Volume 3, 1991

University of Bialystok

Copyright (c) 1991 Association of Mizar Users

### The abstract of the Mizar article:

### Atlas of Midpoint Algebra

**by****Michal Muzalewski**- Received June 21, 1991
- MML identifier: MIDSP_2

- [ Mizar article, MML identifier index ]

environ vocabulary RLVECT_1, MIDSP_1, PRE_TOPC, FUNCT_1, QC_LANG1, ARYTM_1, VECTSP_1, RLVECT_2, BINOP_1, MIDSP_2; notation XBOOLE_0, ZFMISC_1, SUBSET_1, FUNCT_2, BINOP_1, STRUCT_0, PRE_TOPC, RLVECT_1, VECTSP_1, MIDSP_1; constructors BINOP_1, MIDSP_1, VECTSP_1, MEMBERED, XBOOLE_0; clusters MIDSP_1, STRUCT_0, RELSET_1, SUBSET_1, VECTSP_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin :: :: GROUP OF FREE VECTORS :: reserve G for non empty LoopStr; reserve x for Element of G; reserve M for non empty MidStr; reserve p,q,r for Point of M; reserve w for Function of [:the carrier of M,the carrier of M:], the carrier of G; definition let G,x; func Double x -> Element of G equals :: MIDSP_2:def 1 x + x; end; definition let M,G,w; pred M,G are_associated_wrp w means :: MIDSP_2:def 2 p@q = r iff w.(p,r) = w.(r,q); end; theorem :: MIDSP_2:1 M,G are_associated_wrp w implies p@p = p; reserve S for non empty set; reserve a,b,b',c,c',d for Element of S; reserve w for Function of [:S,S:],the carrier of G; definition let S,G,w; pred w is_atlas_of S,G means :: MIDSP_2:def 3 (for a,x ex b st w.(a,b) = x) & (for a,b,c holds w.(a,b) = w.(a,c) implies b = c) & for a,b,c holds w.(a,b) + w.(b,c) = w.(a,c); end; definition let S,G,w,a,x; assume w is_atlas_of S,G; func (a,x).w -> Element of S means :: MIDSP_2:def 4 w.(a,it) = x; end; reserve G for add-associative right_zeroed right_complementable (non empty LoopStr); reserve x for Element of G; reserve w for Function of [:S,S:],the carrier of G; theorem :: MIDSP_2:2 Double 0.G = 0.G; canceled; theorem :: MIDSP_2:4 w is_atlas_of S,G implies w.(a,a) = 0.G; theorem :: MIDSP_2:5 w is_atlas_of S,G & w.(a,b) = 0.G implies a = b; theorem :: MIDSP_2:6 w is_atlas_of S,G implies w.(a,b) = -w.(b,a); theorem :: MIDSP_2:7 w is_atlas_of S,G & w.(a,b) = w.(c,d) implies w.(b,a) = w.(d,c); theorem :: MIDSP_2:8 w is_atlas_of S,G implies for b,x ex a st w.(a,b) = x; theorem :: MIDSP_2:9 w is_atlas_of S,G & w.(b,a) = w.(c,a) implies b = c; theorem :: MIDSP_2:10 for w being Function of [:the carrier of M,the carrier of M:], the carrier of G holds w is_atlas_of the carrier of M,G & M,G are_associated_wrp w implies p@q = q@p; theorem :: MIDSP_2:11 for w being Function of [:the carrier of M,the carrier of M:], the carrier of G holds w is_atlas_of the carrier of M,G & M,G are_associated_wrp w implies ex r st r@p = q; reserve G for add-associative right_zeroed right_complementable Abelian (non empty LoopStr); reserve x for Element of G; canceled; theorem :: MIDSP_2:13 for G being add-associative Abelian (non empty LoopStr), x,y,z,t being Element of G holds (x+y)+(z+t) = (x+z)+(y+t); theorem :: MIDSP_2:14 for G being add-associative Abelian (non empty LoopStr), x,y being Element of G holds Double (x + y) = Double x + Double y; theorem :: MIDSP_2:15 Double (-x) = -Double x; theorem :: MIDSP_2:16 for w being Function of [:the carrier of M,the carrier of M:], the carrier of G holds w is_atlas_of the carrier of M,G & M,G are_associated_wrp w implies for a,b,c,d being Point of M holds (a@b = c@d iff w.(a,d) = w.(c,b)); reserve w for Function of [:S,S:],the carrier of G; theorem :: MIDSP_2:17 w is_atlas_of S,G implies for a,b,b',c,c' holds w.(a,b) = w.(b,c) & w.(a,b') = w.(b',c') implies w.(c,c') = Double w.(b,b'); reserve M for MidSp; reserve p,q,r,s for Point of M; definition let M; cluster vectgroup(M) -> Abelian add-associative right_zeroed right_complementable; end; theorem :: MIDSP_2:18 (for a being set holds a is Element of vectgroup(M) iff a is Vector of M) & 0.vectgroup(M) = ID(M) & for a,b being Element of vectgroup(M), x,y being Vector of M st a=x & b=y holds a+b = x+y; definition let IT be non empty LoopStr; attr IT is midpoint_operator means :: MIDSP_2:def 5 (for a being Element of IT ex x being Element of IT st Double x = a) & for a being Element of IT holds Double a = 0.IT implies a = 0.IT; end; definition cluster midpoint_operator -> Fanoian (non empty LoopStr); end; definition cluster strict midpoint_operator add-associative right_zeroed right_complementable Abelian (non empty LoopStr); end; reserve G for midpoint_operator add-associative right_zeroed right_complementable Abelian (non empty LoopStr); reserve x,y for Element of G; theorem :: MIDSP_2:19 for G being Fanoian add-associative right_zeroed right_complementable (non empty LoopStr), x being Element of G holds x = -x implies x = 0.G; theorem :: MIDSP_2:20 for G being Fanoian add-associative right_zeroed right_complementable Abelian (non empty LoopStr), x,y being Element of G holds Double x = Double y implies x = y; definition let G be midpoint_operator add-associative right_zeroed right_complementable Abelian (non empty LoopStr), x be Element of G; func Half x -> Element of G means :: MIDSP_2:def 6 Double it = x; end; theorem :: MIDSP_2:21 Half 0.G = 0.G & Half (x+y) = Half x + Half y & (Half x = Half y implies x = y) & Half Double x = x; theorem :: MIDSP_2:22 for M being non empty MidStr, w being Function of [:the carrier of M,the carrier of M:],the carrier of G holds w is_atlas_of the carrier of M,G & M,G are_associated_wrp w implies for a,b,c,d being Point of M holds (a@b)@(c@d) = (a@c)@(b@d); theorem :: MIDSP_2:23 for M being non empty MidStr, w being Function of [:the carrier of M,the carrier of M:],the carrier of G holds w is_atlas_of the carrier of M,G & M,G are_associated_wrp w implies M is MidSp; reserve x,y for Element of vectgroup(M); definition let M; cluster vectgroup(M) -> midpoint_operator; end; definition let M,p,q; func vector(p,q) -> Element of vectgroup(M) equals :: MIDSP_2:def 7 vect(p,q); end; definition let M; func vect(M) -> Function of [:the carrier of M,the carrier of M:], the carrier of vectgroup(M) means :: MIDSP_2:def 8 it.(p,q) = vect(p,q); end; theorem :: MIDSP_2:24 vect(M) is_atlas_of the carrier of M, vectgroup(M); theorem :: MIDSP_2:25 vect(p,q) = vect(r,s) iff p@s = q@r; theorem :: MIDSP_2:26 p@q = r iff vect(p,r) = vect(r,q); theorem :: MIDSP_2:27 M,vectgroup(M) are_associated_wrp vect(M); :: :: REPRESENTATION THEOREM :: reserve w for Function of [:S,S:],the carrier of G; definition let S,G,w; assume w is_atlas_of S,G; func @(w) -> BinOp of S means :: MIDSP_2:def 9 w.(a,it.(a,b)) = w.(it.(a,b),b); end; theorem :: MIDSP_2:28 w is_atlas_of S,G implies for a,b,c holds @(w).(a,b) = c iff w.(a,c) = w.(c,b); definition let D be non empty set, M be BinOp of D; cluster MidStr(#D,M#) -> non empty; end; reserve a,b,c for Point of MidStr(#S,@(w)#); definition let S,G,w; func Atlas(w) -> Function of [:the carrier of MidStr(#S,@(w)#),the carrier of MidStr(#S,@(w)#):], the carrier of G equals :: MIDSP_2:def 10 w; end; canceled 3; theorem :: MIDSP_2:32 w is_atlas_of S,G implies MidStr(#S,@(w)#),G are_associated_wrp Atlas(w); definition let S,G,w; assume w is_atlas_of S,G; func MidSp.w -> strict MidSp equals :: MIDSP_2:def 11 MidStr (# S, @(w) #); end; reserve M for non empty MidStr; reserve w for Function of [:the carrier of M,the carrier of M:], the carrier of G; reserve a,b,b1,b2,c for Point of M; theorem :: MIDSP_2:33 M is MidSp iff ex G st ex w st w is_atlas_of the carrier of M,G & M,G are_associated_wrp w; definition let M be non empty MidStr; struct AtlasStr over M (# algebra -> non empty LoopStr, function -> Function of [:the carrier of M,the carrier of M:], the carrier of the algebra #); end; definition let M be non empty MidStr; let IT be AtlasStr over M; attr IT is ATLAS-like means :: MIDSP_2:def 12 the algebra of IT is midpoint_operator add-associative right_zeroed right_complementable Abelian & M,the algebra of IT are_associated_wrp the function of IT & the function of IT is_atlas_of the carrier of M,the algebra of IT; end; definition let M be MidSp; cluster ATLAS-like AtlasStr over M; end; definition let M be non empty MidSp; mode ATLAS of M is ATLAS-like AtlasStr over M; end; definition let M be non empty MidStr, W be AtlasStr over M; mode Vector of W is Element of the algebra of W; end; definition let M be MidSp; let W be AtlasStr over M; let a,b be Point of M; func W.(a,b) -> Element of the algebra of W equals :: MIDSP_2:def 13 (the function of W).(a,b); end; definition let M be MidSp; let W be AtlasStr over M; let a be Point of M; let x be Vector of W; func (a,x).W -> Point of M equals :: MIDSP_2:def 14 (a,x).(the function of W); end; definition let M be MidSp, W be ATLAS of M; func 0.W -> Vector of W equals :: MIDSP_2:def 15 0.(the algebra of W); end; :: SOME USEFUL PROPOSITIONS theorem :: MIDSP_2:34 w is_atlas_of the carrier of M,G & M,G are_associated_wrp w implies (a@c = b1@b2 iff w.(a,c) = w.(a,b1) + w.(a,b2)); theorem :: MIDSP_2:35 w is_atlas_of the carrier of M,G & M,G are_associated_wrp w implies (a@c = b iff w.(a,c) = Double w.(a,b)); reserve M for MidSp; reserve W for ATLAS of M; reserve a,b,b1,b2,c,d for Point of M; reserve x for Vector of W; theorem :: MIDSP_2:36 a@c = b1@b2 iff W.(a,c) = W.(a,b1) + W.(a,b2); theorem :: MIDSP_2:37 a@c = b iff W.(a,c) = Double W.(a,b); theorem :: MIDSP_2:38 (for a,x ex b st W.(a,b) = x) & (for a,b,c holds W.(a,b) = W.(a,c) implies b = c) & for a,b,c holds W.(a,b) + W.(b,c) = W.(a,c); theorem :: MIDSP_2:39 W.(a,a) = 0.W & (W.(a,b) = 0.W implies a = b) & W.(a,b) = -W.(b,a) & (W.(a,b) = W.(c,d) implies W.(b,a) = W.(d,c)) & (for b,x ex a st W.(a,b) = x) & (W.(b,a) = W.(c,a) implies b = c) & (a@b = c iff W.(a,c) = W.(c,b)) & (a@b = c@d iff W.(a,d) = W.(c,b)) & (W.(a,b) = x iff (a,x).W = b); theorem :: MIDSP_2:40 (a,0.W).W = a;

Back to top