:: GROUP_1 semantic presentation
:: deftheorem Def1 defines * GROUP_1:def 1 :
E1:
now
set c
1 =
HGrStr(#
REAL ,
addreal #);
E2:
for b
1, b
2 being
Element of
HGrStr(#
REAL ,
addreal #)
for b
3, b
4 being
Real holds
( b
1 = b
3 & b
2 = b
4 implies b
1 * b
2 = b
3 + b
4 )
by BINOP_2:def 9;
thus
for b
1, b
2, b
3 being
Element of
HGrStr(#
REAL ,
addreal #) holds
(b1 * b2) * b
3 = b
1 * (b2 * b3)
reconsider c
2 = 0 as
Element of
HGrStr(#
REAL ,
addreal #) ;
take c
3 = c
2;
let c
4 be
Element of
HGrStr(#
REAL ,
addreal #);
reconsider c
5 = c
4 as
Real ;
thus c
4 * c
3 =
c
5 + 0
by E2
.=
c
4
;
thus c
3 * c
4 =
0
+ c
5
by E2
.=
c
4
;
reconsider c
6 =
- c
5 as
Element of
HGrStr(#
REAL ,
addreal #) ;
take c
7 = c
6;
thus c
4 * c
7 =
c
5 + (- c5)
by E2
.=
c
3
;
thus c
7 * c
4 =
(- c5) + c
5
by E2
.=
c
3
;
end;
:: deftheorem Def2 defines unital GROUP_1:def 2 :
:: deftheorem Def3 defines Group-like GROUP_1:def 3 :
:: deftheorem Def4 defines associative GROUP_1:def 4 :
theorem Th1: :: GROUP_1:1
canceled;
theorem Th2: :: GROUP_1:2
canceled;
theorem Th3: :: GROUP_1:3
canceled;
theorem Th4: :: GROUP_1:4
canceled;
theorem Th5: :: GROUP_1:5
theorem Th6: :: GROUP_1:6
theorem Th7: :: GROUP_1:7
:: deftheorem Def5 defines 1. GROUP_1:def 5 :
theorem Th8: :: GROUP_1:8
canceled;
theorem Th9: :: GROUP_1:9
canceled;
theorem Th10: :: GROUP_1:10
:: deftheorem Def6 defines " GROUP_1:def 6 :
for b
1 being
Groupfor b
2, b
3 being
Element of b
1 holds
( b
3 = b
2 " iff ( b
2 * b
3 = 1. b
1 & b
3 * b
2 = 1. b
1 ) );
theorem Th11: :: GROUP_1:11
canceled;
theorem Th12: :: GROUP_1:12
theorem Th13: :: GROUP_1:13
canceled;
theorem Th14: :: GROUP_1:14
for b
1 being
Groupfor b
2, b
3, b
4 being
Element of b
1 holds
( ( b
2 * b
3 = b
2 * b
4 or b
3 * b
2 = b
4 * b
2 ) implies b
3 = b
4 )
theorem Th15: :: GROUP_1:15
for b
1 being
Groupfor b
2, b
3 being
Element of b
1 holds
( ( b
2 * b
3 = b
2 or b
3 * b
2 = b
2 ) implies b
3 = 1. b
1 )
theorem Th16: :: GROUP_1:16
theorem Th17: :: GROUP_1:17
for b
1 being
Groupfor b
2, b
3 being
Element of b
1 holds
( b
2 " = b
3 " implies b
2 = b
3 )
theorem Th18: :: GROUP_1:18
theorem Th19: :: GROUP_1:19
theorem Th20: :: GROUP_1:20
for b
1 being
Groupfor b
2, b
3 being
Element of b
1 holds
( b
2 * b
3 = 1. b
1 implies ( b
2 = b
3 " & b
3 = b
2 " ) )
theorem Th21: :: GROUP_1:21
for b
1 being
Groupfor b
2, b
3, b
4 being
Element of b
1 holds
( b
2 * b
3 = b
4 iff b
3 = (b2 " ) * b
4 )
theorem Th22: :: GROUP_1:22
for b
1 being
Groupfor b
2, b
3, b
4 being
Element of b
1 holds
( b
2 * b
3 = b
4 iff b
2 = b
4 * (b3 " ) )
theorem Th23: :: GROUP_1:23
theorem Th24: :: GROUP_1:24
theorem Th25: :: GROUP_1:25
theorem Th26: :: GROUP_1:26
for b
1 being
Groupfor b
2, b
3 being
Element of b
1 holds
( b
2 * b
3 = b
3 * b
2 iff
(b2 * b3) " = (b2 " ) * (b3 " ) )
theorem Th27: :: GROUP_1:27
for b
1 being
Groupfor b
2, b
3 being
Element of b
1 holds
( b
2 * b
3 = b
3 * b
2 iff
(b2 " ) * (b3 " ) = (b3 " ) * (b2 " ) )
theorem Th28: :: GROUP_1:28
for b
1 being
Groupfor b
2, b
3 being
Element of b
1 holds
( b
2 * b
3 = b
3 * b
2 iff b
2 * (b3 " ) = (b3 " ) * b
2 )
:: deftheorem Def7 defines inverse_op GROUP_1:def 7 :
theorem Th29: :: GROUP_1:29
canceled;
theorem Th30: :: GROUP_1:30
canceled;
theorem Th31: :: GROUP_1:31
theorem Th32: :: GROUP_1:32
theorem Th33: :: GROUP_1:33
theorem Th34: :: GROUP_1:34
theorem Th35: :: GROUP_1:35
theorem Th36: :: GROUP_1:36
theorem Th37: :: GROUP_1:37
definition
let c
1 be non
empty HGrStr ;
func power c
1 -> Function of
[:the carrier of a1,NAT :],the
carrier of a
1 means :
Def8:
:: GROUP_1:def 8
for b
1 being
Element of a
1 holds
( a
2 . b
1,0
= 1. a
1 & ( for b
2 being
Nat holds a
2 . b
1,
(b2 + 1) = (a2 . b1,b2) * b
1 ) );
existence
ex b1 being Function of [:the carrier of c1,NAT :],the carrier of c1 st
for b2 being Element of c1 holds
( b1 . b2,0 = 1. c1 & ( for b3 being Nat holds b1 . b2,(b3 + 1) = (b1 . b2,b3) * b2 ) )
uniqueness
for b1, b2 being Function of [:the carrier of c1,NAT :],the carrier of c1 holds
( ( for b3 being Element of c1 holds
( b1 . b3,0 = 1. c1 & ( for b4 being Nat holds b1 . b3,(b4 + 1) = (b1 . b3,b4) * b3 ) ) ) & ( for b3 being Element of c1 holds
( b2 . b3,0 = 1. c1 & ( for b4 being Nat holds b2 . b3,(b4 + 1) = (b2 . b3,b4) * b3 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def8 defines power GROUP_1:def 8 :
:: deftheorem Def9 defines |^ GROUP_1:def 9 :
:: deftheorem Def10 defines |^ GROUP_1:def 10 :
Lemma28:
for b1 being Nat
for b2 being Group
for b3 being Element of b2 holds b3 |^ (b1 + 1) = (b3 |^ b1) * b3
by Def8;
Lemma29:
for b1 being Group
for b2 being Element of b1 holds b2 |^ 0 = 1. b1
by Def8;
theorem Th38: :: GROUP_1:38
canceled;
theorem Th39: :: GROUP_1:39
canceled;
theorem Th40: :: GROUP_1:40
canceled;
theorem Th41: :: GROUP_1:41
canceled;
theorem Th42: :: GROUP_1:42
theorem Th43: :: GROUP_1:43
theorem Th44: :: GROUP_1:44
theorem Th45: :: GROUP_1:45
theorem Th46: :: GROUP_1:46
theorem Th47: :: GROUP_1:47
theorem Th48: :: GROUP_1:48
theorem Th49: :: GROUP_1:49
theorem Th50: :: GROUP_1:50
theorem Th51: :: GROUP_1:51
theorem Th52: :: GROUP_1:52
for b
1 being
Natfor b
2 being
Groupfor b
3, b
4 being
Element of b
2 holds
( b
3 * b
4 = b
4 * b
3 implies b
3 * (b4 |^ b1) = (b4 |^ b1) * b
3 )
theorem Th53: :: GROUP_1:53
for b
1, b
2 being
Natfor b
3 being
Groupfor b
4, b
5 being
Element of b
3 holds
( b
4 * b
5 = b
5 * b
4 implies
(b4 |^ b1) * (b5 |^ b2) = (b5 |^ b2) * (b4 |^ b1) )
theorem Th54: :: GROUP_1:54
for b
1 being
Natfor b
2 being
Groupfor b
3, b
4 being
Element of b
2 holds
( b
3 * b
4 = b
4 * b
3 implies
(b3 * b4) |^ b
1 = (b3 |^ b1) * (b4 |^ b1) )
theorem Th55: :: GROUP_1:55
theorem Th56: :: GROUP_1:56
theorem Th57: :: GROUP_1:57
canceled;
theorem Th58: :: GROUP_1:58
canceled;
theorem Th59: :: GROUP_1:59
theorem Th60: :: GROUP_1:60
theorem Th61: :: GROUP_1:61
theorem Th62: :: GROUP_1:62
Lemma44:
for b1 being Integer
for b2 being Group
for b3 being Element of b2 holds b3 |^ (- b1) = (b3 |^ b1) "
Lemma45:
for b1 being Integer holds
not ( not b1 >= 1 & not b1 = 0 & not b1 < 0 )
Lemma46:
for b1 being Integer
for b2 being Group
for b3 being Element of b2 holds b3 |^ (b1 - 1) = (b3 |^ b1) * (b3 |^ (- 1))
Lemma47:
for b1 being Integer holds
not ( not b1 >= 0 & not b1 = - 1 & not b1 < - 1 )
Lemma48:
for b1 being Integer
for b2 being Group
for b3 being Element of b2 holds b3 |^ (b1 + 1) = (b3 |^ b1) * (b3 |^ 1)
theorem Th63: :: GROUP_1:63
theorem Th64: :: GROUP_1:64
theorem Th65: :: GROUP_1:65
theorem Th66: :: GROUP_1:66
Lemma50:
for b1 being Integer
for b2 being Group
for b3 being Element of b2 holds (b3 " ) |^ b1 = (b3 |^ b1) "
theorem Th67: :: GROUP_1:67
theorem Th68: :: GROUP_1:68
theorem Th69: :: GROUP_1:69
theorem Th70: :: GROUP_1:70
theorem Th71: :: GROUP_1:71
theorem Th72: :: GROUP_1:72
theorem Th73: :: GROUP_1:73
theorem Th74: :: GROUP_1:74
theorem Th75: :: GROUP_1:75
theorem Th76: :: GROUP_1:76
canceled;
theorem Th77: :: GROUP_1:77
:: deftheorem Def11 defines being_of_order_0 GROUP_1:def 11 :
theorem Th78: :: GROUP_1:78
canceled;
theorem Th79: :: GROUP_1:79
:: deftheorem Def12 defines ord GROUP_1:def 12 :
theorem Th80: :: GROUP_1:80
canceled;
theorem Th81: :: GROUP_1:81
canceled;
theorem Th82: :: GROUP_1:82
theorem Th83: :: GROUP_1:83
canceled;
theorem Th84: :: GROUP_1:84
theorem Th85: :: GROUP_1:85
theorem Th86: :: GROUP_1:86
:: deftheorem Def13 defines Ord GROUP_1:def 13 :
:: deftheorem Def14 defines finite GROUP_1:def 14 :
:: deftheorem Def15 defines ord GROUP_1:def 15 :
theorem Th87: :: GROUP_1:87
canceled;
theorem Th88: :: GROUP_1:88
canceled;
theorem Th89: :: GROUP_1:89
canceled;
theorem Th90: :: GROUP_1:90
reconsider c1 = HGrStr(# REAL ,addreal #) as Group by Th7;
:: deftheorem Def16 defines commutative GROUP_1:def 16 :
theorem Th91: :: GROUP_1:91
canceled;
theorem Th92: :: GROUP_1:92
theorem Th93: :: GROUP_1:93
canceled;
theorem Th94: :: GROUP_1:94
theorem Th95: :: GROUP_1:95
theorem Th96: :: GROUP_1:96
theorem Th97: :: GROUP_1:97