:: CAT_3 semantic presentation
:: deftheorem Def1 defines /. CAT_3:def 1 :
theorem Th1: :: CAT_3:1
for b
1 being
set for b
2 being non
empty set for b
3, b
4 being
Function of b
1,b
2 holds
( ( for b
5 being
set holds
( b
5 in b
1 implies b
3 /. b
5 = b
4 /. b
5 ) ) implies b
3 = b
4 )
theorem Th2: :: CAT_3:2
theorem Th3: :: CAT_3:3
canceled;
theorem Th4: :: CAT_3:4
canceled;
theorem Th5: :: CAT_3:5
canceled;
theorem Th6: :: CAT_3:6
canceled;
theorem Th7: :: CAT_3:7
for b
1, b
2 being
set for b
3 being non
empty set holds
( b
1 <> b
2 implies for b
4, b
5 being
Element of b
3 holds
(
(b1,b2 --> b4,b5) /. b
1 = b
4 &
(b1,b2 --> b4,b5) /. b
2 = b
5 ) )
:: deftheorem Def2 CAT_3:def 2 :
canceled;
:: deftheorem Def3 defines doms CAT_3:def 3 :
:: deftheorem Def4 defines cods CAT_3:def 4 :
theorem Th8: :: CAT_3:8
theorem Th9: :: CAT_3:9
theorem Th10: :: CAT_3:10
theorem Th11: :: CAT_3:11
:: deftheorem Def5 defines opp CAT_3:def 5 :
theorem Th12: :: CAT_3:12
theorem Th13: :: CAT_3:13
theorem Th14: :: CAT_3:14
:: deftheorem Def6 defines opp CAT_3:def 6 :
theorem Th15: :: CAT_3:15
theorem Th16: :: CAT_3:16
theorem Th17: :: CAT_3:17
:: deftheorem Def7 defines * CAT_3:def 7 :
:: deftheorem Def8 defines * CAT_3:def 8 :
theorem Th18: :: CAT_3:18
for b
1, b
2 being
set for b
3 being
Categoryfor b
4, b
5, b
6 being
Morphism of b
3 holds
( b
1 <> b
2 implies
(b1,b2 --> b4,b5) * b
6 = b
1,b
2 --> (b4 * b6),
(b5 * b6) )
theorem Th19: :: CAT_3:19
for b
1, b
2 being
set for b
3 being
Categoryfor b
4, b
5, b
6 being
Morphism of b
3 holds
( b
1 <> b
2 implies b
4 * (b1,b2 --> b5,b6) = b
1,b
2 --> (b4 * b5),
(b4 * b6) )
theorem Th20: :: CAT_3:20
theorem Th21: :: CAT_3:21
:: deftheorem Def9 defines "*" CAT_3:def 9 :
theorem Th22: :: CAT_3:22
theorem Th23: :: CAT_3:23
for b
1, b
2 being
set for b
3 being
Categoryfor b
4, b
5, b
6, b
7 being
Morphism of b
3 holds
( b
1 <> b
2 implies
(b1,b2 --> b4,b5) "*" (b1,b2 --> b6,b7) = b
1,b
2 --> (b4 * b6),
(b5 * b7) )
theorem Th24: :: CAT_3:24
theorem Th25: :: CAT_3:25
:: deftheorem Def10 defines retraction CAT_3:def 10 :
:: deftheorem Def11 defines coretraction CAT_3:def 11 :
theorem Th26: :: CAT_3:26
theorem Th27: :: CAT_3:27
theorem Th28: :: CAT_3:28
theorem Th29: :: CAT_3:29
theorem Th30: :: CAT_3:30
theorem Th31: :: CAT_3:31
theorem Th32: :: CAT_3:32
theorem Th33: :: CAT_3:33
theorem Th34: :: CAT_3:34
theorem Th35: :: CAT_3:35
theorem Th36: :: CAT_3:36
theorem Th37: :: CAT_3:37
theorem Th38: :: CAT_3:38
:: deftheorem Def12 defines term CAT_3:def 12 :
theorem Th39: :: CAT_3:39
theorem Th40: :: CAT_3:40
theorem Th41: :: CAT_3:41
:: deftheorem Def13 defines init CAT_3:def 13 :
theorem Th42: :: CAT_3:42
theorem Th43: :: CAT_3:43
theorem Th44: :: CAT_3:44
:: deftheorem Def14 defines Projections_family CAT_3:def 14 :
theorem Th45: :: CAT_3:45
theorem Th46: :: CAT_3:46
theorem Th47: :: CAT_3:47
theorem Th48: :: CAT_3:48
theorem Th49: :: CAT_3:49
canceled;
theorem Th50: :: CAT_3:50
theorem Th51: :: CAT_3:51
theorem Th52: :: CAT_3:52
:: deftheorem Def15 defines is_a_product_wrt CAT_3:def 15 :
theorem Th53: :: CAT_3:53
theorem Th54: :: CAT_3:54
theorem Th55: :: CAT_3:55
theorem Th56: :: CAT_3:56
theorem Th57: :: CAT_3:57
theorem Th58: :: CAT_3:58
:: deftheorem Def16 defines is_a_product_wrt CAT_3:def 16 :
theorem Th59: :: CAT_3:59
theorem Th60: :: CAT_3:60
for b
1 being
Categoryfor b
2, b
3, b
4 being
Object of b
1 holds
(
Hom b
2,b
3 <> {} &
Hom b
2,b
4 <> {} implies for b
5 being
Morphism of b
2,b
3for b
6 being
Morphism of b
2,b
4 holds
( b
2 is_a_product_wrt b
5,b
6 iff for b
7 being
Object of b
1 holds
(
Hom b
7,b
3 <> {} &
Hom b
7,b
4 <> {} implies (
Hom b
7,b
2 <> {} & ( for b
8 being
Morphism of b
7,b
3for b
9 being
Morphism of b
7,b
4 holds
ex b
10 being
Morphism of b
7,b
2 st
for b
11 being
Morphism of b
7,b
2 holds
( ( b
5 * b
11 = b
8 & b
6 * b
11 = b
9 ) iff b
10 = b
11 ) ) ) ) ) )
theorem Th61: :: CAT_3:61
theorem Th62: :: CAT_3:62
theorem Th63: :: CAT_3:63
theorem Th64: :: CAT_3:64
theorem Th65: :: CAT_3:65
theorem Th66: :: CAT_3:66
:: deftheorem Def17 defines Injections_family CAT_3:def 17 :
theorem Th67: :: CAT_3:67
theorem Th68: :: CAT_3:68
theorem Th69: :: CAT_3:69
theorem Th70: :: CAT_3:70
theorem Th71: :: CAT_3:71
canceled;
theorem Th72: :: CAT_3:72
theorem Th73: :: CAT_3:73
theorem Th74: :: CAT_3:74
theorem Th75: :: CAT_3:75
theorem Th76: :: CAT_3:76
:: deftheorem Def18 defines is_a_coproduct_wrt CAT_3:def 18 :
theorem Th77: :: CAT_3:77
theorem Th78: :: CAT_3:78
theorem Th79: :: CAT_3:79
theorem Th80: :: CAT_3:80
theorem Th81: :: CAT_3:81
theorem Th82: :: CAT_3:82
theorem Th83: :: CAT_3:83
:: deftheorem Def19 defines is_a_coproduct_wrt CAT_3:def 19 :
theorem Th84: :: CAT_3:84
theorem Th85: :: CAT_3:85
theorem Th86: :: CAT_3:86
theorem Th87: :: CAT_3:87
for b
1 being
Categoryfor b
2, b
3, b
4 being
Object of b
1 holds
(
Hom b
2,b
3 <> {} &
Hom b
4,b
3 <> {} implies for b
5 being
Morphism of b
2,b
3for b
6 being
Morphism of b
4,b
3 holds
( b
3 is_a_coproduct_wrt b
5,b
6 iff for b
7 being
Object of b
1 holds
(
Hom b
2,b
7 <> {} &
Hom b
4,b
7 <> {} implies (
Hom b
3,b
7 <> {} & ( for b
8 being
Morphism of b
2,b
7for b
9 being
Morphism of b
4,b
7 holds
ex b
10 being
Morphism of b
3,b
7 st
for b
11 being
Morphism of b
3,b
7 holds
( ( b
11 * b
5 = b
8 & b
11 * b
6 = b
9 ) iff b
10 = b
11 ) ) ) ) ) )
theorem Th88: :: CAT_3:88
theorem Th89: :: CAT_3:89
theorem Th90: :: CAT_3:90
theorem Th91: :: CAT_3:91
theorem Th92: :: CAT_3:92