:: The Collinearity Structure
:: by Wojciech Skaba
::
:: Received May 9, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let X be set ;

mode Relation3 of X -> set means :Def1: :: COLLSP:def 1
it c= [:X,X,X:];

existence
ex b₁ being set st b₁ c= [:X,X,X:] ;

end;

:: deftheorem Def1 defines Relation3 COLLSP:def 1 :
for X, b₂ being set holds
( b₂ is Relation3 of X iff b₂ c= [:X,X,X:] );

theorem Th1: :: COLLSP:1

for X being set holds
( X = {} or ex a being set st
( {a} = X or ex a, b being set st
( a <> b & a in X & b in X ) ) )

proof end;

:: *********************
:: * COLLINEARITY *
:: *********************

definition

attr c₁ is strict ;
struct CollStr -> 1-sorted ;
aggr CollStr(# carrier, Collinearity #) -> CollStr ;
sel Collinearity c₁ -> Relation3 of the carrier of c₁;

end;

registration

cluster non empty strict for CollStr ;

existence
ex b₁ being CollStr st
( not b₁ is empty & b₁ is strict )

proof end;

end;

definition

let CS be non empty CollStr ;
let a, b, c be Point of CS;

pred a,b,c are_collinear means :: COLLSP:def 2
[a,b,c] in the Collinearity of CS;

end;

:: deftheorem defines are_collinear COLLSP:def 2 :
for CS being non empty CollStr
for a, b, c being Point of CS holds
( a,b,c are_collinear iff [a,b,c] in the Collinearity of CS );

set Z = {1};

Lm1: 1 in {1}
by TARSKI:def 1;

Lm2: {[1,1,1]} c= [:{1},{1},{1}:]

proof end;

reconsider Z = {1} as non empty set ;

reconsider RR = {[1,1,1]} as Relation3 of Z by Def1, Lm2;

reconsider CLS = CollStr(# Z,RR #) as non empty CollStr ;

Lm3: now :: thesis: for a, b, c, p, q, r being Point of CLS holds
( ( ( a = b or a = c or b = c ) implies [a,b,c] in the Collinearity of CLS ) & ( a <> b & [a,b,p] in the Collinearity of CLS & [a,b,q] in the Collinearity of CLS & [a,b,r] in the Collinearity of CLS implies [p,q,r] in the Collinearity of CLS ) )

let a, b, c, p, q, r be Point of CLS; :: thesis: ( ( ( a = b or a = c or b = c ) implies [a,b,c] in the Collinearity of CLS ) & ( a <> b & [a,b,p] in the Collinearity of CLS & [a,b,q] in the Collinearity of CLS & [a,b,r] in the Collinearity of CLS implies [p,q,r] in the Collinearity of CLS ) )
A1: for z1, z2, z3 being Point of CLS holds [z1,z2,z3] in the Collinearity of CLS

proof

let z1, z2, z3 be Point of CLS; :: thesis: [z1,z2,z3] in the Collinearity of CLS
A2: z3 = 1 by TARSKI:def 1;
( z1 = 1 & z2 = 1 ) by TARSKI:def 1;
hence [z1,z2,z3] in the Collinearity of CLS by A2, TARSKI:def 1; :: thesis: verum

end;

hence ( ( a = b or a = c or b = c ) implies [a,b,c] in the Collinearity of CLS ) ; :: thesis: ( a <> b & [a,b,p] in the Collinearity of CLS & [a,b,q] in the Collinearity of CLS & [a,b,r] in the Collinearity of CLS implies [p,q,r] in the Collinearity of CLS )
thus ( a <> b & [a,b,p] in the Collinearity of CLS & [a,b,q] in the Collinearity of CLS & [a,b,r] in the Collinearity of CLS implies [p,q,r] in the Collinearity of CLS ) by A1; :: thesis: verum

end;

definition

let IT be non empty CollStr ;

attr IT is reflexive means :Def3: :: COLLSP:def 3
for a, b, c being Point of IT st ( a = b or a = c or b = c ) holds
[a,b,c] in the Collinearity of IT;

end;

:: deftheorem Def3 defines reflexive COLLSP:def 3 :
for IT being non empty CollStr holds
( IT is reflexive iff for a, b, c being Point of IT st ( a = b or a = c or b = c ) holds
[a,b,c] in the Collinearity of IT );

definition

let IT be non empty CollStr ;

attr IT is transitive means :Def4: :: COLLSP:def 4
for a, b, p, q, r being Point of IT st a <> b & [a,b,p] in the Collinearity of IT & [a,b,q] in the Collinearity of IT & [a,b,r] in the Collinearity of IT holds
[p,q,r] in the Collinearity of IT;

end;

:: deftheorem Def4 defines transitive COLLSP:def 4 :
for IT being non empty CollStr holds
( IT is transitive iff for a, b, p, q, r being Point of IT st a <> b & [a,b,p] in the Collinearity of IT & [a,b,q] in the Collinearity of IT & [a,b,r] in the Collinearity of IT holds
[p,q,r] in the Collinearity of IT );

registration

cluster non empty strict reflexive transitive for CollStr ;

existence
ex b₁ being non empty CollStr st
( b₁ is strict & b₁ is reflexive & b₁ is transitive )

proof end;

end;

definition

mode CollSp is non empty reflexive transitive CollStr ;

end;

theorem Th2: :: COLLSP:2

for CLSP being CollSp
for a, b, c being Point of CLSP st ( a = b or a = c or b = c ) holds
a,b,c are_collinear by Def3;

theorem Th3: :: COLLSP:3

for CLSP being CollSp
for a, b, p, q, r being Point of CLSP st a <> b & a,b,p are_collinear & a,b,q are_collinear & a,b,r are_collinear holds
p,q,r are_collinear by Def4;

theorem Th4: :: COLLSP:4

for CLSP being CollSp
for a, b, c being Point of CLSP st a,b,c are_collinear holds
( b,a,c are_collinear & a,c,b are_collinear )

proof end;

theorem :: COLLSP:5

for CLSP being CollSp
for a, b being Point of CLSP holds a,b,a are_collinear by Th2;

theorem Th6: :: COLLSP:6

for CLSP being CollSp
for a, b, c, d being Point of CLSP st a <> b & a,b,c are_collinear & a,b,d are_collinear holds
a,c,d are_collinear

proof end;

theorem :: COLLSP:7

for CLSP being CollSp
for a, b, c being Point of CLSP st a,b,c are_collinear holds
b,a,c are_collinear by Th4;

theorem Th8: :: COLLSP:8

for CLSP being CollSp
for a, b, c being Point of CLSP st a,b,c are_collinear holds
b,c,a are_collinear

proof end;

theorem Th9: :: COLLSP:9

for CLSP being CollSp
for a, b, p, q, r being Point of CLSP st p <> q & a,b,p are_collinear & a,b,q are_collinear & p,q,r are_collinear holds
a,b,r are_collinear

proof end;

:: *******************
:: * LINES *
:: *******************

definition

let CLSP be CollSp;
let a, b be Point of CLSP;

func Line (a,b) -> set equals :: COLLSP:def 5
{ p where p is Point of CLSP : a,b,p are_collinear } ;

correctness
coherence
{ p where p is Point of CLSP : a,b,p are_collinear } is set ;
;

end;

:: deftheorem defines Line COLLSP:def 5 :
for CLSP being CollSp
for a, b being Point of CLSP holds Line (a,b) = { p where p is Point of CLSP : a,b,p are_collinear } ;

theorem Th10: :: COLLSP:10

for CLSP being CollSp
for a, b being Point of CLSP holds
( a in Line (a,b) & b in Line (a,b) )

proof end;

theorem Th11: :: COLLSP:11

for CLSP being CollSp
for a, b, r being Point of CLSP holds
( a,b,r are_collinear iff r in Line (a,b) )

proof end;

set Z = {1,2,3};

set RR = { [i,j,k] where i, j, k is Element of NAT : ( ( i = j or j = k or k = i ) & i in {1,2,3} & j in {1,2,3} & k in {1,2,3} ) } ;

Lm4: { [i,j,k] where i, j, k is Element of NAT : ( ( i = j or j = k or k = i ) & i in {1,2,3} & j in {1,2,3} & k in {1,2,3} ) } c= [:{1,2,3},{1,2,3},{1,2,3}:]

proof end;

reconsider Z = {1,2,3} as non empty set by ENUMSET1:def 1;

reconsider RR = { [i,j,k] where i, j, k is Element of NAT : ( ( i = j or j = k or k = i ) & i in {1,2,3} & j in {1,2,3} & k in {1,2,3} ) } as Relation3 of Z by Def1, Lm4;

reconsider CLS = CollStr(# Z,RR #) as non empty CollStr ;

Lm5: for a, b, c being Point of CLS holds
( [a,b,c] in RR iff ( ( a = b or b = c or c = a ) & a in Z & b in Z & c in Z ) )

proof end;

Lm6: for a, b, c, p, q, r being Point of CLS st a <> b & [a,b,p] in the Collinearity of CLS & [a,b,q] in the Collinearity of CLS & [a,b,r] in the Collinearity of CLS holds
[p,q,r] in the Collinearity of CLS

proof end;

Lm7: not for a, b, c being Point of CLS holds a,b,c are_collinear

proof end;

Lm8: CLS is CollSp

proof end;

definition

let IT be non empty CollStr ;

attr IT is proper means :Def6: :: COLLSP:def 6
not for a, b, c being Point of IT holds a,b,c are_collinear ;

end;

:: deftheorem Def6 defines proper COLLSP:def 6 :
for IT being non empty CollStr holds
( IT is proper iff not for a, b, c being Point of IT holds a,b,c are_collinear );

registration

cluster non empty strict reflexive transitive proper for CollStr ;

existence
ex b₁ being CollSp st
( b₁ is strict & b₁ is proper )

proof end;

end;

theorem Th12: :: COLLSP:12

for CLSP being proper CollSp
for p, q being Point of CLSP st p <> q holds
ex r being Point of CLSP st not p,q,r are_collinear

proof end;

definition

let CLSP be proper CollSp;

mode LINE of CLSP -> set means :Def7: :: COLLSP:def 7
ex a, b being Point of CLSP st
( a <> b & it = Line (a,b) );

existence
ex b₁ being set ex a, b being Point of CLSP st
( a <> b & b₁ = Line (a,b) )

proof end;

end;

:: deftheorem Def7 defines LINE COLLSP:def 7 :
for CLSP being proper CollSp
for b₂ being set holds
( b₂ is LINE of CLSP iff ex a, b being Point of CLSP st
( a <> b & b₂ = Line (a,b) ) );

theorem :: COLLSP:13

for CLSP being proper CollSp
for a, b being Point of CLSP st a = b holds
Line (a,b) = the carrier of CLSP

proof end;

theorem :: COLLSP:14

for CLSP being proper CollSp
for P being LINE of CLSP ex a, b being Point of CLSP st
( a <> b & a in P & b in P )

proof end;

theorem :: COLLSP:15

for CLSP being proper CollSp
for a, b being Point of CLSP st a <> b holds
ex P being LINE of CLSP st
( a in P & b in P )

proof end;

theorem Th16: :: COLLSP:16

for CLSP being proper CollSp
for p, q, r being Point of CLSP
for P being LINE of CLSP st p in P & q in P & r in P holds
p,q,r are_collinear

proof end;

Lm9: for CLSP being proper CollSp
for P being LINE of CLSP
for x being set st x in P holds
x is Point of CLSP

proof end;

theorem Th17: :: COLLSP:17

for CLSP being proper CollSp
for P, Q being LINE of CLSP st P c= Q holds
P = Q

proof end;

theorem Th18: :: COLLSP:18

for CLSP being proper CollSp
for p, q being Point of CLSP
for P being LINE of CLSP st p <> q & p in P & q in P holds
Line (p,q) c= P

proof end;

theorem Th19: :: COLLSP:19

for CLSP being proper CollSp
for p, q being Point of CLSP
for P being LINE of CLSP st p <> q & p in P & q in P holds
Line (p,q) = P

proof end;

theorem Th20: :: COLLSP:20

for CLSP being proper CollSp
for p, q being Point of CLSP
for P, Q being LINE of CLSP st p <> q & p in P & q in P & p in Q & q in Q holds
P = Q

proof end;

theorem :: COLLSP:21

for CLSP being proper CollSp
for P, Q being LINE of CLSP holds
( P = Q or P misses Q or ex p being Point of CLSP st P /\ Q = {p} )

proof end;

theorem :: COLLSP:22

for CLSP being proper CollSp
for a, b being Point of CLSP st a <> b holds
Line (a,b) <> the carrier of CLSP

proof end;