### The Mizar article:

### The Collinearity Structure

**by****Wojciech Skaba**

- Received May 9, 1990
Copyright (c) 1990 Association of Mizar Users

- MML identifier: COLLSP
- [ MML identifier index ]

environ vocabulary BOOLE, PRE_TOPC, INCSP_1, RELAT_2, COLLSP; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NAT_1, DOMAIN_1, STRUCT_0, PRE_TOPC; constructors REAL_1, DOMAIN_1, STRUCT_0, MEMBERED, XBOOLE_0; clusters STRUCT_0, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements NUMERALS, SUBSET, BOOLE; definitions TARSKI, STRUCT_0; theorems TARSKI, MCART_1, ENUMSET1, STRUCT_0, XBOOLE_0; begin :: AUXILIARY THEOREMS reserve X for set; definition let X; mode Relation3 of X -> set means :Def1: it c= [:X,X,X:]; existence; end; canceled; theorem Th2: X = {} or ex a be set st {a} = X or ex a,b be set st a<>b & a in X & b in X proof now assume that A1: X <> {} and A2: not ex a,b be set st a<>b & a in X & b in X; consider p be Element of X; for z be set holds z in X iff z = p by A1,A2; then X={p} by TARSKI:def 1; hence ex a be set st {a} = X; end; hence thesis; end; :: ********************* :: * COLLINEARITY * :: ********************* definition struct(1-sorted) CollStr (# carrier -> set, Collinearity -> Relation3 of the carrier #); end; definition cluster non empty strict CollStr; existence proof consider A being non empty set, r being Relation3 of A; take CollStr(#A,r#); thus the carrier of CollStr(#A,r#) is non empty; thus thesis; end; end; reserve CS for non empty CollStr; reserve a,b,c for Point of CS; definition let CS, a,b,c; pred a,b,c is_collinear means :Def2: [a,b,c] in the Collinearity of CS; end; set Z = {1}; Lm1: 1 in Z by TARSKI:def 1; Lm2: {[1,1,1]} c= [:{1},{1},{1}:] proof now let x be set; assume x in {[1,1,1]}; then x = [1,1,1] by TARSKI:def 1; hence x in [:{1},{1},{1}:] by Lm1,MCART_1:73; end; hence thesis by TARSKI:def 3; end; reconsider Z as non empty set by Lm1; reconsider RR = {[1,1,1]} as Relation3 of Z by Def1,Lm2; reconsider CLS = CollStr (# Z, RR #) as non empty CollStr by STRUCT_0:def 1; Lm3: now 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; z1 = 1 & z2 = 1 & z3 = 1 by TARSKI:def 1; hence thesis by TARSKI:def 1; end; let a,b,c,p,q,r be Point of CLS; thus (a=b or a=c or b=c implies [a,b,c] in the Collinearity of CLS) by A1; 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; end; definition let IT be non empty CollStr; attr IT is reflexive means :Def3: 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; definition let IT be non empty CollStr; attr IT is transitive means :Def4: 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; definition cluster strict reflexive transitive (non empty CollStr); existence proof take CLS; thus thesis by Def3,Def4,Lm3; end; end; definition mode CollSp is reflexive transitive (non empty CollStr); end; reserve CLSP for CollSp; reserve a,b,c,d,p,q,r for Point of CLSP; canceled 4; theorem Th7: (a=b or a=c or b=c) implies a,b,c is_collinear proof assume a=b or a=c or b=c; then [a,b,c] in the Collinearity of CLSP by Def3; hence thesis by Def2; end; theorem Th8: a<>b & a,b,p is_collinear & a,b,q is_collinear & a,b,r is_collinear implies p,q,r is_collinear proof assume a<>b & a,b,p is_collinear & a,b,q is_collinear & a,b,r is_collinear; then a<>b & [a,b,p] in the Collinearity of CLSP & [a,b,q] in the Collinearity of CLSP & [a,b,r] in the Collinearity of CLSP by Def2; then [p,q,r] in the Collinearity of CLSP by Def4; hence thesis by Def2; end; theorem Th9: a,b,c is_collinear implies b,a,c is_collinear & a,c,b is_collinear proof assume A1: a,b,c is_collinear; thus b,a,c is_collinear proof a=b or a<>b & a,b,b is_collinear & a,b,a is_collinear & a,b,c is_collinear by A1,Th7; hence thesis by Th7,Th8; end; thus a,c,b is_collinear proof a=b or a<>b & a,b,a is_collinear & a,b,c is_collinear & a,b,b is_collinear by A1,Th7; hence thesis by Th7,Th8; end; end; theorem a,b,a is_collinear by Th7; theorem Th11: a<>b & a,b,c is_collinear & a,b,d is_collinear implies a,c,d is_collinear proof assume A1: a<>b & a,b,c is_collinear & a,b,d is_collinear; a,b,a is_collinear by Th7; hence thesis by A1,Th8; end; theorem a,b,c is_collinear implies b,a,c is_collinear by Th9; theorem Th13: a,b,c is_collinear implies b,c,a is_collinear proof assume a,b,c is_collinear; then b,a,c is_collinear by Th9; hence thesis by Th9; end; theorem Th14: p<>q & a,b,p is_collinear & a,b,q is_collinear & p,q,r is_collinear implies a,b,r is_collinear proof assume A1: p<>q & a,b,p is_collinear & a,b,q is_collinear & p,q,r is_collinear; now assume A2: a<>b; then A3: a,p,q is_collinear by A1,Th11; a,b,b is_collinear by Th7; then p,q,a is_collinear & p,q,b is_collinear by A1,A2,A3,Th8,Th13; hence thesis by A1,Th8; end; hence thesis by Th7; end; :: ******************* :: * LINES * :: ******************* definition let CLSP,a,b; func Line(a,b) -> set equals :Def5: {p: a,b,p is_collinear}; correctness; end; canceled; theorem Th16: a in Line(a,b) & b in Line(a,b) proof thus a in Line(a,b) proof a,b,a is_collinear by Th7; then a in {p: a,b,p is_collinear}; hence thesis by Def5; end; thus b in Line(a,b) proof a,b,b is_collinear by Th7; then b in {p: a,b,p is_collinear}; hence thesis by Def5; end; end; theorem Th17: a,b,r is_collinear iff r in Line(a,b) proof thus a,b,r is_collinear implies r in Line(a,b) proof assume a,b,r is_collinear; then r in {p: a,b,p is_collinear}; hence thesis by Def5; end; thus r in Line(a,b) implies a,b,r is_collinear proof assume r in Line(a,b); then r in {p: a,b,p is_collinear} by Def5; then ex p st r=p & a,b,p is_collinear; hence thesis; end; end; :: ************************************ :: * PROPER COLLINEARITY SPACES * :: ************************************ reserve i,j,k for Nat; set Z = {1, 2, 3}; set RR = { [i,j,k]: (i=j or j=k or k=i) & i in Z & j in Z & k in Z }; Lm4: RR c= [:Z,Z,Z:] proof now let x be set; assume x in RR; then ex i,j,k st x = [i,j,k] & (i=j or j=k or k=i) & i in Z & j in Z & k in Z; hence x in [:Z,Z,Z:] by MCART_1:73; end; hence thesis by TARSKI:def 3; end; reconsider Z as non empty set by ENUMSET1:def 1; reconsider RR as Relation3 of Z by Def1,Lm4; reconsider CLS = CollStr (# Z, RR #) as non empty CollStr by STRUCT_0:def 1; Lm5: for a,b,c be 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 let a,b,c be Point of CLS; thus [a,b,c] in RR implies (a=b or b=c or c =a) & a in Z & b in Z & c in Z proof assume [a,b,c] in RR; then consider i,j,k such that A1: [a,b,c] = [i,j,k] and A2: (i=j or j=k or k=i) & i in Z & j in Z & k in Z; a=i & b=j & c =k by A1,MCART_1:28; hence thesis by A2; end; thus (a=b or b=c or c =a) & a in Z & b in Z & c in Z implies [a,b,c] in RR; end; Lm6: for a,b,c,p,q,r be Point of CLS holds (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) proof let a,b,c,p,q,r be Point of CLS; assume A1: 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; thus [p,q,r] in the Collinearity of CLS proof (a=p or b=p) & (a=q or b=q) & (a=r or b=r) & p in Z & q in Z & r in Z by A1,Lm5; hence thesis; end; end; Lm7: ex a,b,c be Point of CLS st not a,b,c is_collinear proof reconsider a=1,b=2,c =3 as Point of CLS by ENUMSET1:def 1; A1: not [a,b,c] in the Collinearity of CLS by Lm5; take a,b,c; thus thesis by A1,Def2; end; Lm8: CLS is CollSp proof 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) by Lm5,Lm6; hence thesis by Def3,Def4; end; definition let IT be non empty CollStr; attr IT is proper means :Def6: ex a,b,c being Point of IT st not a,b,c is_collinear; end; definition cluster strict proper CollSp; existence proof reconsider CLS as CollSp by Lm8; CLS is proper by Def6,Lm7; hence thesis; end; end; reserve CLSP for proper CollSp; reserve a,b,c,d,p,q,r for Point of CLSP; canceled; theorem Th19: for p,q holds p<>q implies ex r st not p,q,r is_collinear proof let p,q; assume A1: p<>q; consider a,b,c such that A2: not a,b,c is_collinear by Def6; not p,q,a is_collinear or not p,q,b is_collinear or not p,q,c is_collinear by A1,A2,Th8; hence thesis; end; definition let CLSP; mode LINE of CLSP -> set means :Def7: ex a,b st a<>b & it=Line(a,b); existence proof consider a,b,c such that A1: not a,b,c is_collinear by Def6; A2: a<>b by A1,Th7; take Line(a,b); thus thesis by A2; end; end; reserve P,Q for LINE of CLSP; canceled 2; theorem a=b implies Line(a,b) = the carrier of CLSP proof assume A1: a=b; for x be set holds x in Line(a,b) iff x in the carrier of CLSP proof let x be set; thus x in Line(a,b) implies x in the carrier of CLSP proof assume x in Line(a,b); then x in {p: a,b,p is_collinear} by Def5; then ex p st x=p & a,b,p is_collinear; then reconsider x as Point of CLSP; x is Element of CLSP; hence thesis; end; thus x in the carrier of CLSP implies x in Line(a,b) proof assume x in the carrier of CLSP; then reconsider x as Point of CLSP; a,b,x is_collinear by A1,Th7; then x in {p: a,b,p is_collinear}; hence thesis by Def5; end; end; hence thesis by TARSKI:2; end; theorem for P ex a,b st a<>b & a in P & b in P proof let P; consider a,b such that A1: a<>b and A2: P = Line(a,b) by Def7; take a,b; thus thesis by A1,A2,Th16; end; theorem a <> b implies ex P st a in P & b in P proof assume a<>b; then reconsider P = Line(a,b) as LINE of CLSP by Def7; take P; thus thesis by Th16; end; theorem Th25: p in P & q in P & r in P implies p,q,r is_collinear proof assume A1: p in P & q in P & r in P; consider a,b such that A2: a<>b and A3: P = Line(a,b) by Def7; p in {d: a,b,d is_collinear} by A1,A3,Def5; then A4: ex x be Point of CLSP st x=p & a,b,x is_collinear; q in {d: a,b,d is_collinear} by A1,A3,Def5; then A5: ex y be Point of CLSP st y=q & a,b,y is_collinear; r in {d: a,b,d is_collinear} by A1,A3,Def5; then ex z be Point of CLSP st z=r & a,b,z is_collinear; hence thesis by A2,A4,A5,Th8; end; Lm9: for x be set holds x in Line(a,b) implies ex r be Point of CLSP st r=x & a,b,r is_collinear proof let x be set; assume x in Line(a,b); then x in {p: a,b,p is_collinear} by Def5; hence thesis; end; Lm10: for x be set holds x in P implies x is Point of CLSP proof let x be set; assume A1: x in P; consider a,b such that A2: a<>b & P = Line(a,b) by Def7; ex r be Point of CLSP st r=x & a,b,r is_collinear by A1,A2,Lm9; hence thesis; end; theorem Th26: P c= Q implies P = Q proof assume A1: P c= Q; Q c= P proof let r be set; assume A2: r in Q; then reconsider r as Point of CLSP by Lm10; consider p,q such that p<>q and A3: P = Line(p,q) by Def7; p in P & q in P by A3,Th16; then p,q,r is_collinear by A1,A2,Th25; hence thesis by A3,Th17; end; hence thesis by A1,XBOOLE_0:def 10; end; theorem Th27: p<>q & p in P & q in P implies Line(p,q) c= P proof assume that A1: p<>q and A2: p in P and A3: q in P; consider a,b such that a<>b and A4: P = Line(a,b) by Def7; A5: a,b,p is_collinear & a,b,q is_collinear by A2,A3,A4,Th17; let x be set; assume x in Line(p,q); then consider r be Point of CLSP such that A6: r=x and A7: p,q,r is_collinear by Lm9; a,b,r is_collinear by A1,A5,A7,Th14; hence thesis by A4,A6,Th17; end; theorem Th28: p<>q & p in P & q in P implies Line(p,q) = P proof assume A1: p<>q & p in P & q in P; then reconsider Q = Line(p,q) as LINE of CLSP by Def7; Q c= P by A1,Th27; hence thesis by Th26; end; theorem Th29: p<>q & p in P & q in P & p in Q & q in Q implies P = Q proof assume p<>q & p in P & q in P & p in Q & q in Q; then Line(p,q) = P & Line(p,q) = Q by Th28; hence thesis; end; theorem P = Q or P misses Q or ex p st P /\ Q = {p} proof A1: P /\ Q = {} or ex a be set st {a} = P /\ Q or ex a,b be set st a<>b & a in P /\ Q & b in P /\ Q by Th2; A2:(ex a be set st {a} = P /\ Q) implies ex p st P /\ Q = {p} proof given a be set such that A3: {a} = P /\ Q; a in P /\ Q by A3,TARSKI:def 1; then a in P by XBOOLE_0:def 3; then reconsider p=a as Point of CLSP by Lm10; P /\ Q = {p} by A3; hence thesis; end; (ex a,b be set st a<>b & a in P /\ Q & b in P /\ Q) implies P = Q proof given a,b be set such that A4: a<>b & a in P /\ Q & b in P /\ Q; a in P & a in Q & b in P & b in Q by A4,XBOOLE_0:def 3; then reconsider p=a, q=b as Point of CLSP by Lm10; p<>q & p in P & q in P & p in Q & q in Q by A4,XBOOLE_0:def 3; hence thesis by Th29; end; hence thesis by A1,A2,XBOOLE_0:def 7; end; theorem a<>b implies Line(a,b) <> the carrier of CLSP proof assume a<>b; then ex r st not a,b,r is_collinear by Th19; hence thesis by Th17; end;

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