:: Analytical Ordered Affine Spaces :: by Henryk Oryszczyszyn and Krzysztof Pra\.zmowski :: :: Received April 11, 1990 :: Copyright (c) 1990-2016 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies NUMBERS, RLVECT_1, REAL_1, CARD_1, ARYTM_3, RELAT_1, ARYTM_1, SUPINF_2, STRUCT_0, ZFMISC_1, XBOOLE_0, SUBSET_1, ANALOAF; notations TARSKI, XBOOLE_0, ZFMISC_1, ORDINAL1, DOMAIN_1, XXREAL_0, XCMPLX_0, XREAL_0, REAL_1, RELSET_1, NUMBERS, STRUCT_0, RLVECT_1; constructors XXREAL_0, REAL_1, MEMBERED, DOMAIN_1, RLVECT_1; registrations SUBSET_1, RELSET_1, XXREAL_0, STRUCT_0, ZFMISC_1, XREAL_0, ORDINAL1; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin reserve V for RealLinearSpace; reserve p,q,u,v,w,y for VECTOR of V; reserve a,b,c,d for Real; definition let V; let u,v,w,y; pred u,v // w,y means :: ANALOAF:def 1 u=v or w=y or ex a,b st 00 & a*u=v implies u=a"*v; theorem :: ANALOAF:6 (a<>0 & a*u=v implies u=a"*v) & (a<>0 & u=a"*v implies a*u=v); theorem :: ANALOAF:7 u,v // w,y & u<>v & w<>y implies ex a,b st a*(v-u)=b*(y-w) & 0q & p,q // u,v & p,q // w,y implies u,v // w,y; theorem :: ANALOAF:12 u,v // w,y implies v,u // y,w & w,y // u,v; theorem :: ANALOAF:13 u,v // v,w implies u,v // u,w; theorem :: ANALOAF:14 u,v // u,w implies u,v // v,w or u,w // w,v; theorem :: ANALOAF:15 v-u=y-w implies u,v // w,y; theorem :: ANALOAF:16 y=(v+w)-u implies u,v // w,y & u,w // v,y; theorem :: ANALOAF:17 (ex p,q st p<>q) implies for u,v,w ex y st u,v // w,y & u,w // v ,y & v<>y; theorem :: ANALOAF:18 p<>v & v,p // p,w implies ex y st u,p // p,y & u,v // w,y; theorem :: ANALOAF:19 (for a,b st a*u + b*v=0.V holds a=0 & b=0) implies u<>v & u<>0.V & v<>0.V; theorem :: ANALOAF:20 (ex u,v st (for a,b st a*u + b*v=0.V holds a=0 & b=0)) implies ex u,v,w,y st not u,v // w,y & not u,v // y,w; theorem :: ANALOAF:21 (ex p,q st (for w ex a,b st a*p + b*q=w)) implies for u,v,w,y st not u,v // w,y & not u,v // y,w ex z being VECTOR of V st (u,v // u,z or u,v // z,u) & (w,y // w,z or w,y // z,w); definition struct(1-sorted) AffinStruct (#carrier -> set, CONGR -> Relation of [:the carrier,the carrier:]#); end; registration cluster non trivial strict for AffinStruct; end; reserve AS for non empty AffinStruct; reserve a,b,c,d for Element of AS; reserve x,z for object; definition let AS,a,b,c,d; pred a,b // c,d means :: ANALOAF:def 2 [[a,b],[c,d]] in the CONGR of AS; end; definition let V; func DirPar(V) -> Relation of [:the carrier of V,the carrier of V:] means :: ANALOAF:def 3 [x,z] in it iff ex u,v,w,y st x=[u,v] & z=[w,y] & u,v // w,y; end; theorem :: ANALOAF:22 [[u,v],[w,y]] in DirPar(V) iff u,v // w,y; definition let V; func OASpace(V) -> strict AffinStruct equals :: ANALOAF:def 4 AffinStruct (#the carrier of V, DirPar(V)#); end; registration let V; cluster OASpace V -> non empty; end; theorem :: ANALOAF:23 (ex u,v st for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0 ) implies (ex a,b being Element of OASpace(V) st a<>b) & (for a,b,c,d,p,q,r,s being Element of OASpace(V) holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a,b // c,d implies b,a // d,c) & (a,b // b,c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c, b)) & (ex a,b,c,d being Element of OASpace(V) st not a,b // c,d & not a,b // d, c) & (for a,b,c being Element of OASpace(V) ex d being Element of OASpace(V) st a,b // c,d & a,c // b,d & b<>d) & for p,a,b,c being Element of OASpace(V) st p <>b & b,p // p,c ex d being Element of OASpace(V) st a,p // p,d & a,b // c,d; theorem :: ANALOAF:24 (ex p,q being VECTOR of V st (for w being VECTOR of V ex a,b being Real st a*p + b*q=w)) implies for a,b,c,d being Element of OASpace(V) st not a,b // c,d & not a,b // d,c ex t being Element of OASpace(V) st (a,b // a,t or a,b // t,a) & (c,d // c,t or c,d // t,c); definition let IT be non empty AffinStruct; attr IT is OAffinSpace-like means :: ANALOAF:def 5 (for a,b,c,d,p,q,r,s being Element of IT holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a,b // c,d implies b,a // d,c) & (a,b // b,c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c,b)) & (ex a,b,c,d being Element of IT st not a,b // c,d & not a,b // d,c) & (for a,b,c being Element of IT ex d being Element of IT st a,b // c,d & a,c // b,d & b<>d) & for p,a,b,c being Element of IT st p<>b & b,p // p,c ex d being Element of IT st a, p // p,d & a,b // c,d; end; registration cluster strict OAffinSpace-like for non trivial AffinStruct; end; definition mode OAffinSpace is OAffinSpace-like non trivial AffinStruct; end; theorem :: ANALOAF:25 (ex a,b being Element of AS st a<>b) & (for a,b,c,d,p,q,r,s being Element of AS holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a,b // c,d implies b,a // d,c) & (a,b // b, c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c,b)) & (ex a, b,c,d being Element of AS st not a,b // c,d & not a,b // d,c) & (for a,b,c being Element of AS ex d being Element of AS st a,b // c,d & a,c // b,d & b<>d) & (for p,a,b,c being Element of AS st p<>b & b,p // p,c ex d being Element of AS st a,p // p,d & a,b // c,d) iff AS is OAffinSpace; theorem :: ANALOAF:26 (ex u,v st for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0 ) implies OASpace(V) is OAffinSpace; definition let IT be OAffinSpace; attr IT is 2-dimensional means :: ANALOAF:def 6 for a,b,c,d being Element of IT st not a,b // c,d & not a,b // d,c holds ex p being Element of IT st (a,b // a,p or a, b // p,a) & (c,d // c,p or c,d // p,c); end; registration cluster strict 2-dimensional for OAffinSpace; end; definition mode OAffinPlane is 2-dimensional OAffinSpace; end; theorem :: ANALOAF:27 (ex a,b being Element of AS st a<>b) & (for a,b,c,d,p,q,r,s being Element of AS holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a,b // c,d implies b,a // d,c) & (a,b // b, c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c,b)) & (ex a, b,c,d being Element of AS st not a,b // c,d & not a,b // d,c) & (for a,b,c being Element of AS ex d being Element of AS st a,b // c,d & a,c // b,d & b<>d) & (for p,a,b,c being Element of AS st p<>b & b,p // p,c ex d being Element of AS st a,p // p,d & a,b // c,d) & (for a,b,c,d being Element of AS st not a,b // c,d & not a,b // d,c holds ex p being Element of AS st (a,b // a,p or a,b // p, a) & (c,d // c,p or c,d // p,c)) iff AS is OAffinPlane; theorem :: ANALOAF:28 (ex u,v st (for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0) & (for w ex a,b being Real st w = a*u + b*v)) implies OASpace(V) is OAffinPlane;