AFPROJ: A Projective Closure and Projective Horizon of an Affine Space

:: A Projective Closure and Projective Horizon of an Affine Space
:: by Henryk Oryszczyszyn and Krzysztof Pra\.zmowski
::
:: Received December 17, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

:: The aim of this article is to formalize the well known construction of
:: the projective closure of an affine space. We begin with some evident
:: properties of planes in affine planes.

theorem Th1: :: AFPROJ:1

for AS being AffinSpace
for X being Subset of AS st AS is AffinPlane & X = the carrier of AS holds
X is being_plane

proof end;

theorem Th2: :: AFPROJ:2

for AS being AffinSpace
for X being Subset of AS st AS is AffinPlane & X is being_plane holds
X = the carrier of AS

proof end;

theorem Th3: :: AFPROJ:3

for AS being AffinSpace
for X, Y being Subset of AS st AS is AffinPlane & X is being_plane & Y is being_plane holds
X = Y

proof end;

theorem :: AFPROJ:4

for AS being AffinSpace
for X being Subset of AS st X = the carrier of AS & X is being_plane holds
AS is AffinPlane

proof end;

theorem Th5: :: AFPROJ:5

for AS being AffinSpace
for A, K, X, Y being Subset of AS st not A // K & A '||' X & A '||' Y & K '||' X & K '||' Y & A is being_line & K is being_line & X is being_plane & Y is being_plane holds
X '||' Y

proof end;

theorem :: AFPROJ:6

for AS being AffinSpace
for A, X, Y being Subset of AS st X is being_plane & A '||' X & X '||' Y holds
A '||' Y by AFF_4:59, AFF_4:60;

:: Next we distinguish two sets, one consisting of the lines, the second
:: consisting of the planes of a given affine space and we consider two
:: equivalence relations defined on each of these sets; theses relations
:: are in fact the relation of parallelity restricted to suitable area.
:: Their equivalence classes are called directions (of lines and planes,
:: respectively); they are intended to be considered as new objects which
:: extend the given affine space to a projective space.

definition

let AS be AffinSpace;

func AfLines AS -> Subset-Family of AS equals :: AFPROJ:def 1
{ A where A is Subset of AS : A is being_line } ;

proof end;

end;

:: deftheorem defines AfLines AFPROJ:def 1 :

definition

let AS be AffinSpace;

func AfPlanes AS -> Subset-Family of AS equals :: AFPROJ:def 2
{ A where A is Subset of AS : A is being_plane } ;

proof end;

end;

:: deftheorem defines AfPlanes AFPROJ:def 2 :

theorem :: AFPROJ:7

for AS being AffinSpace
for x being set holds
( x in AfLines AS iff ex X being Subset of AS st
( x = X & X is being_line ) ) ;

theorem :: AFPROJ:8

for AS being AffinSpace
for x being set holds
( x in AfPlanes AS iff ex X being Subset of AS st
( x = X & X is being_plane ) ) ;

definition

let AS be AffinSpace;

func LinesParallelity AS -> Equivalence_Relation of (AfLines AS) equals :: AFPROJ:def 3
{ [K,M] where K, M is Subset of AS : ( K is being_line & M is being_line & K '||' M ) } ;

proof end;

end;

:: deftheorem defines LinesParallelity AFPROJ:def 3 :

definition

let AS be AffinSpace;

func PlanesParallelity AS -> Equivalence_Relation of (AfPlanes AS) equals :: AFPROJ:def 4
{ [X,Y] where X, Y is Subset of AS : ( X is being_plane & Y is being_plane & X '||' Y ) } ;

proof end;

end;

:: deftheorem defines PlanesParallelity AFPROJ:def 4 :

definition

let AS be AffinSpace;
let X be Subset of AS;

func LDir X -> Subset of (AfLines AS) equals :: AFPROJ:def 5
Class ((LinesParallelity AS),X);

end;

:: deftheorem defines LDir AFPROJ:def 5 :

definition

let AS be AffinSpace;
let X be Subset of AS;

func PDir X -> Subset of (AfPlanes AS) equals :: AFPROJ:def 6
Class ((PlanesParallelity AS),X);

end;

:: deftheorem defines PDir AFPROJ:def 6 :

theorem Th9: :: AFPROJ:9

for AS being AffinSpace
for X being Subset of AS st X is being_line holds
for x being set holds
( x in LDir X iff ex Y being Subset of AS st
( x = Y & Y is being_line & X '||' Y ) )

proof end;

theorem Th10: :: AFPROJ:10

for AS being AffinSpace
for X being Subset of AS st X is being_plane holds
for x being set holds
( x in PDir X iff ex Y being Subset of AS st
( x = Y & Y is being_plane & X '||' Y ) )

proof end;

theorem Th11: :: AFPROJ:11

for AS being AffinSpace
for X, Y being Subset of AS st X is being_line & Y is being_line holds
( LDir X = LDir Y iff X // Y )

proof end;

theorem Th12: :: AFPROJ:12

for AS being AffinSpace
for X, Y being Subset of AS st X is being_line & Y is being_line holds
( LDir X = LDir Y iff X '||' Y )

proof end;

theorem Th13: :: AFPROJ:13

for AS being AffinSpace
for X, Y being Subset of AS st X is being_plane & Y is being_plane holds
( PDir X = PDir Y iff X '||' Y )

proof end;

definition

let AS be AffinSpace;

func Dir_of_Lines AS -> non empty set equals :: AFPROJ:def 7
Class (LinesParallelity AS);

proof end;

end;

:: deftheorem defines Dir_of_Lines AFPROJ:def 7 :

definition

let AS be AffinSpace;

func Dir_of_Planes AS -> non empty set equals :: AFPROJ:def 8
Class (PlanesParallelity AS);

proof end;

end;

:: deftheorem defines Dir_of_Planes AFPROJ:def 8 :

theorem Th14: :: AFPROJ:14

for AS being AffinSpace
for x being set holds
( x in Dir_of_Lines AS iff ex X being Subset of AS st
( x = LDir X & X is being_line ) )

proof end;

theorem Th15: :: AFPROJ:15

for AS being AffinSpace
for x being set holds
( x in Dir_of_Planes AS iff ex X being Subset of AS st
( x = PDir X & X is being_plane ) )

proof end;

:: The point is to guarantee that the classes of new objects consist of really
:: new objects. Clearly the set of directions of lines and the set of affine
:: points do not intersect. However we cannot claim, in general, that the set
:: of affine lines and the set of directions of planes do not intersect; this
:: is evidently true only in the case of affine planes. Therefore we have to
:: modify (slightly) a construction of the set of lines of the projective
:: closure of affine space, when compared with (naive) intuitions.

theorem Th16: :: AFPROJ:16

for AS being AffinSpace holds the carrier of AS misses Dir_of_Lines AS

proof end;

theorem :: AFPROJ:17

for AS being AffinSpace st AS is AffinPlane holds
AfLines AS misses Dir_of_Planes AS

proof end;

theorem Th18: :: AFPROJ:18

for AS being AffinSpace
for x being set holds
( x in [:(AfLines AS),{1}:] iff ex X being Subset of AS st
( x = [X,1] & X is being_line ) )

proof end;

theorem Th19: :: AFPROJ:19

for AS being AffinSpace
for x being set holds
( x in [:(Dir_of_Planes AS),{2}:] iff ex X being Subset of AS st
( x = [(PDir X),2] & X is being_plane ) )

proof end;

definition

let AS be AffinSpace;

func ProjectivePoints AS -> non empty set equals :: AFPROJ:def 9
the carrier of AS \/ (Dir_of_Lines AS);

end;

:: deftheorem defines ProjectivePoints AFPROJ:def 9 :

definition

let AS be AffinSpace;

func ProjectiveLines AS -> non empty set equals :: AFPROJ:def 10
[:(AfLines AS),{1}:] \/ [:(Dir_of_Planes AS),{2}:];

end;

:: deftheorem defines ProjectiveLines AFPROJ:def 10 :

definition

let AS be AffinSpace;

func Proj_Inc AS -> Relation of (ProjectivePoints AS),(ProjectiveLines AS) means :Def11: :: AFPROJ:def 11
for x, y being object holds
( [x,y] in it iff ( ex K being Subset of AS st
( K is being_line & y = [K,1] & ( ( x in the carrier of AS & x in K ) or x = LDir K ) ) or ex K, X being Subset of AS st
( K is being_line & X is being_plane & x = LDir K & y = [(PDir X),2] & K '||' X ) ) );

proof end;

proof end;

end;

:: deftheorem Def11 defines Proj_Inc AFPROJ:def 11 :

definition

let AS be AffinSpace;

func Inc_of_Dir AS -> Relation of (Dir_of_Lines AS),(Dir_of_Planes AS) means :Def12: :: AFPROJ:def 12
for x, y being object holds
( [x,y] in it iff ex A, X being Subset of AS st
( x = LDir A & y = PDir X & A is being_line & X is being_plane & A '||' X ) );

proof end;

proof end;

end;

:: deftheorem Def12 defines Inc_of_Dir AFPROJ:def 12 :

definition

let AS be AffinSpace;

func IncProjSp_of AS -> strict IncProjStr equals :: AFPROJ:def 13
IncProjStr(# (ProjectivePoints AS),(ProjectiveLines AS),(Proj_Inc AS) #);

end;

:: deftheorem defines IncProjSp_of AFPROJ:def 13 :

definition

let AS be AffinSpace;

func ProjHorizon AS -> strict IncProjStr equals :: AFPROJ:def 14
IncProjStr(# (Dir_of_Lines AS),(Dir_of_Planes AS),(Inc_of_Dir AS) #);

end;

:: deftheorem defines ProjHorizon AFPROJ:def 14 :

theorem Th20: :: AFPROJ:20

for AS being AffinSpace
for x being set holds
( x is POINT of (IncProjSp_of AS) iff ( x is Element of AS or ex X being Subset of AS st
( x = LDir X & X is being_line ) ) )

proof end;

theorem :: AFPROJ:21

for AS being AffinSpace
for x being set holds
( x is Element of the Points of (ProjHorizon AS) iff ex X being Subset of AS st
( x = LDir X & X is being_line ) ) by Th14;

theorem Th22: :: AFPROJ:22

for AS being AffinSpace
for x being set st x is Element of the Points of (ProjHorizon AS) holds
x is POINT of (IncProjSp_of AS)

proof end;

theorem Th23: :: AFPROJ:23

for AS being AffinSpace
for x being set holds
( x is LINE of (IncProjSp_of AS) iff ex X being Subset of AS st
( ( x = [X,1] & X is being_line ) or ( x = [(PDir X),2] & X is being_plane ) ) )

proof end;

theorem :: AFPROJ:24

for AS being AffinSpace
for x being set holds
( x is Element of the Lines of (ProjHorizon AS) iff ex X being Subset of AS st
( x = PDir X & X is being_plane ) ) by Th15;

theorem Th25: :: AFPROJ:25

for AS being AffinSpace
for x being set st x is Element of the Lines of (ProjHorizon AS) holds
[x,2] is LINE of (IncProjSp_of AS)

proof end;

theorem Th26: :: AFPROJ:26

for AS being AffinSpace
for x being Element of AS
for X being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st x = a & [X,1] = A holds
( a on A iff ( X is being_line & x in X ) )

proof end;

theorem Th27: :: AFPROJ:27

for AS being AffinSpace
for x being Element of AS
for X being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st x = a & [(PDir X),2] = A holds
not a on A

proof end;

theorem Th28: :: AFPROJ:28

for AS being AffinSpace
for X, Y being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st a = LDir Y & [X,1] = A & Y is being_line & X is being_line holds
( a on A iff Y '||' X )

proof end;

theorem Th29: :: AFPROJ:29

for AS being AffinSpace
for X, Y being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st a = LDir Y & A = [(PDir X),2] & Y is being_line & X is being_plane holds
( a on A iff Y '||' X )

proof end;

theorem Th30: :: AFPROJ:30

for AS being AffinSpace
for X being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st X is being_line & a = LDir X & A = [X,1] holds
a on A

proof end;

theorem Th31: :: AFPROJ:31

for AS being AffinSpace
for X, Y being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st X is being_line & Y is being_plane & X c= Y & a = LDir X & A = [(PDir Y),2] holds
a on A

proof end;

theorem Th32: :: AFPROJ:32

for AS being AffinSpace
for X, Y, X9 being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st Y is being_plane & X c= Y & X9 // X & a = LDir X9 & A = [(PDir Y),2] holds
a on A

proof end;

theorem :: AFPROJ:33

for AS being AffinSpace
for X being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st A = [(PDir X),2] & X is being_plane & a on A holds
not a is Element of AS by Th27;

theorem Th34: :: AFPROJ:34

for AS being AffinSpace
for X being Subset of AS
for p being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st A = [X,1] & X is being_line & p on A & p is not Element of AS holds
p = LDir X

proof end;

theorem Th35: :: AFPROJ:35

for AS being AffinSpace
for X being Subset of AS
for a, p being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st A = [X,1] & X is being_line & p on A & a on A & a <> p & p is not Element of AS holds
a is Element of AS

proof end;

theorem Th36: :: AFPROJ:36

for AS being AffinSpace
for X, Y being Subset of AS
for a being Element of the Points of (ProjHorizon AS)
for A being Element of the Lines of (ProjHorizon AS) st a = LDir X & A = PDir Y & X is being_line & Y is being_plane holds
( a on A iff X '||' Y )

proof end;

theorem Th37: :: AFPROJ:37

for AS being AffinSpace
for a being Element of the Points of (ProjHorizon AS)
for a9 being Element of the Points of (IncProjSp_of AS)
for A being Element of the Lines of (ProjHorizon AS)
for A9 being LINE of (IncProjSp_of AS) st a9 = a & A9 = [A,2] holds
( a on A iff a9 on A9 )

proof end;

theorem Th38: :: AFPROJ:38

for AS being AffinSpace
for a, b being Element of the Points of (ProjHorizon AS)
for A, K being Element of the Lines of (ProjHorizon AS) st a on A & a on K & b on A & b on K & not a = b holds
A = K

proof end;

theorem Th39: :: AFPROJ:39

for AS being AffinSpace
for A being Element of the Lines of (ProjHorizon AS) ex a, b, c being Element of the Points of (ProjHorizon AS) st
( a on A & b on A & c on A & a <> b & b <> c & c <> a )

proof end;

theorem Th40: :: AFPROJ:40

for AS being AffinSpace
for a, b being Element of the Points of (ProjHorizon AS) ex A being Element of the Lines of (ProjHorizon AS) st
( a on A & b on A )

proof end;

Lm1: for AS being AffinSpace st AS is not AffinPlane holds
ex a being Element of the Points of (ProjHorizon AS) ex A being Element of the Lines of (ProjHorizon AS) st not a on A

proof end;

Lm2: for AS being AffinSpace
for a, b being POINT of (IncProjSp_of AS)
for A, K being LINE of (IncProjSp_of AS) st a on A & a on K & b on A & b on K & not a = b holds
A = K

proof end;

Lm3: for AS being AffinSpace
for A being LINE of (IncProjSp_of AS) ex a, b, c being POINT of (IncProjSp_of AS) st
( a on A & b on A & c on A & a <> b & b <> c & c <> a )

proof end;

Lm4: for AS being AffinSpace
for a, b being POINT of (IncProjSp_of AS) ex A being LINE of (IncProjSp_of AS) st
( a on A & b on A )

proof end;

Lm5: for AS being AffinSpace
for A, K being LINE of (IncProjSp_of AS) st AS is AffinPlane holds
ex a being POINT of (IncProjSp_of AS) st
( a on A & a on K )

proof end;

Lm6: for AS being AffinSpace holds
not for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) holds a on A

proof end;

theorem Th41: :: AFPROJ:41

for AS being AffinSpace
for x, y being Element of the Points of (ProjHorizon AS)
for X being Element of the Lines of (IncProjSp_of AS) st x <> y & [x,X] in the Inc of (IncProjSp_of AS) & [y,X] in the Inc of (IncProjSp_of AS) holds
ex Y being Element of the Lines of (ProjHorizon AS) st X = [Y,2]

proof end;

theorem Th42: :: AFPROJ:42

for AS being AffinSpace
for x being POINT of (IncProjSp_of AS)
for X being Element of the Lines of (ProjHorizon AS) st [x,[X,2]] in the Inc of (IncProjSp_of AS) holds
x is Element of the Points of (ProjHorizon AS)

proof end;

Lm7: for AS being AffinSpace
for X, Y, X9, Y9 being Subset of AS
for b, c being POINT of (IncProjSp_of AS)
for A, K, M being LINE of (IncProjSp_of AS) st X is being_line & X9 is being_line & Y is being_plane & X c= Y & X9 c= Y & A = [X,1] & K = [X9,1] & b on A & c on K & b on M & c on M & b <> c & M = [Y9,1] & Y9 is being_line holds
Y9 c= Y

proof end;

Lm8: for AS being AffinSpace
for X, Y, X9, Y9 being Subset of AS
for b, c being POINT of (IncProjSp_of AS)
for A, K, M being LINE of (IncProjSp_of AS) st X is being_line & X9 is being_line & Y is being_plane & X c= Y & X9 c= Y & A = [X,1] & K = [X9,1] & b on A & c on K & b on M & c on M & b <> c & M = [(PDir Y9),2] & Y9 is being_plane holds
( Y9 '||' Y & Y '||' Y9 )

proof end;

theorem Th43: :: AFPROJ:43

for AS being AffinSpace
for X, Y, X9 being Subset of AS
for P, Q being LINE of (IncProjSp_of AS) st Y is being_plane & X is being_line & X9 is being_line & X c= Y & X9 c= Y & P = [X,1] & Q = [X9,1] holds
ex q being POINT of (IncProjSp_of AS) st
( q on P & q on Q )

proof end;

Lm9: for AS being AffinSpace
for a, b, c, d, p being POINT of (IncProjSp_of AS)
for M, N, P, Q being LINE of (IncProjSp_of AS) st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N & p is Element of AS holds
ex q being POINT of (IncProjSp_of AS) st
( q on P & q on Q )

proof end;

Lm10: for AS being AffinSpace
for a, b, c, d, p being POINT of (IncProjSp_of AS)
for M, N, P, Q being LINE of (IncProjSp_of AS) st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N & p is not Element of AS & a is Element of AS holds
ex q being POINT of (IncProjSp_of AS) st
( q on P & q on Q )

proof end;

Lm11: for AS being AffinSpace
for a, b, c, d, p being POINT of (IncProjSp_of AS)
for M, N, P, Q being LINE of (IncProjSp_of AS) st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N & p is not Element of AS & ( a is Element of AS or b is Element of AS or c is Element of AS or d is Element of AS ) holds
ex q being POINT of (IncProjSp_of AS) st
( q on P & q on Q )

proof end;

Lm12: for AS being AffinSpace
for a, b, c, d, p being POINT of (IncProjSp_of AS)
for M, N, P, Q being LINE of (IncProjSp_of AS) st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N holds
ex q being POINT of (IncProjSp_of AS) st
( q on P & q on Q )

proof end;

theorem Th44: :: AFPROJ:44

for AS being AffinSpace
for a, b, c, d, p being Element of the Points of (ProjHorizon AS)
for M, N, P, Q being Element of the Lines of (ProjHorizon AS) st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N holds
ex q being Element of the Points of (ProjHorizon AS) st
( q on P & q on Q )

proof end;

registration

let AS be AffinSpace;

cluster IncProjSp_of AS -> strict linear partial up-2-dimensional up-3-rank Vebleian ;

proof end;

end;

registration

cluster strict linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional for IncProjStr ;

proof end;

end;

registration

let AS be AffinPlane;

cluster IncProjSp_of AS -> strict 2-dimensional ;

proof end;

end;

theorem :: AFPROJ:45

for AS being AffinSpace st IncProjSp_of AS is 2-dimensional holds
AS is AffinPlane

proof end;

theorem :: AFPROJ:46

for AS being AffinSpace st AS is not AffinPlane holds
ProjHorizon AS is IncProjSp

proof end;

theorem :: AFPROJ:47

for AS being AffinSpace st ProjHorizon AS is IncProjSp holds
not AS is AffinPlane

proof end;

theorem Th48: :: AFPROJ:48

for AS being AffinSpace
for M, N being Subset of AS
for o, a, b, c, a9, b9, c9 being Element of AS st M is being_line & N is being_line & M <> N & o in M & o in N & o <> b & o <> b9 & o <> c9 & b in M & c in M & a9 in N & b9 in N & c9 in N & a,b9 // b,a9 & b,c9 // c,b9 & ( a = b or b = c or a = c ) holds
a,c9 // c,a9

proof end;

theorem :: AFPROJ:49

for AS being AffinSpace st IncProjSp_of AS is Pappian holds
AS is Pappian

proof end;

theorem Th50: :: AFPROJ:50

for AS being AffinSpace
for A, P, C being Subset of AS
for o, a, b, c, a9, b9, c9 being Element of AS st o in A & o in P & o in C & o <> a & o <> b & o <> c & a in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 & ( o = a9 or a = a9 ) holds
b,c // b9,c9

proof end;

theorem :: AFPROJ:51

for AS being AffinSpace st IncProjSp_of AS is Desarguesian holds
AS is Desarguesian

proof end;

theorem :: AFPROJ:52

for AS being AffinSpace st IncProjSp_of AS is Fanoian holds
AS is Fanoian

proof end;