:: AFPROJ semantic presentation
theorem Th1: :: AFPROJ:1
theorem Th2: :: AFPROJ:2
theorem Th3: :: AFPROJ:3
theorem Th4: :: AFPROJ:4
theorem Th5: :: AFPROJ:5
theorem Th6: :: AFPROJ:6
:: deftheorem Def1 defines AfLines AFPROJ:def 1 :
:: deftheorem Def2 defines AfPlanes AFPROJ:def 2 :
theorem Th7: :: AFPROJ:7
theorem Th8: :: AFPROJ:8
:: deftheorem Def3 defines LinesParallelity AFPROJ:def 3 :
:: deftheorem Def4 defines PlanesParallelity AFPROJ:def 4 :
:: deftheorem Def5 defines LDir AFPROJ:def 5 :
:: deftheorem Def6 defines PDir AFPROJ:def 6 :
theorem Th9: :: AFPROJ:9
theorem Th10: :: AFPROJ:10
theorem Th11: :: AFPROJ:11
theorem Th12: :: AFPROJ:12
theorem Th13: :: AFPROJ:13
:: deftheorem Def7 defines Dir_of_Lines AFPROJ:def 7 :
:: deftheorem Def8 defines Dir_of_Planes AFPROJ:def 8 :
theorem Th14: :: AFPROJ:14
theorem Th15: :: AFPROJ:15
theorem Th16: :: AFPROJ:16
theorem Th17: :: AFPROJ:17
theorem Th18: :: AFPROJ:18
theorem Th19: :: AFPROJ:19
:: deftheorem Def9 defines ProjectivePoints AFPROJ:def 9 :
:: deftheorem Def10 defines ProjectiveLines AFPROJ:def 10 :
definition
let c
1 be
AffinSpace;
func Proj_Inc c
1 -> Relation of
ProjectivePoints a
1,
ProjectiveLines a
1 means :
Def11:
:: AFPROJ:def 11
for b
1, b
2 being
set holds
(
[b1,b2] in a
2 iff not ( ( for b
3 being
Subset of a
1 holds
not ( b
3 is_line & b
2 = [b3,1] & ( ( b
1 in the
carrier of a
1 & b
1 in b
3 ) or b
1 = LDir b
3 ) ) ) & ( for b
3, b
4 being
Subset of a
1 holds
not ( b
3 is_line & b
4 is_plane & b
1 = LDir b
3 & b
2 = [(PDir b4),2] & b
3 '||' b
4 ) ) ) );
existence
ex b1 being Relation of ProjectivePoints c1, ProjectiveLines c1 st
for b2, b3 being set holds
( [b2,b3] in b1 iff not ( ( for b4 being Subset of c1 holds
not ( b4 is_line & b3 = [b4,1] & ( ( b2 in the carrier of c1 & b2 in b4 ) or b2 = LDir b4 ) ) ) & ( for b4, b5 being Subset of c1 holds
not ( b4 is_line & b5 is_plane & b2 = LDir b4 & b3 = [(PDir b5),2] & b4 '||' b5 ) ) ) )
uniqueness
for b1, b2 being Relation of ProjectivePoints c1, ProjectiveLines c1 holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff not ( ( for b5 being Subset of c1 holds
not ( b5 is_line & b4 = [b5,1] & ( ( b3 in the carrier of c1 & b3 in b5 ) or b3 = LDir b5 ) ) ) & ( for b5, b6 being Subset of c1 holds
not ( b5 is_line & b6 is_plane & b3 = LDir b5 & b4 = [(PDir b6),2] & b5 '||' b6 ) ) ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff not ( ( for b5 being Subset of c1 holds
not ( b5 is_line & b4 = [b5,1] & ( ( b3 in the carrier of c1 & b3 in b5 ) or b3 = LDir b5 ) ) ) & ( for b5, b6 being Subset of c1 holds
not ( b5 is_line & b6 is_plane & b3 = LDir b5 & b4 = [(PDir b6),2] & b5 '||' b6 ) ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def11 defines Proj_Inc AFPROJ:def 11 :
definition
let c
1 be
AffinSpace;
func Inc_of_Dir c
1 -> Relation of
Dir_of_Lines a
1,
Dir_of_Planes a
1 means :
Def12:
:: AFPROJ:def 12
for b
1, b
2 being
set holds
(
[b1,b2] in a
2 iff ex b
3, b
4 being
Subset of a
1 st
( b
1 = LDir b
3 & b
2 = PDir b
4 & b
3 is_line & b
4 is_plane & b
3 '||' b
4 ) );
existence
ex b1 being Relation of Dir_of_Lines c1, Dir_of_Planes c1 st
for b2, b3 being set holds
( [b2,b3] in b1 iff ex b4, b5 being Subset of c1 st
( b2 = LDir b4 & b3 = PDir b5 & b4 is_line & b5 is_plane & b4 '||' b5 ) )
uniqueness
for b1, b2 being Relation of Dir_of_Lines c1, Dir_of_Planes c1 holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff ex b5, b6 being Subset of c1 st
( b3 = LDir b5 & b4 = PDir b6 & b5 is_line & b6 is_plane & b5 '||' b6 ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ex b5, b6 being Subset of c1 st
( b3 = LDir b5 & b4 = PDir b6 & b5 is_line & b6 is_plane & b5 '||' b6 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def12 defines Inc_of_Dir AFPROJ:def 12 :
:: deftheorem Def13 defines IncProjSp_of AFPROJ:def 13 :
:: deftheorem Def14 defines ProjHorizon AFPROJ:def 14 :
theorem Th20: :: AFPROJ:20
theorem Th21: :: AFPROJ:21
theorem Th22: :: AFPROJ:22
theorem Th23: :: AFPROJ:23
theorem Th24: :: AFPROJ:24
theorem Th25: :: AFPROJ:25
theorem Th26: :: AFPROJ:26
theorem Th27: :: AFPROJ:27
theorem Th28: :: AFPROJ:28
theorem Th29: :: AFPROJ:29
theorem Th30: :: AFPROJ:30
theorem Th31: :: AFPROJ:31
theorem Th32: :: AFPROJ:32
theorem Th33: :: AFPROJ:33
theorem Th34: :: AFPROJ:34
theorem Th35: :: AFPROJ:35
theorem Th36: :: AFPROJ:36
theorem Th37: :: AFPROJ:37
theorem Th38: :: AFPROJ:38
theorem Th39: :: AFPROJ:39
theorem Th40: :: AFPROJ:40
Lemma42:
for b1 being AffinSpace holds
not ( not b1 is AffinPlane & ( for b2 being Element of the Points of (ProjHorizon b1)
for b3 being Element of the Lines of (ProjHorizon b1) holds b2 on b3 ) )
Lemma43:
for b1 being AffinSpace
for b2, b3 being POINT of (IncProjSp_of b1)
for b4, b5 being LINE of (IncProjSp_of b1) holds
not ( b2 on b4 & b2 on b5 & b3 on b4 & b3 on b5 & not b2 = b3 & not b4 = b5 )
Lemma44:
for b1 being AffinSpace
for b2 being LINE of (IncProjSp_of b1) holds
ex b3, b4, b5 being POINT of (IncProjSp_of b1) st
( b3 on b2 & b4 on b2 & b5 on b2 & b3 <> b4 & b4 <> b5 & b5 <> b3 )
Lemma45:
for b1 being AffinSpace
for b2, b3 being POINT of (IncProjSp_of b1) holds
ex b4 being LINE of (IncProjSp_of b1) st
( b2 on b4 & b3 on b4 )
Lemma46:
for b1 being AffinSpace
for b2, b3 being LINE of (IncProjSp_of b1) holds
not ( b1 is AffinPlane & ( for b4 being POINT of (IncProjSp_of b1) holds
not ( b4 on b2 & b4 on b3 ) ) )
Lemma47:
for b1 being AffinSpace holds
not for b2 being POINT of (IncProjSp_of b1)
for b3 being LINE of (IncProjSp_of b1) holds b2 on b3
theorem Th41: :: AFPROJ:41
theorem Th42: :: AFPROJ:42
Lemma50:
for b1 being AffinSpace
for b2 being Element of b1
for b3, b4, b5, b6 being Subset of b1
for b7, b8, b9 being POINT of (IncProjSp_of b1)
for b10, b11, b12 being LINE of (IncProjSp_of b1) holds
( b3 is_line & b4 is_line & b5 is_plane & b3 c= b5 & b4 c= b5 & b2 = b7 & b10 = [b3,1] & b11 = [b4,1] & b10 <> b11 & b8 on b10 & b9 on b11 & b8 on b12 & b9 on b12 & b8 <> b9 & b12 = [b6,1] & b6 is_line implies b6 c= b5 )
Lemma51:
for b1 being AffinSpace
for b2 being Element of b1
for b3, b4, b5, b6 being Subset of b1
for b7, b8, b9 being POINT of (IncProjSp_of b1)
for b10, b11, b12 being LINE of (IncProjSp_of b1) holds
( b3 is_line & b4 is_line & b5 is_plane & b3 c= b5 & b4 c= b5 & b2 = b7 & b10 = [b3,1] & b11 = [b4,1] & b10 <> b11 & b8 on b10 & b9 on b11 & b8 on b12 & b9 on b12 & b8 <> b9 & b12 = [(PDir b6),2] & b6 is_plane implies ( b6 '||' b5 & b5 '||' b6 ) )
theorem Th43: :: AFPROJ:43
Lemma53:
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being POINT of (IncProjSp_of b1)
for b7, b8, b9, b10 being LINE of (IncProjSp_of b1) holds
not ( b2 on b7 & b3 on b7 & b4 on b8 & b5 on b8 & b6 on b7 & b6 on b8 & b2 on b9 & b4 on b9 & b3 on b10 & b5 on b10 & not b6 on b9 & not b6 on b10 & b7 <> b8 & b6 is Element of b1 & ( for b11 being POINT of (IncProjSp_of b1) holds
not ( b11 on b9 & b11 on b10 ) ) )
Lemma54:
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being POINT of (IncProjSp_of b1)
for b7, b8, b9, b10 being LINE of (IncProjSp_of b1) holds
not ( b2 on b7 & b3 on b7 & b4 on b8 & b5 on b8 & b6 on b7 & b6 on b8 & b2 on b9 & b4 on b9 & b3 on b10 & b5 on b10 & not b6 on b9 & not b6 on b10 & b7 <> b8 & not b6 is Element of b1 & b2 is Element of b1 & ( for b11 being POINT of (IncProjSp_of b1) holds
not ( b11 on b9 & b11 on b10 ) ) )
Lemma55:
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being POINT of (IncProjSp_of b1)
for b7, b8, b9, b10 being LINE of (IncProjSp_of b1) holds
not ( b2 on b7 & b3 on b7 & b4 on b8 & b5 on b8 & b6 on b7 & b6 on b8 & b2 on b9 & b4 on b9 & b3 on b10 & b5 on b10 & not b6 on b9 & not b6 on b10 & b7 <> b8 & not b6 is Element of b1 & not ( not b2 is Element of b1 & not b3 is Element of b1 & not b4 is Element of b1 & not b5 is Element of b1 ) & ( for b11 being POINT of (IncProjSp_of b1) holds
not ( b11 on b9 & b11 on b10 ) ) )
Lemma56:
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being POINT of (IncProjSp_of b1)
for b7, b8, b9, b10 being LINE of (IncProjSp_of b1) holds
not ( b2 on b7 & b3 on b7 & b4 on b8 & b5 on b8 & b6 on b7 & b6 on b8 & b2 on b9 & b4 on b9 & b3 on b10 & b5 on b10 & not b6 on b9 & not b6 on b10 & b7 <> b8 & ( for b11 being POINT of (IncProjSp_of b1) holds
not ( b11 on b9 & b11 on b10 ) ) )
theorem Th44: :: AFPROJ:44
theorem Th45: :: AFPROJ:45
theorem Th46: :: AFPROJ:46
theorem Th47: :: AFPROJ:47
theorem Th48: :: AFPROJ:48
for b
1 being
AffinSpacefor b
2, b
3 being
Subset of b
1for b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of the
carrier of b
1 holds
( b
2 is_line & b
3 is_line & b
2 <> b
3 & b
4 in b
2 & b
4 in b
3 & b
4 <> b
5 & b
4 <> b
8 & b
4 <> b
6 & b
4 <> b
9 & b
4 <> b
7 & b
4 <> b
10 & b
5 in b
2 & b
6 in b
2 & b
7 in b
2 & b
8 in b
3 & b
9 in b
3 & b
10 in b
3 & b
5,b
9 // b
6,b
8 & b
6,b
10 // b
7,b
9 & not ( not b
5 = b
6 & not b
6 = b
7 & not b
5 = b
7 ) implies b
5,b
10 // b
7,b
8 )
theorem Th49: :: AFPROJ:49
theorem Th50: :: AFPROJ:50
for b
1 being
AffinSpacefor b
2, b
3, b
4 being
Subset of b
1for b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Element of the
carrier of b
1 holds
( b
5 in b
2 & b
5 in b
3 & b
5 in b
4 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 in b
2 & b
9 in b
2 & b
7 in b
3 & b
10 in b
3 & b
8 in b
4 & b
11 in b
4 & b
2 is_line & b
3 is_line & b
4 is_line & b
2 <> b
3 & b
2 <> b
4 & b
6,b
7 // b
9,b
10 & b
6,b
8 // b
9,b
11 & ( b
5 = b
9 or b
6 = b
9 ) implies b
7,b
8 // b
10,b
11 )
theorem Th51: :: AFPROJ:51
theorem Th52: :: AFPROJ:52