Volume 1, 1989

University of Bialystok

Copyright (c) 1989 Association of Mizar Users

### The abstract of the Mizar article:

### Axioms of Incidency

**by****Wojciech A. Trybulec**- Received April 14, 1989
- MML identifier: INCSP_1

- [ Mizar article, MML identifier index ]

environ vocabulary RELAT_1, BOOLE, INCSP_1; notation TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, DOMAIN_1, RELSET_1; constructors DOMAIN_1, RELSET_1, MEMBERED, XBOOLE_0; clusters SUBSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin definition struct IncProjStr (# Points, Lines -> non empty set, Inc -> Relation of the Points, the Lines #); end; definition struct (IncProjStr) IncStruct (# Points, Lines, Planes -> non empty set, Inc -> Relation of the Points,the Lines, Inc2 -> Relation of the Points,the Planes, Inc3 -> Relation of the Lines,the Planes #); end; definition let S be IncProjStr; mode POINT of S is Element of the Points of S; mode LINE of S is Element of the Lines of S; end; definition let S be IncStruct; mode PLANE of S is Element of the Planes of S; end; reserve S for IncStruct; reserve A,B,C,D for POINT of S; reserve L for LINE of S; reserve P for PLANE of S; reserve F,G for Subset of the Points of S; :: Definitions of predicates: on, is_collinear, is_coplanar definition let S be IncProjStr; let A be POINT of S, L be LINE of S; pred A on L means :: INCSP_1:def 1 [A,L] in the Inc of S; end; definition let S; let A be POINT of S, P be PLANE of S; pred A on P means :: INCSP_1:def 2 [A,P] in the Inc2 of S; end; definition let S; let L be LINE of S, P be PLANE of S; pred L on P means :: INCSP_1:def 3 [L,P] in the Inc3 of S; end; definition let S be IncProjStr; let F be Subset of the Points of S, L be LINE of S; pred F on L means :: INCSP_1:def 4 for A being POINT of S st A in F holds A on L; end; definition let S; let F be Subset of the Points of S, P be PLANE of S; pred F on P means :: INCSP_1:def 5 for A st A in F holds A on P; end; definition let S be IncProjStr; let F be Subset of the Points of S; attr F is linear means :: INCSP_1:def 6 ex L being LINE of S st F on L; synonym F is_collinear; end; definition let S be IncStruct; let F be Subset of the Points of S; attr F is planar means :: INCSP_1:def 7 ex P be PLANE of S st F on P; synonym F is_coplanar; end; :: Definitional theorems of predicates: on, is_collinear, is_coplanar canceled 10; theorem :: INCSP_1:11 {A,B} on L iff A on L & B on L; theorem :: INCSP_1:12 {A,B,C} on L iff A on L & B on L & C on L; theorem :: INCSP_1:13 {A,B} on P iff A on P & B on P; theorem :: INCSP_1:14 {A,B,C} on P iff A on P & B on P & C on P; theorem :: INCSP_1:15 {A,B,C,D} on P iff A on P & B on P & C on P & D on P; theorem :: INCSP_1:16 G c= F & F on L implies G on L; theorem :: INCSP_1:17 G c= F & F on P implies G on P; theorem :: INCSP_1:18 F on L & A on L iff F \/ {A} on L; theorem :: INCSP_1:19 F on P & A on P iff F \/ {A} on P; theorem :: INCSP_1:20 F \/ G on L iff F on L & G on L; theorem :: INCSP_1:21 F \/ G on P iff F on P & G on P; theorem :: INCSP_1:22 G c= F & F is_collinear implies G is_collinear; theorem :: INCSP_1:23 G c= F & F is_coplanar implies G is_coplanar; reserve a,b,c for Element of Po; reserve k,l for Element of Li; reserve p,q for Element of Pl; definition let IT be IncStruct; attr IT is IncSpace-like means :: INCSP_1:def 8 (for L being LINE of IT ex A,B being POINT of IT st A <> B & {A,B} on L) & (for A,B being POINT of IT ex L being LINE of IT st {A,B} on L) & (for A,B being (POINT of IT), K,L being LINE of IT st A <> B & {A,B} on K & {A,B} on L holds K = L) & (for P being PLANE of IT ex A being POINT of IT st A on P) & (for A,B,C being POINT of IT ex P being PLANE of IT st {A,B,C} on P) & (for A,B,C being (POINT of IT), P,Q being PLANE of IT st not {A,B,C} is_collinear & {A,B,C} on P & {A,B,C} on Q holds P = Q) & (for L being (LINE of IT), P being PLANE of IT st ex A,B being POINT of IT st A <> B & {A,B} on L & {A,B} on P holds L on P) & (for A being (POINT of IT), P,Q being PLANE of IT st A on P & A on Q ex B being POINT of IT st A <> B & B on P & B on Q) & (ex A,B,C,D being POINT of IT st not {A,B,C,D} is_coplanar) & (for A being (POINT of IT), L being (LINE of IT), P being PLANE of IT st A on L & L on P holds A on P); end; definition cluster strict IncSpace-like IncStruct; end; definition mode IncSpace is IncSpace-like IncStruct; end; reserve S for IncSpace; reserve A,B,C,D,E for POINT of S; reserve K,L,L1,L2 for LINE of S; reserve P,P1,P2,Q for PLANE of S; reserve F for Subset of the Points of S; :: Axioms of Incidency canceled 11; theorem :: INCSP_1:35 F on L & L on P implies F on P; :: Collinearity of points & coplanariy of points & lines theorem :: INCSP_1:36 {A,A,B} is_collinear; theorem :: INCSP_1:37 {A,A,B,C} is_coplanar; theorem :: INCSP_1:38 {A,B,C} is_collinear implies {A,B,C,D} is_coplanar; theorem :: INCSP_1:39 A <> B & {A,B} on L & not C on L implies not {A,B,C} is_collinear; theorem :: INCSP_1:40 not {A,B,C} is_collinear & {A,B,C} on P & not D on P implies not {A,B,C,D} is_coplanar; theorem :: INCSP_1:41 not(ex P st K on P & L on P) implies K <> L; theorem :: INCSP_1:42 not(ex P st L on P & L1 on P & L2 on P) & (ex A st A on L & A on L1 & A on L2) implies L <> L1; theorem :: INCSP_1:43 L1 on P & L2 on P & not L on P & L1 <> L2 implies not(ex Q st L on Q & L1 on Q & L2 on Q); :: Lines & planes theorem :: INCSP_1:44 ex P st A on P & L on P; theorem :: INCSP_1:45 (ex A st A on K & A on L) implies (ex P st K on P & L on P); theorem :: INCSP_1:46 A <> B implies ex L st for K holds {A,B} on K iff K = L; theorem :: INCSP_1:47 not {A,B,C} is_collinear implies ex P st for Q holds {A,B,C} on Q iff P = Q; theorem :: INCSP_1:48 not A on L implies ex P st for Q holds A on Q & L on Q iff P = Q; theorem :: INCSP_1:49 K <>L & (ex A st A on K & A on L) implies ex P st for Q holds K on Q & L on Q iff P = Q; :: Definitions of functions: Line, Plane definition let S; let A,B; assume A <> B; func Line(A,B) -> LINE of S means :: INCSP_1:def 9 {A,B} on it; end; definition let S; let A,B,C; assume not {A,B,C} is_collinear; func Plane(A,B,C) -> PLANE of S means :: INCSP_1:def 10 {A,B,C} on it; end; definition let S; let A,L; assume not A on L; func Plane(A,L) -> PLANE of S means :: INCSP_1:def 11 A on it & L on it; end; definition let S; let K,L; assume that K <> L and (ex A st A on K & A on L); func Plane(K,L) -> PLANE of S means :: INCSP_1:def 12 K on it & L on it; end; :: Definitional theorems of functions: Line, Plane canceled 7; theorem :: INCSP_1:57 A <> B implies Line(A,B) = Line(B,A); theorem :: INCSP_1:58 not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(A,C,B); theorem :: INCSP_1:59 not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(B,A,C); theorem :: INCSP_1:60 not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(B,C,A); theorem :: INCSP_1:61 not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(C,A,B); theorem :: INCSP_1:62 not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(C,B,A); canceled; theorem :: INCSP_1:64 K <> L & (ex A st A on K & A on L) implies Plane(K,L) = Plane(L,K); theorem :: INCSP_1:65 A <> B & C on Line(A,B) implies {A,B,C} is_collinear; theorem :: INCSP_1:66 A <> B & A <> C & {A,B,C} is_collinear implies Line(A,B) = Line(A,C); theorem :: INCSP_1:67 not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(C,Line(A,B)); theorem :: INCSP_1:68 not {A,B,C} is_collinear & D on Plane(A,B,C) implies {A,B,C,D} is_coplanar; theorem :: INCSP_1:69 not C on L & {A,B} on L & A <> B implies Plane(C,L) = Plane(A,B,C); theorem :: INCSP_1:70 not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(Line(A,B),Line(A,C)); :: The fourth axiom of incidency theorem :: INCSP_1:71 ex A,B,C st {A,B,C} on P & not {A,B,C} is_collinear; :: Fundamental existence theorems theorem :: INCSP_1:72 ex A,B,C,D st A on P & not {A,B,C,D} is_coplanar; theorem :: INCSP_1:73 ex B st A <> B & B on L; theorem :: INCSP_1:74 A <> B implies ex C st C on P & not {A,B,C} is_collinear; theorem :: INCSP_1:75 not {A,B,C} is_collinear implies ex D st not {A,B,C,D} is_coplanar; theorem :: INCSP_1:76 ex B,C st {B,C} on P & not {A,B,C} is_collinear; theorem :: INCSP_1:77 A <> B implies (ex C,D st not {A,B,C,D} is_coplanar); theorem :: INCSP_1:78 ex B,C,D st not {A,B,C,D} is_coplanar; theorem :: INCSP_1:79 ex L st not A on L & L on P; theorem :: INCSP_1:80 A on P implies (ex L,L1,L2 st L1 <> L2 & L1 on P & L2 on P & not L on P & A on L & A on L1 & A on L2); theorem :: INCSP_1:81 ex L,L1,L2 st A on L & A on L1 & A on L2 & not(ex P st L on P & L1 on P & L2 on P); theorem :: INCSP_1:82 ex P st A on P & not L on P; theorem :: INCSP_1:83 ex A st A on P & not A on L; theorem :: INCSP_1:84 ex K st not(ex P st L on P & K on P); theorem :: INCSP_1:85 ex P,Q st P <> Q & L on P & L on Q; :: Intersection of lines and planes canceled; theorem :: INCSP_1:87 not L on P & {A,B} on L & {A,B} on P implies A = B; theorem :: INCSP_1:88 P <> Q implies not(ex A st A on P & A on Q) or (ex L st for B holds B on P & B on Q iff B on L);

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