:: CARD_FIL semantic presentation
theorem Th1: :: CARD_FIL:1
theorem Th2: :: CARD_FIL:2
:: deftheorem Def1 defines Filter CARD_FIL:def 1 :
theorem Th3: :: CARD_FIL:3
theorem Th4: :: CARD_FIL:4
theorem Th5: :: CARD_FIL:5
theorem Th6: :: CARD_FIL:6
theorem Th7: :: CARD_FIL:7
for b
1 being non
empty set for b
2 being
Filter of b
1for b
3 being non
empty Subset-Family of b
1 holds
( ( for b
4 being
Subset of b
1 holds
( b
4 in b
3 iff b
4 ` in b
2 ) ) implies ( not b
1 in b
3 & ( for b
4, b
5 being
Subset of b
1 holds
( ( b
4 in b
3 & b
5 in b
3 implies b
4 \/ b
5 in b
3 ) & ( b
4 in b
3 & b
5 c= b
4 implies b
5 in b
3 ) ) ) ) )
theorem Th8: :: CARD_FIL:8
theorem Th9: :: CARD_FIL:9
:: deftheorem Def2 defines Ideal CARD_FIL:def 2 :
theorem Th10: :: CARD_FIL:10
theorem Th11: :: CARD_FIL:11
:: deftheorem Def3 defines is_multiplicative_with CARD_FIL:def 3 :
:: deftheorem Def4 defines is_additive_with CARD_FIL:def 4 :
theorem Th12: :: CARD_FIL:12
:: deftheorem Def5 defines uniform CARD_FIL:def 5 :
:: deftheorem Def6 defines principal CARD_FIL:def 6 :
:: deftheorem Def7 defines being_ultrafilter CARD_FIL:def 7 :
:: deftheorem Def8 defines Extend_Filter CARD_FIL:def 8 :
theorem Th13: :: CARD_FIL:13
theorem Th14: :: CARD_FIL:14
:: deftheorem Def9 defines Filters CARD_FIL:def 9 :
theorem Th15: :: CARD_FIL:15
theorem Th16: :: CARD_FIL:16
theorem Th17: :: CARD_FIL:17
:: deftheorem Def10 defines Frechet_Filter CARD_FIL:def 10 :
:: deftheorem Def11 defines Frechet_Ideal CARD_FIL:def 11 :
theorem Th18: :: CARD_FIL:18
theorem Th19: :: CARD_FIL:19
theorem Th20: :: CARD_FIL:20
theorem Th21: :: CARD_FIL:21
theorem Th22: :: CARD_FIL:22
theorem Th23: :: CARD_FIL:23
theorem Th24: :: CARD_FIL:24
:: deftheorem Def12 defines GCH CARD_FIL:def 12 :
:: deftheorem Def13 defines inaccessible CARD_FIL:def 13 :
theorem Th25: :: CARD_FIL:25
:: deftheorem Def14 defines strong_limit CARD_FIL:def 14 :
theorem Th26: :: CARD_FIL:26
theorem Th27: :: CARD_FIL:27
theorem Th28: :: CARD_FIL:28
:: deftheorem Def15 defines strongly_inaccessible CARD_FIL:def 15 :
theorem Th29: :: CARD_FIL:29
theorem Th30: :: CARD_FIL:30
theorem Th31: :: CARD_FIL:31
:: deftheorem Def16 defines measurable CARD_FIL:def 16 :
theorem Th32: :: CARD_FIL:32
theorem Th33: :: CARD_FIL:33
:: deftheorem Def17 defines predecessor CARD_FIL:def 17 :
definition
let c
1 be non
limit Aleph;
let c
2 be
Inf_Matrix of
(predecessor c1),c
1,
bool c
1;
pred c
2 is_Ulam_Matrix_of c
1 means :
Def18:
:: CARD_FIL:def 18
( ( for b
1 being
Element of
predecessor a
1for b
2, b
3 being
Element of a
1 holds
( b
2 <> b
3 implies
(a2 . b1,b2) /\ (a2 . b1,b3) is
empty ) ) & ( for b
1 being
Element of a
1for b
2, b
3 being
Element of
predecessor a
1 holds
( b
2 <> b
3 implies
(a2 . b2,b1) /\ (a2 . b3,b1) is
empty ) ) & ( for b
1 being
Element of
predecessor a
1 holds
Card (a1 \ (union { (a2 . b1,b2) where B is Element of a1 : b2 in a1 } )) <=` predecessor a
1 ) & ( for b
1 being
Element of a
1 holds
Card (a1 \ (union { (a2 . b2,b1) where B is Element of predecessor a1 : b2 in predecessor a1 } )) <=` predecessor a
1 ) );
end;
:: deftheorem Def18 defines is_Ulam_Matrix_of CARD_FIL:def 18 :
for b
1 being non
limit Alephfor b
2 being
Inf_Matrix of
(predecessor b1),b
1,
bool b
1 holds
( b
2 is_Ulam_Matrix_of b
1 iff ( ( for b
3 being
Element of
predecessor b
1for b
4, b
5 being
Element of b
1 holds
( b
4 <> b
5 implies
(b2 . b3,b4) /\ (b2 . b3,b5) is
empty ) ) & ( for b
3 being
Element of b
1for b
4, b
5 being
Element of
predecessor b
1 holds
( b
4 <> b
5 implies
(b2 . b4,b3) /\ (b2 . b5,b3) is
empty ) ) & ( for b
3 being
Element of
predecessor b
1 holds
Card (b1 \ (union { (b2 . b3,b4) where B is Element of b1 : b4 in b1 } )) <=` predecessor b
1 ) & ( for b
3 being
Element of b
1 holds
Card (b1 \ (union { (b2 . b4,b3) where B is Element of predecessor b1 : b4 in predecessor b1 } )) <=` predecessor b
1 ) ) );
theorem Th34: :: CARD_FIL:34
theorem Th35: :: CARD_FIL:35
theorem Th36: :: CARD_FIL:36
theorem Th37: :: CARD_FIL:37
theorem Th38: :: CARD_FIL:38
theorem Th39: :: CARD_FIL:39
theorem Th40: :: CARD_FIL:40
theorem Th41: :: CARD_FIL:41