite set
for b5 being Nat holds
( b1 <> b2 & not b3 in b4 implies ( { b6 where B is Function of b4 \/ {b3},{b1,b2} : ( Card (b6 " {b1}) = b5 + 1 & b6 . b3 = b1 ) } \/ { b6 where B is Function of b4 \/ {b3},{b1,b2} : ( Card (b6 " {b1}) = b5 + 1 & b6 . b3 = b2 ) } = Choose (b4 \/ {b3}),(b5 + 1),b1,b2 & { b6 where B is Function of b4 \/ {b3},{b1,b2} : ( Card (b6 " {b1}) = b5 + 1 & b6 . b3 = b1 ) } misses { b6 where B is Function of b4 \/ {b3},{b1,b2} : ( Card (b6 " {b1}) = b5 + 1 & b6 . b3 = b2 ) } ) )
theorem Th17: :: CARD_FIN:17
for b
1, b
2, b
3 being
set for b
4 being
finite set for b
5 being
Nat holds
( b
1 <> b
2 & not b
3 in b
4 implies
card (Choose (b4 \/ {b3}),(b5 + 1),b1,b2) = (card (Choose b4,(b5 + 1),b1,b2)) + (card (Choose b4,b5,b1,b2)) )
theorem Th18: :: CARD_FIN:18
theorem Th19: :: CARD_FIN:19
theorem Th20: :: CARD_FIN:20
:: deftheorem Def2 defines Intersection CARD_FIN:def 2 :
theorem Th21: :: CARD_FIN:21
theorem Th22: :: CARD_FIN:22
theorem Th23: :: CARD_FIN:23
theorem Th24: :: CARD_FIN:24
theorem Th25: :: CARD_FIN:25
theorem Th26: :: CARD_FIN:26
theorem Th27: :: CARD_FIN:27
theorem Th28: :: CARD_FIN:28
theorem Th29: :: CARD_FIN:29
theorem Th30: :: CARD_FIN:30
theorem Th31: :: CARD_FIN:31
theorem Th32: :: CARD_FIN:32
theorem Th33: :: CARD_FIN:33
theorem Th34: :: CARD_FIN:34
theorem Th35: :: CARD_FIN:35
theorem Th36: :: CARD_FIN:36
theorem Th37: :: CARD_FIN:37
theorem Th38: :: CARD_FIN:38
theorem Th39: :: CARD_FIN:39
:: deftheorem Def3 defines finite-yielding CARD_FIN:def 3 :
theorem Th40: :: CARD_FIN:40
theorem Th41: :: CARD_FIN:41
Lemma41:
for b1 being Nat
for b2 being XFinSequence holds
( b1 <= len b2 implies ( len (b2 | b1) = b1 & dom (b2 | b1) = b1 ) )
theorem Th42: :: CARD_FIN:42
theorem Th43: :: CARD_FIN:43
theorem Th44: :: CARD_FIN:44
theorem Th45: :: CARD_FIN:45
theorem Th46: :: CARD_FIN:46
theorem Th47: :: CARD_FIN:47
theorem Th48: :: CARD_FIN:48
Lemma47:
for b1 being finite set holds
ex b2 being Function of card b1,b1 st b2 is one-to-one
definition
let c
1 be
Nat;
let c
2 be
finite-yielding Function;
assume E48:
dom c
2 is
finite
;
func Card_Intersection c
2,c
1 -> Nat means :
Def4:
:: CARD_FIN:def 4
for b
1, b
2 being
set for b
3 being
finite set for b
4 being
Function of
card (Choose b3,a1,b1,b2),
Choose b
3,a
1,b
1,b
2 holds
not (
dom a
2 = b
3 & b
4 is
one-to-one & b
1 <> b
2 & ( for b
5 being
XFinSequence of
NAT holds
not (
dom b
5 = dom b
4 & ( for b
6 being
set for b
7 being
Function holds
( b
6 in dom b
5 & b
7 = b
4 . b
6 implies b
5 . b
6 = Card (Intersection a2,b7,b1) ) ) & a
3 = Sum b
5 ) ) );
existence
ex b1 being Nat st
for b2, b3 being set
for b4 being finite set
for b5 being Function of card (Choose b4,c1,b2,b3), Choose b4,c1,b2,b3 holds
not ( dom c2 = b4 & b5 is one-to-one & b2 <> b3 & ( for b6 being XFinSequence of NAT holds
not ( dom b6 = dom b5 & ( for b7 being set
for b8 being Function holds
( b7 in dom b6 & b8 = b5 . b7 implies b6 . b7 = Card (Intersection c2,b8,b2) ) ) & b1 = Sum b6 ) ) )
uniqueness
for b1, b2 being Nat holds
( ( for b3, b4 being set
for b5 being finite set
for b6 being Function of card (Choose b5,c1,b3,b4), Choose b5,c1,b3,b4 holds
not ( dom c2 = b5 & b6 is one-to-one & b3 <> b4 & ( for b7 being XFinSequence of NAT holds
not ( dom b7 = dom b6 & ( for b8 being set
for b9 being Function holds
( b8 in dom b7 & b9 = b6 . b8 implies b7 . b8 = Card (Intersection c2,b9,b3) ) ) & b1 = Sum b7 ) ) ) ) & ( for b3, b4 being set
for b5 being finite set
for b6 being Function of card (Choose b5,c1,b3,b4), Choose b5,c1,b3,b4 holds
not ( dom c2 = b5 & b6 is one-to-one & b3 <> b4 & ( for b7 being XFinSequence of NAT holds
not ( dom b7 = dom b6 & ( for b8 being set
for b9 being Function holds
( b8 in dom b7 & b9 = b6 . b8 implies b7 . b8 = Card (Intersection c2,b9,b3) ) ) & b2 = Sum b7 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines Card_Intersection CARD_FIN:def 4 :
for b
1 being
Natfor b
2 being
finite-yielding Function holds
(
dom b
2 is
finite implies for b
3 being
Nat holds
( b
3 = Card_Intersection b
2,b
1 iff for b
4, b
5 being
set for b
6 being
finite set for b
7 being
Function of
card (Choose b6,b1,b4,b5),
Choose b
6,b
1,b
4,b
5 holds
not (
dom b
2 = b
6 & b
7 is
one-to-one & b
4 <> b
5 & ( for b
8 being
XFinSequence of
NAT holds
not (
dom b
8 = dom b
7 & ( for b
9 being
set for b
10 being
Function holds
( b
9 in dom b
8 & b
10 = b
7 . b
9 implies b
8 . b
9 = Card (Intersection b2,b10,b4) ) ) & b
3 = Sum b
8 ) ) ) ) );
theorem Th49: :: CARD_FIN:49
for b
1 being
Natfor b
2 being
finite-yielding Functionfor b
3, b
4 being
set for b
5 being
finite set for b
6 being
Function of
card (Choose b5,b1,b3,b4),
Choose b
5,b
1,b
3,b
4 holds
(
dom b
2 = b
5 & b
6 is
one-to-one & b
3 <> b
4 implies for b
7 being
XFinSequence of
NAT holds
(
dom b
7 = dom b
6 & ( for b
8 being
set for b
9 being
Function holds
( b
8 in dom b
7 & b
9 = b
6 . b
8 implies b
7 . b
8 = Card (Intersection b2,b9,b3) ) ) implies
Card_Intersection b
2,b
1 = Sum b
7 ) )
theorem Th50: :: CARD_FIN:50
theorem Th51: :: CARD_FIN:51
theorem Th52: :: CARD_FIN:52
theorem Th53: :: CARD_FIN:53
theorem Th54: :: CARD_FIN:54
theorem Th55: :: CARD_FIN:55
theorem Th56: :: CARD_FIN:56
theorem Th57: :: CARD_FIN:57
theorem Th58: :: CARD_FIN:58
theorem Th59: :: CARD_FIN:59
theorem Th60: :: CARD_FIN:60
theorem Th61: :: CARD_FIN:61
theorem Th62: :: CARD_FIN:62
:: deftheorem Def5 defines Sum CARD_FIN:def 5 :
theorem Th63: :: CARD_FIN:63
theorem Th64: :: CARD_FIN:64
theorem Th65: :: CARD_FIN:65
theorem Th66: :: CARD_FIN:66
theorem Th67: :: CARD_FIN:67
theorem Th68: :: CARD_FIN:68
theorem Th69: :: CARD_FIN:69
Lemma67:
for b1, b2 being finite set holds
not ( not b1 is empty & not b2 is empty & ( for b3 being XFinSequence of INT holds
not ( dom b3 = card b2 & ((card b2) |^ (card b1)) - (Sum b3) = Card { b4 where B is Function of b1,b2 : b4 is onto } & ( for b4 being Nat holds
( b4 in dom b3 implies b3 . b4 = (((- 1) |^ b4) * ((card b2) choose (b4 + 1))) * ((((card b2) - b4) - 1) |^ (card b1)) ) ) ) ) )
theorem Th70: :: CARD_FIN:70
theorem Th71: :: CARD_FIN:71
theorem Th72: :: CARD_FIN:72
Lemma70:
for b1, b2 being finite set
for b3 being Function of b1,b2 holds
not ( dom b3 = b1 & b3 is one-to-one & ( for b4 being XFinSequence of INT holds
not ( dom b4 = card b1 & ((card b1) ! ) - (Sum b4) = Card { b5 where B is Function of b1, rng b3 : ( b5 is one-to-one & ( for b1 being set holds
not ( b6 in b1 & not b5 . b6 <> b3 . b6 ) ) ) } & ( for b5 being Nat holds
( b5 in dom b4 implies b4 . b5 = (((- 1) |^ b5) * ((card b1) ! )) / ((b5 + 1) ! ) ) ) ) ) )
theorem Th73: :: CARD_FIN:73
theorem Th74: :: CARD_FIN:74