:: TRANSGEO semantic presentation
theorem Th1: :: TRANSGEO:1
canceled;
theorem Th2: :: TRANSGEO:2
theorem Th3: :: TRANSGEO:3
canceled;
theorem Th4: :: TRANSGEO:4
:: deftheorem Def1 defines \ TRANSGEO:def 1 :
scheme :: TRANSGEO:sch 1
s1{ F
1()
-> non
empty set , P
1[
set ,
set ] } :
ex b
1 being
Permutation of F
1() st
for b
2, b
3 being
Element of F
1() holds
( b
1 . b
2 = b
3 iff P
1[b
2,b
3] )
provided
E3:
for b
1 being
Element of F
1() holds
ex b
2 being
Element of F
1() st P
1[b
1,b
2]
and
E4:
for b
1 being
Element of F
1() holds
ex b
2 being
Element of F
1() st P
1[b
2,b
1]
and
E5:
for b
1, b
2, b
3 being
Element of F
1() holds
( P
1[b
1,b
2] & P
1[b
3,b
2] implies b
1 = b
3 )
and
E6:
for b
1, b
2, b
3 being
Element of F
1() holds
( P
1[b
1,b
2] & P
1[b
1,b
3] implies b
2 = b
3 )
theorem Th5: :: TRANSGEO:5
canceled;
theorem Th6: :: TRANSGEO:6
canceled;
theorem Th7: :: TRANSGEO:7
canceled;
theorem Th8: :: TRANSGEO:8
canceled;
theorem Th9: :: TRANSGEO:9
theorem Th10: :: TRANSGEO:10
canceled;
theorem Th11: :: TRANSGEO:11
Lemma3:
for b1 being non empty set
for b2, b3, b4 being Permutation of b1 holds
( b2 * b3 = b2 * b4 implies b3 = b4 )
Lemma4:
for b1 being non empty set
for b2, b3, b4 being Permutation of b1 holds
( b2 * b3 = b4 * b3 implies b2 = b4 )
theorem Th12: :: TRANSGEO:12
canceled;
theorem Th13: :: TRANSGEO:13
theorem Th14: :: TRANSGEO:14
canceled;
theorem Th15: :: TRANSGEO:15
canceled;
theorem Th16: :: TRANSGEO:16
theorem Th17: :: TRANSGEO:17
theorem Th18: :: TRANSGEO:18
theorem Th19: :: TRANSGEO:19
theorem Th20: :: TRANSGEO:20
theorem Th21: :: TRANSGEO:21
:: deftheorem Def2 defines is_FormalIz_of TRANSGEO:def 2 :
theorem Th22: :: TRANSGEO:22
canceled;
theorem Th23: :: TRANSGEO:23
theorem Th24: :: TRANSGEO:24
theorem Th25: :: TRANSGEO:25
theorem Th26: :: TRANSGEO:26
for b
1 being non
empty set for b
2, b
3 being
Permutation of b
1for b
4 being
Relation of
[:b1,b1:] holds
( ( for b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1 holds
(
[[b7,b8],[b5,b6]] in b
4 &
[[b5,b6],[b9,b10]] in b
4 & b
5 <> b
6 implies
[[b7,b8],[b9,b10]] in b
4 ) ) & ( for b
5, b
6, b
7 being
Element of b
1 holds
[[b5,b5],[b6,b7]] in b
4 ) & b
2 is_FormalIz_of b
4 & b
3 is_FormalIz_of b
4 implies b
2 * b
3 is_FormalIz_of b
4 )
definition
let c
1 be non
empty set ;
let c
2 be
Permutation of c
1;
let c
3 be
Relation of
[:c1,c1:];
pred c
2 is_automorphism_of c
3 means :
Def3:
:: TRANSGEO:def 3
for b
1, b
2, b
3, b
4 being
Element of a
1 holds
(
[[b1,b2],[b3,b4]] in a
3 iff
[[(a2 . b1),(a2 . b2)],[(a2 . b3),(a2 . b4)]] in a
3 );
end;
:: deftheorem Def3 defines is_automorphism_of TRANSGEO:def 3 :
for b
1 being non
empty set for b
2 being
Permutation of b
1for b
3 being
Relation of
[:b1,b1:] holds
( b
2 is_automorphism_of b
3 iff for b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
[[b4,b5],[b6,b7]] in b
3 iff
[[(b2 . b4),(b2 . b5)],[(b2 . b6),(b2 . b7)]] in b
3 ) );
theorem Th27: :: TRANSGEO:27
canceled;
theorem Th28: :: TRANSGEO:28
theorem Th29: :: TRANSGEO:29
theorem Th30: :: TRANSGEO:30
theorem Th31: :: TRANSGEO:31
theorem Th32: :: TRANSGEO:32
for b
1 being non
empty set for b
2 being
Permutation of b
1for b
3 being
Relation of
[:b1,b1:] holds
( ( for b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
(
[[b6,b7],[b4,b5]] in b
3 &
[[b4,b5],[b8,b9]] in b
3 & b
4 <> b
5 implies
[[b6,b7],[b8,b9]] in b
3 ) ) & ( for b
4, b
5, b
6 being
Element of b
1 holds
[[b4,b4],[b5,b6]] in b
3 ) & b
3 is_symmetric_in [:b1,b1:] & b
2 is_FormalIz_of b
3 implies b
2 is_automorphism_of b
3 )
theorem Th33: :: TRANSGEO:33
:: deftheorem Def4 defines is_DIL_of TRANSGEO:def 4 :
theorem Th34: :: TRANSGEO:34
canceled;
theorem Th35: :: TRANSGEO:35
definition
let c
1 be non
empty AffinStruct ;
attr a
1 is
CongrSpace-like means :
Def5:
:: TRANSGEO:def 5
( ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( b
1,b
2 // b
5,b
6 & b
5,b
6 // b
3,b
4 & b
5 <> b
6 implies b
1,b
2 // b
3,b
4 ) ) & ( for b
1, b
2, b
3 being
Element of a
1 holds b
1,b
1 // b
2,b
3 ) & ( for b
1, b
2, b
3, b
4 being
Element of a
1 holds
( b
1,b
2 // b
3,b
4 implies b
3,b
4 // b
1,b
2 ) ) & ( for b
1, b
2 being
Element of a
1 holds b
1,b
2 // b
1,b
2 ) );
end;
:: deftheorem Def5 defines CongrSpace-like TRANSGEO:def 5 :
for b
1 being non
empty AffinStruct holds
( b
1 is
CongrSpace-like iff ( ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
6,b
7 & b
6,b
7 // b
4,b
5 & b
6 <> b
7 implies b
2,b
3 // b
4,b
5 ) ) & ( for b
2, b
3, b
4 being
Element of b
1 holds b
2,b
2 // b
3,b
4 ) & ( for b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 implies b
4,b
5 // b
2,b
3 ) ) & ( for b
2, b
3 being
Element of b
1 holds b
2,b
3 // b
2,b
3 ) ) );
Lemma16:
for b1 being CongrSpace holds the CONGR of b1 is_reflexive_in [:the carrier of b1,the carrier of b1:]
Lemma17:
for b1 being CongrSpace holds the CONGR of b1 is_symmetric_in [:the carrier of b1,the carrier of b1:]
theorem Th36: :: TRANSGEO:36
canceled;
theorem Th37: :: TRANSGEO:37
theorem Th38: :: TRANSGEO:38
theorem Th39: :: TRANSGEO:39
theorem Th40: :: TRANSGEO:40
:: deftheorem Def6 defines positive_dilatation TRANSGEO:def 6 :
theorem Th41: :: TRANSGEO:41
canceled;
theorem Th42: :: TRANSGEO:42
:: deftheorem Def7 defines negative_dilatation TRANSGEO:def 7 :
theorem Th43: :: TRANSGEO:43
canceled;
theorem Th44: :: TRANSGEO:44
theorem Th45: :: TRANSGEO:45
theorem Th46: :: TRANSGEO:46
theorem Th47: :: TRANSGEO:47
theorem Th48: :: TRANSGEO:48
theorem Th49: :: TRANSGEO:49
:: deftheorem Def8 defines dilatation TRANSGEO:def 8 :
theorem Th50: :: TRANSGEO:50
canceled;
theorem Th51: :: TRANSGEO:51
theorem Th52: :: TRANSGEO:52
theorem Th53: :: TRANSGEO:53
theorem Th54: :: TRANSGEO:54
theorem Th55: :: TRANSGEO:55
theorem Th56: :: TRANSGEO:56
theorem Th57: :: TRANSGEO:57
theorem Th58: :: TRANSGEO:58
theorem Th59: :: TRANSGEO:59
theorem Th60: :: TRANSGEO:60
theorem Th61: :: TRANSGEO:61
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2,b
3 '||' b
4,b
5 & not b
2,b
4 '||' b
3,b
5 & ( for b
6 being
Element of b
1 holds
not (
LIN b
2,b
4,b
6 &
LIN b
3,b
5,b
6 ) ) )
theorem Th62: :: TRANSGEO:62
theorem Th63: :: TRANSGEO:63
theorem Th64: :: TRANSGEO:64
theorem Th65: :: TRANSGEO:65
:: deftheorem Def9 defines translation TRANSGEO:def 9 :
theorem Th66: :: TRANSGEO:66
canceled;
theorem Th67: :: TRANSGEO:67
theorem Th68: :: TRANSGEO:68
canceled;
theorem Th69: :: TRANSGEO:69
theorem Th70: :: TRANSGEO:70
theorem Th71: :: TRANSGEO:71
theorem Th72: :: TRANSGEO:72
Lemma42:
for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Tr & not LIN b2,b4 . b2,b3 implies ( b2,b3 // b4 . b2,b4 . b3 & b2,b4 . b2 // b3,b4 . b3 ) )
Lemma43:
for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Tr & b2 <> b4 . b2 & LIN b2,b4 . b2,b3 implies b2,b4 . b2 // b3,b4 . b3 )
Lemma44:
for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Tr & Mid b2,b4 . b2,b3 & b2 <> b4 . b2 implies b2,b3 // b4 . b2,b4 . b3 )
Lemma45:
for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Tr & b2 <> b4 . b2 & b3 <> b4 . b2 & Mid b2,b3,b4 . b2 implies b2,b3 // b4 . b2,b4 . b3 )
Lemma46:
for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Tr & b2 <> b4 . b2 & LIN b2,b4 . b2,b3 implies b2,b3 // b4 . b2,b4 . b3 )
theorem Th73: :: TRANSGEO:73
theorem Th74: :: TRANSGEO:74
theorem Th75: :: TRANSGEO:75
theorem Th76: :: TRANSGEO:76
theorem Th77: :: TRANSGEO:77
theorem Th78: :: TRANSGEO:78
theorem Th79: :: TRANSGEO:79
theorem Th80: :: TRANSGEO:80
theorem Th81: :: TRANSGEO:81
canceled;
theorem Th82: :: TRANSGEO:82
theorem Th83: :: TRANSGEO:83
:: deftheorem Def10 defines dilatation TRANSGEO:def 10 :
theorem Th84: :: TRANSGEO:84
canceled;
theorem Th85: :: TRANSGEO:85
theorem Th86: :: TRANSGEO:86
theorem Th87: :: TRANSGEO:87
theorem Th88: :: TRANSGEO:88
theorem Th89: :: TRANSGEO:89
theorem Th90: :: TRANSGEO:90
theorem Th91: :: TRANSGEO:91
theorem Th92: :: TRANSGEO:92
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2,b
3 // b
4,b
5 & not b
2,b
4 // b
3,b
5 & ( for b
6 being
Element of b
1 holds
not (
LIN b
2,b
4,b
6 &
LIN b
3,b
5,b
6 ) ) )
theorem Th93: :: TRANSGEO:93
theorem Th94: :: TRANSGEO:94
theorem Th95: :: TRANSGEO:95
theorem Th96: :: TRANSGEO:96
theorem Th97: :: TRANSGEO:97
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 & b
2,b
3 // b
4,b
5 & b
2,b
3 // b
4,b
6 & b
2,b
4 // b
3,b
5 & b
2,b
4 // b
3,b
6 implies b
5 = b
6 )
:: deftheorem Def11 defines translation TRANSGEO:def 11 :
theorem Th98: :: TRANSGEO:98
canceled;
theorem Th99: :: TRANSGEO:99
theorem Th100: :: TRANSGEO:100
theorem Th101: :: TRANSGEO:101
canceled;
theorem Th102: :: TRANSGEO:102
theorem Th103: :: TRANSGEO:103
theorem Th104: :: TRANSGEO:104
theorem Th105: :: TRANSGEO:105
:: deftheorem Def12 defines collineation TRANSGEO:def 12 :
theorem Th106: :: TRANSGEO:106
canceled;
theorem Th107: :: TRANSGEO:107
theorem Th108: :: TRANSGEO:108
theorem Th109: :: TRANSGEO:109
theorem Th110: :: TRANSGEO:110
theorem Th111: :: TRANSGEO:111
theorem Th112: :: TRANSGEO:112
theorem Th113: :: TRANSGEO:113
theorem Th114: :: TRANSGEO:114
theorem Th115: :: TRANSGEO:115
theorem Th116: :: TRANSGEO:116
theorem Th117: :: TRANSGEO:117