:: SETWOP_2 semantic presentation
Lemma1:
for b1 being Nat holds
Seg b1 is Element of Fin NAT
by FINSUB_1:def 5;
theorem Th1: :: SETWOP_2:1
canceled;
theorem Th2: :: SETWOP_2:2
canceled;
theorem Th3: :: SETWOP_2:3
theorem Th4: :: SETWOP_2:4
theorem Th5: :: SETWOP_2:5
theorem Th6: :: SETWOP_2:6
theorem Th7: :: SETWOP_2:7
theorem Th8: :: SETWOP_2:8
theorem Th9: :: SETWOP_2:9
theorem Th10: :: SETWOP_2:10
theorem Th11: :: SETWOP_2:11
for b
1, b
2 being non
empty set for b
3 being
Element of
Fin b
1for b
4 being
Element of b
2for b
5, b
6 being
BinOp of b
2for b
7, b
8 being
Function of b
1,b
2 holds
( b
5 is
commutative & b
5 is
associative & b
5 has_a_unity & b
4 = the_unity_wrt b
5 & b
6 . b
4,b
4 = b
4 & ( for b
9, b
10, b
11, b
12 being
Element of b
2 holds b
5 . (b6 . b9,b10),
(b6 . b11,b12) = b
6 . (b5 . b9,b11),
(b5 . b10,b12) ) implies b
6 . (b5 $$ b3,b7),
(b5 $$ b3,b8) = b
5 $$ b
3,
(b6 .: b7,b8) )
Lemma7:
for b1 being non empty set
for b2 being BinOp of b1 holds
( b2 is commutative & b2 is associative implies for b3, b4, b5, b6 being Element of b1 holds b2 . (b2 . b3,b4),(b2 . b5,b6) = b2 . (b2 . b3,b5),(b2 . b4,b6) )
theorem Th12: :: SETWOP_2:12
theorem Th13: :: SETWOP_2:13
for b
1, b
2 being non
empty set for b
3 being
Element of
Fin b
1for b
4, b
5 being
BinOp of b
2for b
6, b
7 being
Function of b
1,b
2 holds
( b
4 is
commutative & b
4 is
associative & b
4 has_a_unity & b
4 has_an_inverseOp & b
5 = b
4 * (id b2),
(the_inverseOp_wrt b4) implies b
5 . (b4 $$ b3,b6),
(b4 $$ b3,b7) = b
4 $$ b
3,
(b5 .: b6,b7) )
theorem Th14: :: SETWOP_2:14
theorem Th15: :: SETWOP_2:15
theorem Th16: :: SETWOP_2:16
theorem Th17: :: SETWOP_2:17
theorem Th18: :: SETWOP_2:18
theorem Th19: :: SETWOP_2:19
theorem Th20: :: SETWOP_2:20
theorem Th21: :: SETWOP_2:21
:: deftheorem Def1 defines [#] SETWOP_2:def 1 :
theorem Th22: :: SETWOP_2:22
theorem Th23: :: SETWOP_2:23
theorem Th24: :: SETWOP_2:24
theorem Th25: :: SETWOP_2:25
theorem Th26: :: SETWOP_2:26
Lemma12:
for b1 being non empty set
for b2 being Element of b1
for b3 being BinOp of b1
for b4, b5 being FinSequence of b1 holds
( len b4 = len b5 & b3 . b2,b2 = b2 implies b3 .: ([#] b4,b2),([#] b5,b2) = [#] (b3 .: b4,b5),b2 )
Lemma13:
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being BinOp of b1
for b5 being FinSequence of b1 holds
( b4 . b2,b3 = b2 implies b4 [:] ([#] b5,b2),b3 = [#] (b4 [:] b5,b3),b2 )
Lemma14:
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being BinOp of b1
for b5 being FinSequence of b1 holds
( b4 . b2,b3 = b3 implies b4 [;] b2,([#] b5,b3) = [#] (b4 [;] b2,b5),b3 )
:: deftheorem Def2 defines $$ SETWOP_2:def 2 :
theorem Th27: :: SETWOP_2:27
canceled;
theorem Th28: :: SETWOP_2:28
canceled;
theorem Th29: :: SETWOP_2:29
canceled;
theorem Th30: :: SETWOP_2:30
canceled;
theorem Th31: :: SETWOP_2:31
canceled;
theorem Th32: :: SETWOP_2:32
canceled;
theorem Th33: :: SETWOP_2:33
canceled;
theorem Th34: :: SETWOP_2:34
canceled;
theorem Th35: :: SETWOP_2:35
theorem Th36: :: SETWOP_2:36
canceled;
theorem Th37: :: SETWOP_2:37
theorem Th38: :: SETWOP_2:38
theorem Th39: :: SETWOP_2:39
theorem Th40: :: SETWOP_2:40
theorem Th41: :: SETWOP_2:41
theorem Th42: :: SETWOP_2:42
theorem Th43: :: SETWOP_2:43
for b
1 being non
empty set for b
2 being
Element of b
1for b
3, b
4 being
BinOp of b
1for b
5, b
6 being
FinSequence of b
1 holds
( b
3 is
commutative & b
3 is
associative & b
3 has_a_unity & b
2 = the_unity_wrt b
3 & b
4 . b
2,b
2 = b
2 & ( for b
7, b
8, b
9, b
10 being
Element of b
1 holds b
3 . (b4 . b7,b8),
(b4 . b9,b10) = b
4 . (b3 . b7,b9),
(b3 . b8,b10) ) &
len b
5 = len b
6 implies b
4 . (b3 "**" b5),
(b3 "**" b6) = b
3 "**" (b4 .: b5,b6) )
theorem Th44: :: SETWOP_2:44
for b
1 being non
empty set for b
2 being
Element of b
1for b
3, b
4 being
BinOp of b
1for b
5 being
Natfor b
6, b
7 being
Element of b
5 -tuples_on b
1 holds
( b
3 is
commutative & b
3 is
associative & b
3 has_a_unity & b
2 = the_unity_wrt b
3 & b
4 . b
2,b
2 = b
2 & ( for b
8, b
9, b
10, b
11 being
Element of b
1 holds b
3 . (b4 . b8,b9),
(b4 . b10,b11) = b
4 . (b3 . b8,b10),
(b3 . b9,b11) ) implies b
4 . (b3 "**" b6),
(b3 "**" b7) = b
3 "**" (b4 .: b6,b7) )
theorem Th45: :: SETWOP_2:45
theorem Th46: :: SETWOP_2:46
theorem Th47: :: SETWOP_2:47
theorem Th48: :: SETWOP_2:48
theorem Th49: :: SETWOP_2:49
theorem Th50: :: SETWOP_2:50
theorem Th51: :: SETWOP_2:51
theorem Th52: :: SETWOP_2:52