:: AFVECT0 semantic presentation
definition
let c
1 be non
empty AffinStruct ;
attr a
1 is
WeakAffVect-like means :
Def1:
:: AFVECT0:def 1
( ( for b
1, b
2, b
3 being
Element of a
1 holds
( b
1,b
2 // b
3,b
3 implies b
1 = b
2 ) ) & ( 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
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
ex b
4 being
Element of a
1 st b
1,b
2 // b
3,b
4 ) & ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( b
1,b
2 // b
4,b
5 & b
1,b
3 // b
4,b
6 implies b
2,b
3 // b
5,b
6 ) ) & ( for b
1, b
2 being
Element of a
1 holds
ex b
3 being
Element of a
1 st b
1,b
3 // b
3,b
2 ) & ( 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
1,b
3 // b
2,b
4 ) ) );
end;
:: deftheorem Def1 defines WeakAffVect-like AFVECT0:def 1 :
for b
1 being non
empty AffinStruct holds
( b
1 is
WeakAffVect-like iff ( ( for b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3 // b
4,b
4 implies b
2 = b
3 ) ) & ( 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
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
ex b
5 being
Element of b
1 st b
2,b
3 // b
4,b
5 ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
5,b
6 & b
2,b
4 // b
5,b
7 implies b
3,b
4 // b
6,b
7 ) ) & ( for b
2, b
3 being
Element of b
1 holds
ex b
4 being
Element of b
1 st b
2,b
4 // b
4,b
3 ) & ( 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
2,b
4 // b
3,b
5 ) ) ) );
theorem Th1: :: AFVECT0:1
canceled;
theorem Th2: :: AFVECT0:2
theorem Th3: :: AFVECT0:3
theorem Th4: :: AFVECT0:4
for b
1 being
WeakAffVectfor 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 )
theorem Th5: :: AFVECT0:5
theorem Th6: :: AFVECT0:6
for b
1 being
WeakAffVectfor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 & b
2,b
3 // b
4,b
6 implies b
5 = b
6 )
theorem Th7: :: AFVECT0:7
theorem Th8: :: AFVECT0:8
for b
1 being
WeakAffVectfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 implies b
3,b
2 // b
5,b
4 )
theorem Th9: :: AFVECT0:9
for b
1 being
WeakAffVectfor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 & b
2,b
4 // b
6,b
5 implies b
3 = b
6 )
theorem Th10: :: AFVECT0:10
for b
1 being
WeakAffVectfor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 & b
6,b
7 // b
2,b
3 & b
6,b
8 // b
4,b
5 implies b
7 = b
8 )
theorem Th11: :: AFVECT0:11
for b
1 being
WeakAffVectfor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 & b
6,b
7 // b
3,b
2 & b
6,b
8 // b
5,b
4 implies b
7 = b
8 )
theorem Th12: :: AFVECT0:12
for b
1 being
WeakAffVectfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 & b
6,b
7 // b
8,b
9 & b
3,b
10 // b
6,b
7 & b
5,b
11 // b
8,b
9 implies b
2,b
10 // b
4,b
11 )
theorem Th13: :: AFVECT0:13
for b
1 being
WeakAffVectfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 & b
2,b
6 // b
7,b
5 implies b
3,b
6 // b
7,b
4 )
definition
let c
1 be
WeakAffVect;
let c
2, c
3 be
Element of c
1;
pred MDist c
2,c
3 means :
Def2:
:: AFVECT0:def 2
( a
2,a
3 // a
3,a
2 & a
2 <> a
3 );
irreflexivity
for b1 being Element of c1 holds
not ( b1,b1 // b1,b1 & b1 <> b1 )
;
symmetry
for b1, b2 being Element of c1 holds
( b1,b2 // b2,b1 & b1 <> b2 implies ( b2,b1 // b1,b2 & b2 <> b1 ) )
by Th4;
end;
:: deftheorem Def2 defines MDist AFVECT0:def 2 :
theorem Th14: :: AFVECT0:14
canceled;
theorem Th15: :: AFVECT0:15
canceled;
theorem Th16: :: AFVECT0:16
theorem Th17: :: AFVECT0:17
canceled;
theorem Th18: :: AFVECT0:18
theorem Th19: :: AFVECT0:19
:: deftheorem Def3 defines Mid AFVECT0:def 3 :
theorem Th20: :: AFVECT0:20
canceled;
theorem Th21: :: AFVECT0:21
theorem Th22: :: AFVECT0:22
theorem Th23: :: AFVECT0:23
theorem Th24: :: AFVECT0:24
theorem Th25: :: AFVECT0:25
theorem Th26: :: AFVECT0:26
theorem Th27: :: AFVECT0:27
theorem Th28: :: AFVECT0:28
theorem Th29: :: AFVECT0:29
theorem Th30: :: AFVECT0:30
for b
1 being
WeakAffVectfor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
Mid b
2,b
3,b
4 &
Mid b
5,b
3,b
6 implies b
2,b
5 // b
6,b
4 )
theorem Th31: :: AFVECT0:31
for b
1 being
WeakAffVectfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
Mid b
2,b
3,b
4 &
Mid b
5,b
6,b
7 &
MDist b
3,b
6 implies b
2,b
5 // b
7,b
4 )
definition
let c
1 be
WeakAffVect;
let c
2, c
3 be
Element of c
1;
func PSym c
2,c
3 -> Element of a
1 means :
Def4:
:: AFVECT0:def 4
Mid a
3,a
2,a
4;
correctness
existence
ex b1 being Element of c1 st Mid c3,c2,b1;
uniqueness
for b1, b2 being Element of c1 holds
( Mid c3,c2,b1 & Mid c3,c2,b2 implies b1 = b2 );
by Th26, Th27;
end;
:: deftheorem Def4 defines PSym AFVECT0:def 4 :
theorem Th32: :: AFVECT0:32
canceled;
theorem Th33: :: AFVECT0:33
theorem Th34: :: AFVECT0:34
canceled;
theorem Th35: :: AFVECT0:35
theorem Th36: :: AFVECT0:36
theorem Th37: :: AFVECT0:37
theorem Th38: :: AFVECT0:38
theorem Th39: :: AFVECT0:39
theorem Th40: :: AFVECT0:40
for b
1 being
WeakAffVectfor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 iff
PSym b
6,b
2,
PSym b
6,b
3 // PSym b
6,b
4,
PSym b
6,b
5 )
theorem Th41: :: AFVECT0:41
theorem Th42: :: AFVECT0:42
for b
1 being
WeakAffVectfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
Mid b
2,b
3,b
4 iff
Mid PSym b
5,b
2,
PSym b
5,b
3,
PSym b
5,b
4 )
theorem Th43: :: AFVECT0:43
theorem Th44: :: AFVECT0:44
theorem Th45: :: AFVECT0:45
theorem Th46: :: AFVECT0:46
theorem Th47: :: AFVECT0:47
theorem Th48: :: AFVECT0:48
definition
let c
1 be
WeakAffVect;
let c
2, c
3, c
4 be
Element of c
1;
func Padd c
2,c
3,c
4 -> Element of a
1 means :
Def5:
:: AFVECT0:def 5
a
2,a
3 // a
4,a
5;
correctness
existence
ex b1 being Element of c1 st c2,c3 // c4,b1;
uniqueness
for b1, b2 being Element of c1 holds
( c2,c3 // c4,b1 & c2,c3 // c4,b2 implies b1 = b2 );
by Def1, Th6;
end;
:: deftheorem Def5 defines Padd AFVECT0:def 5 :
Lemma31:
for b1 being WeakAffVect
for b2, b3, b4 being Element of b1 holds
( Pcom b2,b3 = b4 iff b3,b2 // b2,b4 )
definition
let c
1 be
WeakAffVect;
let c
2 be
Element of c
1;
canceled;func Padd c
2 -> BinOp of the
carrier of a
1 means :
Def7:
:: AFVECT0:def 7
for b
1, b
2 being
Element of a
1 holds a
3 . b
1,b
2 = Padd a
2,b
1,b
2;
existence
ex b1 being BinOp of the carrier of c1 st
for b2, b3 being Element of c1 holds b1 . b2,b3 = Padd c2,b2,b3
uniqueness
for b1, b2 being BinOp of the carrier of c1 holds
( ( for b3, b4 being Element of c1 holds b1 . b3,b4 = Padd c2,b3,b4 ) & ( for b3, b4 being Element of c1 holds b2 . b3,b4 = Padd c2,b3,b4 ) implies b1 = b2 )
end;
:: deftheorem Def6 AFVECT0:def 6 :
canceled;
:: deftheorem Def7 defines Padd AFVECT0:def 7 :
:: deftheorem Def8 defines Pcom AFVECT0:def 8 :
:: deftheorem Def9 defines GroupVect AFVECT0:def 9 :
theorem Th49: :: AFVECT0:49
canceled;
theorem Th50: :: AFVECT0:50
canceled;
theorem Th51: :: AFVECT0:51
canceled;
theorem Th52: :: AFVECT0:52
canceled;
theorem Th53: :: AFVECT0:53
canceled;
theorem Th54: :: AFVECT0:54
canceled;
theorem Th55: :: AFVECT0:55
theorem Th56: :: AFVECT0:56
canceled;
theorem Th57: :: AFVECT0:57
Lemma34:
for b1 being WeakAffVect
for b2 being Element of b1
for b3, b4 being Element of (GroupVect b1,b2) holds b3 + b4 = b4 + b3
Lemma35:
for b1 being WeakAffVect
for b2 being Element of b1
for b3, b4, b5 being Element of (GroupVect b1,b2) holds (b3 + b4) + b5 = b3 + (b4 + b5)
Lemma36:
for b1 being WeakAffVect
for b2 being Element of b1
for b3 being Element of (GroupVect b1,b2) holds b3 + (0. (GroupVect b1,b2)) = b3
Lemma37:
for b1 being WeakAffVect
for b2 being Element of b1 holds
( GroupVect b1,b2 is Abelian & GroupVect b1,b2 is add-associative & GroupVect b1,b2 is right_zeroed )
Lemma38:
for b1 being WeakAffVect
for b2 being Element of b1 holds GroupVect b1,b2 is right_complementable
theorem Th58: :: AFVECT0:58
theorem Th59: :: AFVECT0:59
theorem Th60: :: AFVECT0:60
canceled;
theorem Th61: :: AFVECT0:61
canceled;
theorem Th62: :: AFVECT0:62
canceled;
theorem Th63: :: AFVECT0:63
canceled;
theorem Th64: :: AFVECT0:64
canceled;
theorem Th65: :: AFVECT0:65
canceled;
theorem Th66: :: AFVECT0:66
theorem Th67: :: AFVECT0:67
theorem Th68: :: AFVECT0:68
canceled;
theorem Th69: :: AFVECT0:69
theorem Th70: :: AFVECT0:70
theorem Th71: :: AFVECT0:71
theorem Th72: :: AFVECT0:72
:: deftheorem Def10 defines is_Iso_of AFVECT0:def 10 :
:: deftheorem Def11 defines are_Iso AFVECT0:def 11 :
theorem Th73: :: AFVECT0:73
canceled;
theorem Th74: :: AFVECT0:74
canceled;
theorem Th75: :: AFVECT0:75
theorem Th76: :: AFVECT0:76
theorem Th77: :: AFVECT0:77
theorem Th78: :: AFVECT0:78