:: GROUP_4 semantic presentation
:: deftheorem Def1 GROUP_4:def 1 :
canceled;
:: deftheorem Def2 defines @ GROUP_4:def 2 :
theorem Th1: :: GROUP_4:1
canceled;
theorem Th2: :: GROUP_4:2
canceled;
theorem Th3: :: GROUP_4:3
theorem Th4: :: GROUP_4:4
theorem Th5: :: GROUP_4:5
theorem Th6: :: GROUP_4:6
:: deftheorem Def3 defines Product GROUP_4:def 3 :
theorem Th7: :: GROUP_4:7
canceled;
theorem Th8: :: GROUP_4:8
theorem Th9: :: GROUP_4:9
theorem Th10: :: GROUP_4:10
theorem Th11: :: GROUP_4:11
theorem Th12: :: GROUP_4:12
theorem Th13: :: GROUP_4:13
theorem Th14: :: GROUP_4:14
theorem Th15: :: GROUP_4:15
theorem Th16: :: GROUP_4:16
Lemma12:
for b1 being FinSequence
for b2 being Nat st b2 in dom b1 holds
( ((len b1) - b2) + 1 is Nat & ((len b1) - b2) + 1 >= 1 & ((len b1) - b2) + 1 <= len b1 )
theorem Th17: :: GROUP_4:17
theorem Th18: :: GROUP_4:18
theorem Th19: :: GROUP_4:19
theorem Th20: :: GROUP_4:20
theorem Th21: :: GROUP_4:21
:: deftheorem Def4 defines |^ GROUP_4:def 4 :
theorem Th22: :: GROUP_4:22
canceled;
theorem Th23: :: GROUP_4:23
canceled;
theorem Th24: :: GROUP_4:24
canceled;
theorem Th25: :: GROUP_4:25
theorem Th26: :: GROUP_4:26
theorem Th27: :: GROUP_4:27
theorem Th28: :: GROUP_4:28
theorem Th29: :: GROUP_4:29
theorem Th30: :: GROUP_4:30
for
b1,
b2,
b3 being
Integer for
b4 being
Group for
b5,
b6,
b7 being
Element of
b4 holds
<*b5,b6,b7*> |^ <*(@ b1),(@ b2),(@ b3)*> = <*(b5 |^ b1),(b6 |^ b2),(b7 |^ b3)*>
theorem Th31: :: GROUP_4:31
theorem Th32: :: GROUP_4:32
theorem Th33: :: GROUP_4:33
:: deftheorem Def5 defines gr GROUP_4:def 5 :
theorem Th34: :: GROUP_4:34
canceled;
theorem Th35: :: GROUP_4:35
canceled;
theorem Th36: :: GROUP_4:36
canceled;
theorem Th37: :: GROUP_4:37
theorem Th38: :: GROUP_4:38
theorem Th39: :: GROUP_4:39
theorem Th40: :: GROUP_4:40
theorem Th41: :: GROUP_4:41
theorem Th42: :: GROUP_4:42
theorem Th43: :: GROUP_4:43
theorem Th44: :: GROUP_4:44
:: deftheorem Def6 defines generating GROUP_4:def 6 :
theorem Th45: :: GROUP_4:45
canceled;
theorem Th46: :: GROUP_4:46
:: deftheorem Def7 defines maximal GROUP_4:def 7 :
theorem Th47: :: GROUP_4:47
canceled;
theorem Th48: :: GROUP_4:48
:: deftheorem Def8 defines Phi GROUP_4:def 8 :
theorem Th49: :: GROUP_4:49
canceled;
theorem Th50: :: GROUP_4:50
canceled;
theorem Th51: :: GROUP_4:51
canceled;
theorem Th52: :: GROUP_4:52
theorem Th53: :: GROUP_4:53
theorem Th54: :: GROUP_4:54
theorem Th55: :: GROUP_4:55
theorem Th56: :: GROUP_4:56
:: deftheorem Def9 defines * GROUP_4:def 9 :
theorem Th57: :: GROUP_4:57
theorem Th58: :: GROUP_4:58
theorem Th59: :: GROUP_4:59
for
b1 being
Group for
b2,
b3,
b4 being
Subgroup of
b1 holds
(b2 * b3) * b4 = b2 * (b3 * b4)
theorem Th60: :: GROUP_4:60
theorem Th61: :: GROUP_4:61
theorem Th62: :: GROUP_4:62
theorem Th63: :: GROUP_4:63
theorem Th64: :: GROUP_4:64
theorem Th65: :: GROUP_4:65
:: deftheorem Def10 defines "\/" GROUP_4:def 10 :
theorem Th66: :: GROUP_4:66
canceled;
theorem Th67: :: GROUP_4:67
Lemma39:
for b1 being natural number holds
( b1 mod 2 = 0 or b1 mod 2 = 1 )
by NAT_1:62;
Lemma40:
for b1, b2 being natural number holds (b1 * b2) mod b1 = 0
by NAT_1:63;
Lemma41:
for b1, b2 being natural number st b1 > 1 holds
1 mod b1 = 1
by NAT_1:64;
Lemma42:
for b1, b2, b3, b4 being natural number st b1 mod b3 = 0 & b2 = b1 - (b4 * b3) holds
b2 mod b3 = 0
by NAT_1:65;
Lemma43:
for b1, b2, b3 being natural number st b3 <> 0 & b1 mod b3 = 0 & b2 < b3 holds
(b1 + b2) mod b3 = b2
by NAT_1:66;
Lemma44:
for b1, b2 being natural number st b1 <> 0 holds
(b1 * b2) div b1 = b2
by NAT_1:68;
Lemma45:
for b1, b2, b3 being natural number st b1 mod b2 = 0 holds
(b1 + b3) div b2 = (b1 div b2) + (b3 div b2)
by NAT_1:69;
theorem Th68: :: GROUP_4:68
theorem Th69: :: GROUP_4:69
theorem Th70: :: GROUP_4:70
theorem Th71: :: GROUP_4:71
theorem Th72: :: GROUP_4:72
theorem Th73: :: GROUP_4:73
theorem Th74: :: GROUP_4:74
Lemma49:
for b1 being Group
for b2, b3 being Subgroup of b1 holds b2 is Subgroup of b2 "\/" b3
Lemma50:
for b1 being Group
for b2, b3, b4 being Subgroup of b1 holds (b2 "\/" b3) "\/" b4 is Subgroup of b2 "\/" (b3 "\/" b4)
theorem Th75: :: GROUP_4:75
theorem Th76: :: GROUP_4:76
theorem Th77: :: GROUP_4:77
theorem Th78: :: GROUP_4:78
theorem Th79: :: GROUP_4:79
theorem Th80: :: GROUP_4:80
theorem Th81: :: GROUP_4:81
theorem Th82: :: GROUP_4:82
theorem Th83: :: GROUP_4:83
theorem Th84: :: GROUP_4:84
theorem Th85: :: GROUP_4:85
theorem Th86: :: GROUP_4:86
definition
let c1 be
Group;
func SubJoin c1 -> BinOp of
Subgroups a1 means :
Def11:
:: GROUP_4:def 11
for
b1,
b2 being
Element of
Subgroups a1 for
b3,
b4 being
Subgroup of
a1 st
b1 = b3 &
b2 = b4 holds
a2 . b1,
b2 = b3 "\/" b4;
existence
ex b1 being BinOp of Subgroups c1 st
for b2, b3 being Element of Subgroups c1
for b4, b5 being Subgroup of c1 st b2 = b4 & b3 = b5 holds
b1 . b2,b3 = b4 "\/" b5
uniqueness
for b1, b2 being BinOp of Subgroups c1 st ( for b3, b4 being Element of Subgroups c1
for b5, b6 being Subgroup of c1 st b3 = b5 & b4 = b6 holds
b1 . b3,b4 = b5 "\/" b6 ) & ( for b3, b4 being Element of Subgroups c1
for b5, b6 being Subgroup of c1 st b3 = b5 & b4 = b6 holds
b2 . b3,b4 = b5 "\/" b6 ) holds
b1 = b2
end;
:: deftheorem Def11 defines SubJoin GROUP_4:def 11 :
definition
let c1 be
Group;
func SubMeet c1 -> BinOp of
Subgroups a1 means :
Def12:
:: GROUP_4:def 12
for
b1,
b2 being
Element of
Subgroups a1 for
b3,
b4 being
Subgroup of
a1 st
b1 = b3 &
b2 = b4 holds
a2 . b1,
b2 = b3 /\ b4;
existence
ex b1 being BinOp of Subgroups c1 st
for b2, b3 being Element of Subgroups c1
for b4, b5 being Subgroup of c1 st b2 = b4 & b3 = b5 holds
b1 . b2,b3 = b4 /\ b5
uniqueness
for b1, b2 being BinOp of Subgroups c1 st ( for b3, b4 being Element of Subgroups c1
for b5, b6 being Subgroup of c1 st b3 = b5 & b4 = b6 holds
b1 . b3,b4 = b5 /\ b6 ) & ( for b3, b4 being Element of Subgroups c1
for b5, b6 being Subgroup of c1 st b3 = b5 & b4 = b6 holds
b2 . b3,b4 = b5 /\ b6 ) holds
b1 = b2
end;
:: deftheorem Def12 defines SubMeet GROUP_4:def 12 :
Lemma59:
for b1 being Group holds
( LattStr(# (Subgroups b1),(SubJoin b1),(SubMeet b1) #) is Lattice & LattStr(# (Subgroups b1),(SubJoin b1),(SubMeet b1) #) is 0_Lattice & LattStr(# (Subgroups b1),(SubJoin b1),(SubMeet b1) #) is 1_Lattice )
:: deftheorem Def13 defines lattice GROUP_4:def 13 :
theorem Th87: :: GROUP_4:87
canceled;
theorem Th88: :: GROUP_4:88
canceled;
theorem Th89: :: GROUP_4:89
canceled;
theorem Th90: :: GROUP_4:90
canceled;
theorem Th91: :: GROUP_4:91
canceled;
theorem Th92: :: GROUP_4:92
theorem Th93: :: GROUP_4:93
theorem Th94: :: GROUP_4:94
theorem Th95: :: GROUP_4:95
canceled;
theorem Th96: :: GROUP_4:96
canceled;
theorem Th97: :: GROUP_4:97
canceled;
theorem Th98: :: GROUP_4:98
theorem Th99: :: GROUP_4:99