:: FINSEQOP semantic presentation
theorem Th1: :: FINSEQOP:1
theorem Th2: :: FINSEQOP:2
theorem Th3: :: FINSEQOP:3
canceled;
theorem Th4: :: FINSEQOP:4
theorem Th5: :: FINSEQOP:5
theorem Th6: :: FINSEQOP:6
theorem Th7: :: FINSEQOP:7
theorem Th8: :: FINSEQOP:8
theorem Th9: :: FINSEQOP:9
theorem Th10: :: FINSEQOP:10
Lemma6:
for b1, b2 being non empty set
for b3 being Function of b1,b2
for b4 being Element of 0 -tuples_on b1 holds b3 * b4 = <*> b2
Lemma7:
for b1, b2, b3 being non empty set
for b4 being Nat
for b5 being Function of [:b2,b1:],b3
for b6 being Element of b4 -tuples_on b1
for b7 being Element of 0 -tuples_on b2 holds b5 .: b7,b6 = <*> b3
Lemma8:
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Function of [:b1,b2:],b3
for b6 being Element of 0 -tuples_on b2 holds b5 [;] b4,b6 = <*> b3
Lemma9:
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Function of [:b2,b1:],b3
for b6 being Element of 0 -tuples_on b2 holds b5 [:] b6,b4 = <*> b3
theorem Th11: :: FINSEQOP:11
theorem Th12: :: FINSEQOP:12
theorem Th13: :: FINSEQOP:13
theorem Th14: :: FINSEQOP:14
theorem Th15: :: FINSEQOP:15
theorem Th16: :: FINSEQOP:16
theorem Th17: :: FINSEQOP:17
theorem Th18: :: FINSEQOP:18
theorem Th19: :: FINSEQOP:19
theorem Th20: :: FINSEQOP:20
theorem Th21: :: FINSEQOP:21
theorem Th22: :: FINSEQOP:22
theorem Th23: :: FINSEQOP:23
theorem Th24: :: FINSEQOP:24
Lemma14:
for b1 being non empty set
for b2 being Nat
for b3 being Element of b2 -tuples_on b1 holds
b3 is Function of Seg b2,b1
theorem Th25: :: FINSEQOP:25
theorem Th26: :: FINSEQOP:26
theorem Th27: :: FINSEQOP:27
theorem Th28: :: FINSEQOP:28
theorem Th29: :: FINSEQOP:29
theorem Th30: :: FINSEQOP:30
theorem Th31: :: FINSEQOP:31
theorem Th32: :: FINSEQOP:32
theorem Th33: :: FINSEQOP:33
theorem Th34: :: FINSEQOP:34
theorem Th35: :: FINSEQOP:35
theorem Th36: :: FINSEQOP:36
theorem Th37: :: FINSEQOP:37
theorem Th38: :: FINSEQOP:38
for b
1, b
2, b
3 being non
empty set for b
4, b
5 being
Function of b
1,b
3for b
6 being
Function of b
3,b
2for b
7 being
BinOp of b
3for b
8 being
BinOp of b
2 holds
( ( for b
9, b
10 being
Element of b
3 holds b
6 . (b7 . b9,b10) = b
8 . (b6 . b9),
(b6 . b10) ) implies b
6 * (b7 .: b4,b5) = b
8 .: (b6 * b4),
(b6 * b5) )
theorem Th39: :: FINSEQOP:39
for b
1, b
2, b
3 being non
empty set for b
4 being
Element of b
3for b
5 being
Function of b
1,b
3for b
6 being
Function of b
3,b
2for b
7 being
BinOp of b
3for b
8 being
BinOp of b
2 holds
( ( for b
9, b
10 being
Element of b
3 holds b
6 . (b7 . b9,b10) = b
8 . (b6 . b9),
(b6 . b10) ) implies b
6 * (b7 [;] b4,b5) = b
8 [;] (b6 . b4),
(b6 * b5) )
theorem Th40: :: FINSEQOP:40
for b
1, b
2, b
3 being non
empty set for b
4 being
Element of b
3for b
5 being
Function of b
1,b
3for b
6 being
Function of b
3,b
2for b
7 being
BinOp of b
3for b
8 being
BinOp of b
2 holds
( ( for b
9, b
10 being
Element of b
3 holds b
6 . (b7 . b9,b10) = b
8 . (b6 . b9),
(b6 . b10) ) implies b
6 * (b7 [:] b5,b4) = b
8 [:] (b6 * b5),
(b6 . b4) )
theorem Th41: :: FINSEQOP:41
theorem Th42: :: FINSEQOP:42
theorem Th43: :: FINSEQOP:43
theorem Th44: :: FINSEQOP:44
theorem Th45: :: FINSEQOP:45
theorem Th46: :: FINSEQOP:46
theorem Th47: :: FINSEQOP:47
theorem Th48: :: FINSEQOP:48
theorem Th49: :: FINSEQOP:49
theorem Th50: :: FINSEQOP:50
theorem Th51: :: FINSEQOP:51
theorem Th52: :: FINSEQOP:52
theorem Th53: :: FINSEQOP:53
theorem Th54: :: FINSEQOP:54
theorem Th55: :: FINSEQOP:55
theorem Th56: :: FINSEQOP:56
theorem Th57: :: FINSEQOP:57
theorem Th58: :: FINSEQOP:58
theorem Th59: :: FINSEQOP:59
:: deftheorem Def1 defines is_an_inverseOp_wrt FINSEQOP:def 1 :
:: deftheorem Def2 defines having_an_inverseOp FINSEQOP:def 2 :
:: deftheorem Def3 defines the_inverseOp_wrt FINSEQOP:def 3 :
theorem Th60: :: FINSEQOP:60
canceled;
theorem Th61: :: FINSEQOP:61
canceled;
theorem Th62: :: FINSEQOP:62
canceled;
theorem Th63: :: FINSEQOP:63
theorem Th64: :: FINSEQOP:64
theorem Th65: :: FINSEQOP:65
theorem Th66: :: FINSEQOP:66
theorem Th67: :: FINSEQOP:67
theorem Th68: :: FINSEQOP:68
theorem Th69: :: FINSEQOP:69
theorem Th70: :: FINSEQOP:70
theorem Th71: :: FINSEQOP:71
theorem Th72: :: FINSEQOP:72
theorem Th73: :: FINSEQOP:73
theorem Th74: :: FINSEQOP:74
theorem Th75: :: FINSEQOP:75
theorem Th76: :: FINSEQOP:76
theorem Th77: :: FINSEQOP:77
theorem Th78: :: FINSEQOP:78
theorem Th79: :: FINSEQOP:79
theorem Th80: :: FINSEQOP:80
:: deftheorem Def4 defines * FINSEQOP:def 4 :
theorem Th81: :: FINSEQOP:81
canceled;
theorem Th82: :: FINSEQOP:82
for b
1, b
2 being
set for b
3, b
4, b
5 being
Function holds
(
[b1,b2] in dom (b3 * b4,b5) implies
(b3 * b4,b5) . [b1,b2] = b
3 . [(b4 . b1),(b5 . b2)] )
theorem Th83: :: FINSEQOP:83
for b
1, b
2 being
set for b
3, b
4, b
5 being
Function holds
(
[b1,b2] in dom (b3 * b4,b5) implies
(b3 * b4,b5) . b
1,b
2 = b
3 . (b4 . b1),
(b5 . b2) )
by Th82;
theorem Th84: :: FINSEQOP:84
for b
1, b
2, b
3, b
4, b
5 being non
empty set for b
6 being
Function of
[:b1,b2:],b
3for b
7 being
Function of b
4,b
1for b
8 being
Function of b
5,b
2 holds
b
6 * b
7,b
8 is
Function of
[:b4,b5:],b
3
theorem Th85: :: FINSEQOP:85
theorem Th86: :: FINSEQOP:86
for b
1, b
2, b
3, b
4, b
5 being non
empty set for b
6 being
Element of b
4for b
7 being
Element of b
5for b
8 being
Function of
[:b1,b2:],b
3for b
9 being
Function of b
4,b
1for b
10 being
Function of b
5,b
2 holds
(b8 * b9,b10) . b
6,b
7 = b
8 . (b9 . b6),
(b10 . b7)
theorem Th87: :: FINSEQOP:87
for b
1 being non
empty set for b
2, b
3 being
Element of b
1for b
4 being
BinOp of b
1for b
5 being
Function of b
1,b
1 holds
(
(b4 * (id b1),b5) . b
2,b
3 = b
4 . b
2,
(b5 . b3) &
(b4 * b5,(id b1)) . b
2,b
3 = b
4 . (b5 . b2),b
3 )
theorem Th88: :: FINSEQOP:88
theorem Th89: :: FINSEQOP:89
theorem Th90: :: FINSEQOP:90
theorem Th91: :: FINSEQOP:91
theorem Th92: :: FINSEQOP:92
theorem Th93: :: FINSEQOP:93
theorem Th94: :: FINSEQOP:94
for b
1 being non
empty set for b
2, b
3 being
BinOp of b
1 holds
( b
2 is
commutative & b
2 is
associative & b
2 has_a_unity & b
2 has_an_inverseOp & b
3 = b
2 * (id b1),
(the_inverseOp_wrt b2) implies for b
4, b
5, b
6, b
7 being
Element of b
1 holds b
2 . (b3 . b4,b5),
(b3 . b6,b7) = b
3 . (b2 . b4,b6),
(b2 . b5,b7) )