Volume 1, 1989

University of Bialystok

Copyright (c) 1989 Association of Mizar Users

### The abstract of the Mizar article:

### Midpoint algebras

**by****Michal Muzalewski**- Received November 26, 1989
- MML identifier: MIDSP_1

- [ Mizar article, MML identifier index ]

environ vocabulary BINOP_1, QC_LANG1, FUNCT_1, BOOLE, MCART_1, RELAT_1, VECTSP_1, ARYTM_1, RLVECT_1, MIDSP_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, BINOP_1, STRUCT_0, RLVECT_1, MCART_1, FUNCT_1, FUNCT_2; constructors BINOP_1, VECTSP_1, MCART_1, MEMBERED, XBOOLE_0; clusters STRUCT_0, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin :: PRELIMINARY definition struct(1-sorted) MidStr (# carrier -> set, MIDPOINT -> BinOp of the carrier #); end; definition cluster non empty MidStr; end; reserve MS for non empty MidStr; reserve a, b for Element of MS; definition let MS,a,b; func a@b -> Element of MS equals :: MIDSP_1:def 1 (the MIDPOINT of MS).(a,b); end; definition func op2 -> BinOp of {{}} means :: MIDSP_1:def 2 not contradiction; end; definition func Example -> MidStr equals :: MIDSP_1:def 3 MidStr (# {{}}, op2 #); end; definition cluster Example -> strict non empty; end; canceled 4; theorem :: MIDSP_1:5 the carrier of Example = {{}}; theorem :: MIDSP_1:6 the MIDPOINT of Example = op2; theorem :: MIDSP_1:7 for a being Element of Example holds a = {}; theorem :: MIDSP_1:8 for a,b being Element of Example holds a@b = op2.(a,b); canceled; theorem :: MIDSP_1:10 for a,b,c,d being Element of Example holds a@a = a & a@b = b@a & (a@b)@(c@d) = (a@c)@(b@d) & ex x being Element of Example st x@a = b; :: A. MIDPOINT ALGEBRAS definition let IT be non empty MidStr; attr IT is MidSp-like means :: MIDSP_1:def 4 for a,b,c,d being Element of IT holds a@a = a & a@b = b@a & (a@b)@(c@d) = (a@c)@(b@d) & ex x being Element of IT st x@a = b; end; definition cluster strict MidSp-like (non empty MidStr); end; definition mode MidSp is MidSp-like (non empty MidStr); end; definition let M be MidSp, a, b be Element of M; redefine func a@b; commutativity; end; reserve M for MidSp; reserve a,b,c,d,a',b',c',d',x,y,x' for Element of M; canceled 4; theorem :: MIDSP_1:15 (a@b)@c = (a@c)@(b@c); theorem :: MIDSP_1:16 a@(b@c) = (a@b)@(a@c); theorem :: MIDSP_1:17 a@b = a implies a = b; theorem :: MIDSP_1:18 x@a = x'@a implies x = x'; theorem :: MIDSP_1:19 a@x = a@x' implies x = x'; :: left-cancellation-law :: B. CONGRUENCE RELATION definition let M,a,b,c,d; pred a,b @@ c,d :: bound-vectors ab, cd are equivallent means :: MIDSP_1:def 5 a@d = b@c; end; canceled; theorem :: MIDSP_1:21 a,a @@ b,b; theorem :: MIDSP_1:22 a,b @@ c,d implies c,d @@ a,b; theorem :: MIDSP_1:23 a,a @@ b,c implies b = c; theorem :: MIDSP_1:24 a,b @@ c,c implies a = b; theorem :: MIDSP_1:25 a,b @@ a,b; theorem :: MIDSP_1:26 ex d st a,b @@ c,d; theorem :: MIDSP_1:27 a,b @@ c,d & a,b @@ c,d' implies d = d'; theorem :: MIDSP_1:28 x,y @@ a,b & x,y @@ c,d implies a,b @@ c,d; theorem :: MIDSP_1:29 a,b @@ a',b' & b,c @@ b',c' implies a,c @@ a',c'; :: C. BOUND-VECTORS reserve p,q,r,p',q' for Element of [:the carrier of M,the carrier of M:]; definition let M,p; redefine func p`1 -> Element of M; end; definition let M,p; redefine func p`2 -> Element of M; end; definition let M,p,q; pred p ## q means :: MIDSP_1:def 6 p`1,p`2 @@ q`1,q`2; reflexivity; symmetry; end; definition let M,a,b; redefine func [a,b] -> Element of [:the carrier of M,the carrier of M:]; end; canceled; theorem :: MIDSP_1:31 a,b @@ c,d implies [a,b] ## [c,d]; theorem :: MIDSP_1:32 [a,b] ## [c,d] implies a,b @@ c,d; canceled 2; theorem :: MIDSP_1:35 p ## q & p ## r implies q ## r; theorem :: MIDSP_1:36 p ## r & q ## r implies p ## q; theorem :: MIDSP_1:37 p ## q & q ## r implies p ## r; theorem :: MIDSP_1:38 p ## q implies (r ## p iff r ## q); theorem :: MIDSP_1:39 for p holds { q : q ## p } is non empty Subset of [:the carrier of M,the carrier of M:]; :: D. ( FREE ) VECTORS definition let M,p; func p~ -> Subset of [:the carrier of M,the carrier of M:] equals :: MIDSP_1:def 7 { q : q ## p}; end; definition let M,p; cluster p~ -> non empty; end; canceled; theorem :: MIDSP_1:41 for p holds r in p~ iff r ## p; theorem :: MIDSP_1:42 p ## q implies p~ = q~; theorem :: MIDSP_1:43 p~ = q~ implies p ## q; theorem :: MIDSP_1:44 [a,b]~ = [c,d]~ implies a@d = b@c; theorem :: MIDSP_1:45 p in p~; definition let M; mode Vector of M -> non empty Subset of [:the carrier of M,the carrier of M:] means :: MIDSP_1:def 8 ex p st it = p~; end; reserve u,v,w,u',w' for Vector of M; definition let M,p; redefine func p~ -> Vector of M; end; canceled 2; theorem :: MIDSP_1:48 ex u st for p holds p in u iff p`1 = p`2; definition let M; func ID(M) -> Vector of M :: zero vector equals :: MIDSP_1:def 9 { p : p`1 = p`2 }; end; canceled; theorem :: MIDSP_1:50 ID(M) = [b,b]~; theorem :: MIDSP_1:51 ex w,p,q st u = p~ & v = q~ & p`2 = q`1 & w = [p`1,q`2]~; theorem :: MIDSP_1:52 (ex p,q st u = p~ & v = q~ & p`2 = q`1 & w = [p`1,q`2]~)& (ex p,q st u = p~ & v = q~ & p`2 = q`1 & w' = [p`1,q`2]~) implies w = w'; definition let M,u,v; func u + v -> Vector of M means :: MIDSP_1:def 10 ex p,q st (u = p~ & v = q~ & p`2 = q`1 & it = [p`1,q`2]~); end; :: E. ATLAS theorem :: MIDSP_1:53 ex b st u = [a,b]~; definition let M,a,b; func vect(a,b) -> Vector of M equals :: MIDSP_1:def 11 [a,b]~; end; canceled; theorem :: MIDSP_1:55 ex b st u = vect(a,b); theorem :: MIDSP_1:56 [a,b] ## [c,d] implies vect(a,b) = vect(c,d); theorem :: MIDSP_1:57 vect(a,b) = vect(c,d) implies a@d = b@c; theorem :: MIDSP_1:58 ID(M) = vect(b,b); theorem :: MIDSP_1:59 vect(a,b) = vect(a,c) implies b = c; theorem :: MIDSP_1:60 vect(a,b) + vect(b,c) = vect(a,c); :: F. VECTOR GROUPS theorem :: MIDSP_1:61 [a,a@b] ## [a@b,b]; theorem :: MIDSP_1:62 vect(a,a@b) + vect(a,a@b) = vect(a,b); theorem :: MIDSP_1:63 (u+v)+w = u+(v+w); theorem :: MIDSP_1:64 u+ID(M) = u; theorem :: MIDSP_1:65 ex v st u+v = ID(M); theorem :: MIDSP_1:66 u+v = v+u; theorem :: MIDSP_1:67 u + v = u + w implies v = w; definition let M,u; func -u -> Vector of M means :: MIDSP_1:def 12 u + it = ID(M); end; reserve X for Element of bool [:the carrier of M,the carrier of M:]; definition let M; func setvect(M) -> set equals :: MIDSP_1:def 13 { X : X is Vector of M}; end; reserve x for set; canceled 3; theorem :: MIDSP_1:71 x is Vector of M iff x in setvect(M); definition let M; cluster setvect(M) -> non empty; end; reserve u1,v1,w1,W,W1,W2,T for Element of setvect(M); definition let M,u1,v1; func u1 + v1 -> Element of setvect(M) means :: MIDSP_1:def 14 for u,v holds u1 = u & v1 = v implies it = u + v; end; canceled 2; theorem :: MIDSP_1:74 u1 + v1 = v1 + u1; theorem :: MIDSP_1:75 (u1 + v1) + w1 = u1 + (v1 + w1); definition let M; func addvect(M) -> BinOp of setvect(M) means :: MIDSP_1:def 15 for u1,v1 holds it.(u1,v1) = u1 + v1; end; canceled; theorem :: MIDSP_1:77 for W ex T st W + T = ID(M); theorem :: MIDSP_1:78 for W,W1,W2 st W + W1 = ID(M) & W + W2 = ID(M) holds W1 = W2; definition let M; func complvect(M) -> UnOp of setvect(M) means :: MIDSP_1:def 16 for W holds W + it.W = ID(M); end; definition let M; func zerovect(M) -> Element of setvect(M) equals :: MIDSP_1:def 17 ID(M); end; definition let M; func vectgroup(M) -> LoopStr equals :: MIDSP_1:def 18 LoopStr (# setvect(M), addvect(M), zerovect(M) #); end; definition let M; cluster vectgroup M -> strict non empty; end; canceled 3; theorem :: MIDSP_1:82 the carrier of vectgroup(M) = setvect(M); theorem :: MIDSP_1:83 the add of vectgroup(M) = addvect(M); canceled; theorem :: MIDSP_1:85 the Zero of vectgroup(M) = zerovect(M); theorem :: MIDSP_1:86 vectgroup(M) is add-associative right_zeroed right_complementable Abelian;

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