Volume 1, 1989

University of Bialystok

Copyright (c) 1989 Association of Mizar Users

### The abstract of the Mizar article:

### Binary Operations Applied to Functions

**by****Andrzej Trybulec**- Received September 4, 1989
- MML identifier: FUNCOP_1

- [ Mizar article, MML identifier index ]

environ vocabulary RELAT_1, BOOLE, FUNCT_1, FUNCT_3, PARTFUN1, BINOP_1, MCART_1, FUNCOP_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, MCART_1, RELSET_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, FUNCT_3, BINOP_1; constructors MCART_1, FUNCT_3, BINOP_1, PARTFUN1, RELAT_2, XBOOLE_0; clusters FUNCT_1, RELSET_1, SUBSET_1, XBOOLE_0, ZFMISC_1, PARTFUN1, FUNCT_2; requirements SUBSET, BOOLE; begin ::::::::::::::::::::::::::::::::::::::::::::::::::: :: A u x i l i a r y t h e o r e m s ::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: FUNCOP_1:1 for R being Relation, A,B being set st A <> {} & B <> {} & R = [:A,B:] holds dom R = A & rng R = B; reserve f,g,h for Function, A for set; theorem :: FUNCOP_1:2 delta A = <:id A, id A:>; theorem :: FUNCOP_1:3 dom f = dom g implies dom(f*h) = dom (g*h); theorem :: FUNCOP_1:4 dom f = {} & dom g = {} implies f = g; ::::::::::::::::::::::::::::::::::::::::::::::::::: :: M a i n p a r t ::::::::::::::::::::::::::::::::::::::::::::::::::: reserve F for Function, B,x,y,y1,y2,z for set; definition let f; func f~ -> Function means :: FUNCOP_1:def 1 dom it = dom f & for x st x in dom f holds (for y,z st f.x = [y,z] holds it.x = [z,y]) & (f.x = it.x or ex y,z st f.x =[y,z]); end; canceled; theorem :: FUNCOP_1:6 <:f,g:> = <:g,f:>~; theorem :: FUNCOP_1:7 (f|A)~ = f~|A; theorem :: FUNCOP_1:8 f~~ = f; theorem :: FUNCOP_1:9 (delta A)~ = delta A; theorem :: FUNCOP_1:10 <:f,g:>|A = <:f|A,g:>; theorem :: FUNCOP_1:11 <:f,g:>|A = <:f,g|A:>; definition let A, z be set; func A --> z -> set equals :: FUNCOP_1:def 2 [:A, {z}:]; end; definition let A, z be set; cluster A --> z -> Function-like Relation-like; end; canceled; theorem :: FUNCOP_1:13 x in A implies (A --> z).x = z; theorem :: FUNCOP_1:14 A <> {} implies rng (A --> x) = {x}; theorem :: FUNCOP_1:15 rng f = {x} implies f = (dom f) --> x; theorem :: FUNCOP_1:16 dom ({} --> x) = {} & rng ({} --> x) = {}; theorem :: FUNCOP_1:17 (for z st z in dom f holds f.z = x) implies f = dom f --> x; theorem :: FUNCOP_1:18 (A --> x)|B = A /\ B --> x; theorem :: FUNCOP_1:19 dom (A --> x) = A & rng (A --> x) c= {x}; theorem :: FUNCOP_1:20 x in B implies (A --> x)"B = A; theorem :: FUNCOP_1:21 (A --> x)"{x} = A; theorem :: FUNCOP_1:22 not x in B implies (A --> x)"B = {}; theorem :: FUNCOP_1:23 x in dom h implies h*(A --> x) = A --> h.x; theorem :: FUNCOP_1:24 A <> {} & x in dom h implies dom(h*(A --> x)) <> {}; theorem :: FUNCOP_1:25 (A --> x)*h = h"A --> x; theorem :: FUNCOP_1:26 (A --> [x,y])~ = A --> [y,x]; definition let F,f,g; func F.:(f,g) -> set equals :: FUNCOP_1:def 3 F * <:f,g:>; end; definition let F,f,g; cluster F.:(f,g) -> Function-like Relation-like; end; theorem :: FUNCOP_1:27 for h st dom h = dom(F.:(f,g)) & for z being set st z in dom (F.:(f,g)) holds h.z = F.(f.z,g.z) holds h = F.:(f,g); theorem :: FUNCOP_1:28 x in dom (F.:(f,g)) implies (F.:(f,g)).x = F.(f.x,g.x); theorem :: FUNCOP_1:29 f|A = g|A implies (F.:(f,h))|A = (F.:(g,h))|A; theorem :: FUNCOP_1:30 f|A = g|A implies (F.:(h,f))|A = (F.:(h,g))|A; theorem :: FUNCOP_1:31 F.:(f,g)*h = F.:(f*h, g*h); theorem :: FUNCOP_1:32 h*F.:(f,g) = (h*F).:(f,g); definition let F,f,x; func F[:](f,x) -> set equals :: FUNCOP_1:def 4 F * <:f, dom f --> x:>; end; definition let F,f,x; cluster F[:](f,x) -> Function-like Relation-like; end; canceled; theorem :: FUNCOP_1:34 F[:](f,x) = F.:(f, dom f --> x); theorem :: FUNCOP_1:35 x in dom (F[:](f,z)) implies (F[:](f,z)).x = F.(f.x,z); theorem :: FUNCOP_1:36 f|A = g|A implies (F[:](f,x))|A = (F[:](g,x))|A; theorem :: FUNCOP_1:37 F[:](f,x)*h = F[:](f*h,x); theorem :: FUNCOP_1:38 h*F[:](f,x) = (h*F)[:](f,x); theorem :: FUNCOP_1:39 F[:](f,x)*id A = F[:](f|A,x); definition let F,x,g; func F[;](x,g) -> set equals :: FUNCOP_1:def 5 F * <:dom g --> x, g:>; end; definition let F,x,g; cluster F[;](x,g) -> Function-like Relation-like; end; canceled; theorem :: FUNCOP_1:41 F[;](x,g) = F.:(dom g --> x, g); theorem :: FUNCOP_1:42 x in dom (F[;](z,f)) implies (F[;](z,f)).x = F.(z,f.x); theorem :: FUNCOP_1:43 f|A = g|A implies (F[;](x,f))|A = (F[;](x,g))|A; theorem :: FUNCOP_1:44 F[;](x,f)*h = F[;](x,f*h); theorem :: FUNCOP_1:45 h*F[;](x,f) = (h*F)[;](x,f); theorem :: FUNCOP_1:46 F[;](x,f)*id A = F[;](x,f|A); reserve X,Y for non empty set, F for BinOp of X, f,g,h for Function of Y,X, x,x1,x2 for Element of X; theorem :: FUNCOP_1:47 F.:(f,g) is Function of Y,X; definition let X,Z be non empty set; let F be BinOp of X, f,g be Function of Z,X; redefine func F.:(f,g) -> Function of Z,X; end; theorem :: FUNCOP_1:48 for z being Element of Y holds (F.:(f,g)).z = F.(f.z,g.z); theorem :: FUNCOP_1:49 for h being Function of Y,X holds (for z being Element of Y holds h.z = F.(f.z,g.z)) implies h = F.:(f,g); canceled; theorem :: FUNCOP_1:51 for g being Function of X,X holds F.:(id X, g)*f = F.:(f,g*f); theorem :: FUNCOP_1:52 for g being Function of X,X holds F.:(g, id X)*f = F.:(g*f,f); theorem :: FUNCOP_1:53 F.:(id X, id X)*f = F.:(f,f); theorem :: FUNCOP_1:54 for g being Function of X,X holds F.:(id X, g).x = F.(x,g.x); theorem :: FUNCOP_1:55 for g being Function of X,X holds F.:(g, id X).x = F.(g.x,x); theorem :: FUNCOP_1:56 F.:(id X, id X).x = F.(x,x); theorem :: FUNCOP_1:57 for A,B for x being set st x in B holds A --> x is Function of A, B; theorem :: FUNCOP_1:58 for A,X,x holds A --> x is Function of A, X; theorem :: FUNCOP_1:59 F[:](f,x) is Function of Y,X; definition let X,Z be non empty set; let F be BinOp of X, f be Function of Z,X, x be Element of X; redefine func F[:](f,x) -> Function of Z,X; end; theorem :: FUNCOP_1:60 for y being Element of Y holds (F[:](f,x)).y = F.(f.y,x); theorem :: FUNCOP_1:61 (for y being Element of Y holds g.y = F.(f.y,x)) implies g = F[:](f,x); canceled; theorem :: FUNCOP_1:63 F[:](id X, x)*f = F[:](f,x); theorem :: FUNCOP_1:64 F[:](id X, x).x = F.(x,x); theorem :: FUNCOP_1:65 F[;](x,g) is Function of Y,X; definition let X,Z be non empty set; let F be BinOp of X, x be Element of X; let g be Function of Z,X; redefine func F[;](x,g) -> Function of Z,X; end; theorem :: FUNCOP_1:66 for y being Element of Y holds (F[;](x,f)).y = F.(x,f.y); theorem :: FUNCOP_1:67 (for y being Element of Y holds g.y = F.(x,f.y)) implies g = F[;](x,f); canceled; theorem :: FUNCOP_1:69 F[;](x, id X)*f = F[;](x,f); theorem :: FUNCOP_1:70 F[;](x, id X).x = F.(x,x); theorem :: FUNCOP_1:71 for X,Y,Z being non empty set for f being Function of X, [:Y,Z:] for x being Element of X holds f~.x =[(f.x)`2,(f.x)`1]; theorem :: FUNCOP_1:72 for X,Y,Z being non empty set for f being Function of X, [:Y,Z:] holds rng f is Relation of Y,Z; definition let X,Y,Z be non empty set; let f be Function of X, [:Y,Z:]; redefine func rng f -> Relation of Y,Z; end; definition let X,Y,Z be non empty set; let f be Function of X, [:Y,Z:]; redefine func f~ -> Function of X, [:Z,Y:]; end; theorem :: FUNCOP_1:73 for X,Y,Z being non empty set for f being Function of X, [:Y,Z:] holds rng (f~) = (rng f)~; reserve y for Element of Y; theorem :: FUNCOP_1:74 F is associative implies F[:](F[;](x1,f),x2) = F[;](x1,F[:](f,x2)); theorem :: FUNCOP_1:75 F is associative implies F.:(F[:](f,x),g) = F.:(f,F[;](x,g)); theorem :: FUNCOP_1:76 F is associative implies F.:(F.:(f,g),h) = F.:(f,F.:(g,h)); theorem :: FUNCOP_1:77 F is associative implies F[;](F.(x1,x2),f) = F[;](x1,F[;](x2,f)); theorem :: FUNCOP_1:78 F is associative implies F[:](f, F.(x1,x2)) = F[:](F[:](f,x1),x2); theorem :: FUNCOP_1:79 F is commutative implies F[;](x,f) = F[:](f,x); theorem :: FUNCOP_1:80 F is commutative implies F.:(f,g) = F.:(g,f); theorem :: FUNCOP_1:81 F is idempotent implies F.:(f,f) = f; theorem :: FUNCOP_1:82 F is idempotent implies F[;](f.y,f).y = f.y; theorem :: FUNCOP_1:83 F is idempotent implies F[:](f,f.y).y = f.y; :: Addendum, 2002.07.08 theorem :: FUNCOP_1:84 for F,f,g being Function st [:rng f, rng g:] c= dom F holds dom(F.:(f,g)) = dom f /\ dom g;

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