### The Mizar article:

### Binary Operations Applied to Functions

**by****Andrzej Trybulec**

- Received September 4, 1989
Copyright (c) 1989 Association of Mizar Users

- MML identifier: FUNCOP_1
- [ 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; definitions TARSKI, RELAT_1; theorems ZFMISC_1, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_3, TARSKI, BINOP_1, MCART_1, XBOOLE_0, XBOOLE_1; schemes FUNCT_1; begin ::::::::::::::::::::::::::::::::::::::::::::::::::: :: A u x i l i a r y t h e o r e m s ::::::::::::::::::::::::::::::::::::::::::::::::::: theorem Th1: for R being Relation, A,B being set st A <> {} & B <> {} & R = [:A,B:] holds dom R = A & rng R = B proof let R be Relation, A,B be set such that A1: A <> {} and A2: B <> {} and A3: R = [:A,B:]; consider x0 being Element of A; consider y0 being Element of B; A4: R is Relation of A, B by A3,RELSET_1:def 1; now let x be set; assume A5: x in A; reconsider y0 as set; take y = y0; thus [x,y] in R by A2,A3,A5,ZFMISC_1:106; end; hence dom R = A by A4,RELSET_1:22; now let y be set such that A6: y in B; reconsider x0 as set; take x = x0; thus [x,y] in R by A1,A3,A6,ZFMISC_1:106; end; hence rng R = B by A4,RELSET_1:23; end; reserve f,g,h for Function, A for set; theorem Th2: delta A = <:id A, id A:> proof A1: A = {} implies A = {}; [:A,A:] = {} implies A = {} by ZFMISC_1:113; hence delta A = id [:A,A:]*delta A by FUNCT_2:23 .= [:id A, id A:]*delta A by FUNCT_3:90 .= <:id A, id A:> by A1,FUNCT_3:99; end; theorem Th3: dom f = dom g implies dom(f*h) = dom (g*h) proof assume A1: dom f = dom g; thus dom (f*h) = h"dom f by RELAT_1:182 .= dom (g*h) by A1,RELAT_1:182; end; theorem dom f = {} & dom g = {} implies f = g proof assume A1: dom f = {} & dom g = {}; then for x being set st x in dom f holds f.x = g.x; hence f = g by A1,FUNCT_1:9; end; ::::::::::::::::::::::::::::::::::::::::::::::::::: :: 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 :Def1: 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]); existence proof defpred P[set,set] means (for y,z st f.$1 = [y,z] holds $2 = [z,y]) & (f.$1 = $2 or ex y,z st f.$1 = [y,z]); A1: now let x,y1,y2 such that x in dom f; assume A2: P[x,y1] & P[x,y2]; now per cases; suppose ex y,z st f.x = [y,z]; then consider y,z such that A3: f.x = [y,z]; y1 = [z,y] & y2 = [z,y] by A2,A3; hence y1 = y2; suppose not ex y,z st f.x = [y,z]; hence y1 = y2 by A2; end; hence y1 = y2; end; A4: now let x such that x in dom f; now per cases; suppose A5: ex y,z st f.x = [y,z]; then consider y,z such that A6: f.x = [y,z]; take y1 = [z,y]; thus for y,z st f.x = [y,z] holds y1 = [z,y] proof let y',z' be set; assume f.x = [y',z']; then z = z' & y = y' by A6,ZFMISC_1:33; hence y1 = [z',y']; end; thus f.x = y1 or ex y,z st f.x = [y,z] by A5; suppose A7: not ex y,z st f.x = [y,z]; take y1 = f.x; thus (for y,z st f.x = [y,z] holds y1 = [z,y]) & (f.x = y1 or ex y,z st f.x = [y,z]) by A7; end; hence ex y st P[x,y]; end; thus ex g st dom g = dom f & for x st x in dom f holds P[x,g.x] from FuncEx(A1,A4); end; uniqueness proof defpred P[Function] means for x st x in dom f holds (for y,z st f.x = [y,z] holds $1.x = [z,y]) & (f.x = $1.x or ex y,z st f.x =[y,z]); let g1,g2 be Function such that A8: dom g1 = dom f and A9: P[g1] and A10: dom g2 = dom f and A11: P[g2]; now let x; assume A12: x in dom f; now per cases; suppose ex y,z st f.x = [y,z]; then consider y,z such that A13: f.x = [y,z]; g1.x = [z,y] & g2.x = [z,y] by A9,A11,A12,A13; hence g1.x = g2.x; suppose not ex y,z st f.x = [y,z]; then g1.x = f.x & g2.x = f.x by A9,A11,A12; hence g1.x = g2.x; end; hence g1.x = g2.x; end; hence g1 = g2 by A8,A10,FUNCT_1:9; end; end; canceled; theorem Th6: <:f,g:> = <:g,f:>~ proof A1: dom <:f,g:> = dom g /\ dom f by FUNCT_3:def 8 .= dom <:g,f:> by FUNCT_3:def 8; then A2: dom <:f,g:> = dom (<:g,f:>~) by Def1; now let x; assume A3: x in dom <:f,g:>; then A4: <:g,f:>.x = [g.x, f.x] by A1,FUNCT_3:def 8; thus <:f,g:>.x = [f.x, g.x] by A3,FUNCT_3:def 8 .= <:g,f:>~.x by A1,A3,A4,Def1; end; hence thesis by A2,FUNCT_1:9; end; theorem Th7: (f|A)~ = f~|A proof A1: dom ((f|A)~) = dom (f|A) by Def1; A2: dom (f|A) = dom f /\ A by RELAT_1:90 .= dom (f~) /\ A by Def1 .= dom (f~|A) by RELAT_1:90; now let x such that A3: x in dom(f~|A); A4: dom (f|A) c= dom f by FUNCT_1:76; now per cases; suppose ex y,z st (f|A).x = [y,z]; then consider y,z such that A5: (f|A).x = [y,z]; A6: f.x = [y,z] by A2,A3,A5,FUNCT_1:70; thus (f|A)~.x = [z,y] by A2,A3,A5,Def1 .= f~.x by A2,A3,A4,A6,Def1 .= (f~|A).x by A3,FUNCT_1:70; suppose A7: not ex y,z st (f|A).x = [y,z]; then A8: (f|A)~.x = (f|A).x by A2,A3,Def1; (f|A).x = f.x by A2,A3,FUNCT_1:70; hence (f|A)~.x = f~.x by A2,A3,A4,A7,A8,Def1 .= (f~|A).x by A3,FUNCT_1:70; end; hence (f|A)~.x = (f~|A).x; end; hence thesis by A1,A2,FUNCT_1:9; end; theorem Th8: f~~ = f proof A1: dom (f~~) = dom (f~) by Def1; A2: dom (f~) = dom f by Def1; now let x such that A3: x in dom f; now per cases; suppose ex y,z st f.x = [y,z]; then consider y,z such that A4: f.x = [y,z]; f~.x = [z,y] by A3,A4,Def1; hence f~~.x = f.x by A2,A3,A4,Def1; suppose A5: not ex y,z st f.x = [y,z]; then f~.x = f.x by A3,Def1; hence f~~.x = f.x by A2,A3,A5,Def1; end; hence f~~.x = f.x; end; hence thesis by A1,A2,FUNCT_1:9; end; theorem (delta A)~ = delta A proof thus (delta A)~ = <:id A, id A:>~ by Th2 .= <:id A, id A:> by Th6 .= delta A by Th2; end; theorem Th10: <:f,g:>|A = <:f|A,g:> proof A1: dom (<:f,g:>|A) = dom <:f,g:> /\ A by RELAT_1:90 .= dom f /\ dom g /\ A by FUNCT_3:def 8 .= dom f /\ A /\ dom g by XBOOLE_1:16 .= dom (f|A) /\ dom g by RELAT_1:90; now let x such that A2: x in dom (<:f,g:>|A); A3: dom (<:f,g:>|A) c= dom <:f,g:> by FUNCT_1:76; A4: x in dom (f|A) by A1,A2,XBOOLE_0:def 3; thus (<:f,g:>|A).x = <:f,g:>.x by A2,FUNCT_1:70 .= [f.x, g.x] by A2,A3,FUNCT_3:def 8 .= [(f|A).x, g.x] by A4,FUNCT_1:70; end; hence thesis by A1,FUNCT_3:def 8; end; theorem Th11: <:f,g:>|A = <:f,g|A:> proof thus <:f,g:>|A = <:g,f:>~|A by Th6 .= (<:g,f:>|A)~ by Th7 .= <:g|A,f:>~ by Th10 .= <:f,g|A:> by Th6; end; definition let A, z be set; func A --> z -> set equals :Def2: [:A, {z}:]; coherence; end; definition let A, z be set; cluster A --> z -> Function-like Relation-like; coherence proof A1: for x st x in [: A, {z} :] ex y1,y2 st [y1,y2] =x by ZFMISC_1:102; now let x,y1,y2; assume [x,y1] in [: A, {z} :] & [x,y2] in [: A, {z} :]; then y1 in {z} & y2 in {z} by ZFMISC_1:106; then y1 = z & y2 = z by TARSKI:def 1; hence y1 = y2; end; then [:A, {z}:] is Function by A1,FUNCT_1:2; hence thesis by Def2; end; end; canceled; theorem Th13: x in A implies (A --> z).x = z proof A1: z in {z} by TARSKI:def 1; assume x in A; then [x,z] in [:A, {z}:] by A1,ZFMISC_1:106; then [x,z] in (A --> z) by Def2; hence thesis by FUNCT_1:8; end; Lm1: A <> {} implies dom (A --> x) = A proof assume A1: A <> {}; set f = A --> x; f = [: A, {x} :] by Def2; hence thesis by A1,Th1; end; theorem Th14: A <> {} implies rng (A --> x) = {x} proof assume A1: A <> {}; set f = A --> x; f = [: A, {x} :] by Def2; hence rng f = {x} by A1,Th1; end; theorem Th15: rng f = {x} implies f = (dom f) --> x proof assume A1: rng f = {x}; then dom f <> {} by RELAT_1:65; then dom((dom f) --> x) = dom f & rng((dom f) -->x) = {x} by Lm1,Th14; hence thesis by A1,FUNCT_1:17; end; theorem Th16: dom ({} --> x) = {} & rng ({} --> x) = {} proof A1: ({} --> x) = [: {}, {x}:] by Def2 .= {} by ZFMISC_1:113; hence dom ({} --> x) = {} by RELAT_1:60; thus rng ({} --> x) = {} by A1,RELAT_1:60; end; theorem Th17: (for z st z in dom f holds f.z = x) implies f = dom f --> x proof assume A1: for z st z in dom f holds f.z = x; now per cases; suppose A2: dom f = {}; then A3: dom (dom f --> x) = {} by Th16; for z st z in {} holds f.z = (dom f --> x).z; hence f = dom f --> x by A2,A3,FUNCT_1:9; suppose A4: dom f <> {}; consider z being Element of dom f; now let y; thus y in {x} implies ex y1 st y1 in dom f & y = f.y1 proof assume y in {x}; then y = x by TARSKI:def 1; then f.z = y by A1,A4; hence ex y1 st y1 in dom f & y = f.y1 by A4; end; assume ex y1 st y1 in dom f & y = f.y1; then y = x by A1; hence y in {x} by TARSKI:def 1; end; then rng f = {x} by FUNCT_1:def 5; hence f = dom f --> x by Th15; end; hence f = dom f --> x; end; theorem Th18: (A --> x)|B = A /\ B --> x proof A1: A = {} or A <> {}; A2: A /\ B = {} or A /\ B <> {}; A3:dom ((A --> x)|B) = dom (A --> x) /\ B by RELAT_1:90 .= A /\ B by A1,Lm1,Th16 .= dom (A /\ B --> x) by A2,Lm1,Th16; now let z such that A4: z in dom (A /\ B --> x); A /\ B = {} or A /\ B <> {}; then A5: z in A /\ B by A4,Lm1,Th16; then A6: z in A by XBOOLE_0:def 3; thus ((A --> x)|B).z = (A --> x).z by A3,A4,FUNCT_1:70 .= x by A6,Th13 .= (A /\ B --> x).z by A5,Th13; end; hence thesis by A3,FUNCT_1:9; end; theorem Th19: dom (A --> x) = A & rng (A --> x) c= {x} proof now per cases; suppose A = {}; then dom (A --> x) = A & rng (A --> x) = {} by Th16; hence thesis by XBOOLE_1:2; suppose A <> {}; hence thesis by Lm1,Th14; end; hence thesis; end; theorem Th20: x in B implies (A --> x)"B = A proof assume A1: x in B; now per cases; suppose A2: A = {}; then rng (A --> x) = {} by Th16; then (rng (A -->x) /\ B) = {}; then rng (A -->x) misses B by XBOOLE_0:def 7; hence (A --> x)"B = A by A2,RELAT_1:173; suppose A <> {}; then A3: rng (A --> x) = {x} by Th14; {x} c= B by A1,ZFMISC_1:37; then {x} /\ B = {x} by XBOOLE_1:28; hence (A --> x)"B = (A --> x)"{x} by A3,RELAT_1:168 .= dom (A -->x) by A3,RELAT_1:169 .= A by Th19; end; hence (A --> x)"B = A; end; theorem (A --> x)"{x} = A proof x in {x} by TARSKI:def 1; hence thesis by Th20; end; theorem not x in B implies (A --> x)"B = {} proof A1:rng (A --> x) c= {x} by Th19; assume not x in B; then {x} misses B by ZFMISC_1:56; then rng (A --> x) misses B by A1,XBOOLE_1:63; hence (A --> x)"B = {} by RELAT_1:173; end; theorem x in dom h implies h*(A --> x) = A --> h.x proof assume A1: x in dom h; A2: dom (h*(A --> x)) = (A --> x)"dom h by RELAT_1:182 .= A by A1,Th20 .= dom (A --> h.x) by Th19; now let z; assume A3: z in dom (h*(A --> x)); then z in dom (A --> x) by FUNCT_1:21; then A4: z in A by Th19; thus (h*(A --> x)).z = h.((A --> x).z) by A3,FUNCT_1:22 .= h.x by A4,Th13 .= (A --> h.x).z by A4,Th13; end; hence h*(A --> x) = A --> h.x by A2,FUNCT_1:9; end; theorem A <> {} & x in dom h implies dom(h*(A --> x)) <> {} proof assume that A1: A <> {} and A2: x in dom h; consider y being Element of A; y in A by A1; then A3: y in dom (A -->x) by Th19; (A --> x).y = x by A1,Th13; hence dom (h*(A --> x)) <> {} by A2,A3,FUNCT_1:21; end; theorem (A --> x)*h = h"A --> x proof A1: dom ((A --> x)*h) = h"dom(A --> x) by RELAT_1:182 .= h"A by Th19; then A2: dom ((A --> x)*h) = dom (h"A --> x) by Th19; now let z; assume A3: z in dom ((A --> x)*h); then h.z in dom (A --> x) by FUNCT_1:21; then A4: h.z in A by Th19; thus ((A --> x)*h).z = (A --> x).(h.z) by A3,FUNCT_1:22 .= x by A4,Th13 .= (h"A --> x).z by A1,A3,Th13; end; hence (A --> x)*h = h"A --> x by A2,FUNCT_1:9; end; theorem (A --> [x,y])~ = A --> [y,x] proof A1: dom ((A --> [x,y])~) = dom (A --> [x,y]) by Def1; then A2: dom ((A --> [x,y])~) = A by Th19; then A3: dom ((A --> [x,y])~) = dom (A --> [y,x]) by Th19; now let z; assume A4: z in dom ((A --> [x,y])~); then (A --> [x,y]).z = [x,y] by A2,Th13; hence ((A --> [x,y])~).z = [y,x] by A1,A4,Def1 .= (A --> [y,x]).z by A2,A4,Th13; end; hence (A --> [x,y])~ = A --> [y,x] by A3,FUNCT_1:9; end; definition let F,f,g; func F.:(f,g) -> set equals :Def3: F * <:f,g:>; correctness; end; definition let F,f,g; cluster F.:(f,g) -> Function-like Relation-like; coherence proof F * <:f,g:>= F.:(f,g) by Def3; hence thesis; end; end; Lm2: x in dom (F*<:f,g:>) implies (F*<:f,g:>).x = F.(f.x,g.x) proof assume A1: x in dom (F*<:f,g:>); then A2: x in dom <:f,g:> by FUNCT_1:21; thus (F*<:f,g:>).x = F.(<:f,g:>.x) by A1,FUNCT_1:22 .= F.[f.x,g.x] by A2,FUNCT_3:def 8 .= F.(f.x,g.x) by BINOP_1:def 1; end; Lm3: x in dom (F.:(f,g)) implies (F.:(f,g)).x = F.(f.x,g.x) proof assume x in dom (F.:(f,g)); then A1: x in dom (F * <:f,g:>) by Def3; thus (F.:(f,g)).x = (F * <:f,g:>).x by Def3 .= F.(f.x,g.x) by A1,Lm2; end; theorem 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) proof let h; assume that A1: dom h = dom(F.:(f,g)) and A2: for z being set st z in dom (F.:(f,g)) holds h.z = F.(f.z,g.z); now let z be set; assume A3: z in dom (F.:(f,g)); hence h.z = F.(f.z,g.z) by A2 .= (F.:(f,g)).z by A3,Lm3; end; hence h = F.:(f,g) by A1,FUNCT_1:9; end; theorem x in dom (F.:(f,g)) implies (F.:(f,g)).x = F.(f.x,g.x) by Lm3; theorem Th29: f|A = g|A implies (F.:(f,h))|A = (F.:(g,h))|A proof assume A1: f|A = g|A; thus (F.:(f,h))|A = (F*<:f,h:>)|A by Def3 .= F*<:f,h:>|A by RELAT_1:112 .= F*<:f|A,h:> by Th10 .= F*<:g,h:>|A by A1,Th10 .= (F*<:g,h:>)|A by RELAT_1:112 .= (F.:(g,h))|A by Def3; end; theorem Th30: f|A = g|A implies (F.:(h,f))|A = (F.:(h,g))|A proof assume A1: f|A = g|A; thus (F.:(h,f))|A = (F*<:h,f:>)|A by Def3 .= F*<:h,f:>|A by RELAT_1:112 .= F*<:h,f|A:> by Th11 .= F*<:h,g:>|A by A1,Th11 .= (F*<:h,g:>)|A by RELAT_1:112 .= (F.:(h,g))|A by Def3; end; theorem Th31: F.:(f,g)*h = F.:(f*h, g*h) proof thus F.:(f,g)*h = F*<:f,g:>*h by Def3 .= F*(<:f,g:>*h) by RELAT_1:55 .= F*<:f*h, g*h:> by FUNCT_3:75 .= F.:(f*h, g*h) by Def3; end; theorem h*F.:(f,g) = (h*F).:(f,g) proof thus h*F.:(f,g) = h*(F*<:f,g:>) by Def3 .= h*F*<:f,g:> by RELAT_1:55 .= (h*F).:(f,g) by Def3; end; definition let F,f,x; func F[:](f,x) -> set equals :Def4: F * <:f, dom f --> x:>; correctness; end; definition let F,f,x; cluster F[:](f,x) -> Function-like Relation-like; coherence proof F * <:f, dom f --> x:> = F[:](f,x) by Def4; hence thesis; end; end; canceled; theorem Th34: F[:](f,x) = F.:(f, dom f --> x) proof thus F[:](f,x) = F * <:f, dom f --> x:> by Def4 .= F.:(f, dom f --> x) by Def3; end; theorem Th35: x in dom (F[:](f,z)) implies (F[:](f,z)).x = F.(f.x,z) proof assume x in dom (F[:](f,z)); then A1: x in dom (F * <:f, dom f --> z:>) by Def4; then A2: x in dom <:f, dom f --> z:> by FUNCT_1:21; dom <:f, dom f --> z:> = dom f /\ dom (dom f --> z) by FUNCT_3:def 8; then A3: x in dom f by A2,XBOOLE_0:def 3; thus (F[:](f,z)).x = (F * <:f, dom f --> z:>).x by Def4 .= F.(f.x,(dom f --> z).x) by A1,Lm2 .= F.(f.x,z) by A3,Th13; end; theorem f|A = g|A implies (F[:](f,x))|A = (F[:](g,x))|A proof assume A1: f|A = g|A; dom f /\ A = dom (f|A) by RELAT_1:90 .= dom g /\ A by A1,RELAT_1:90; then A2: (dom f --> x)|A = dom g /\ A --> x by Th18 .= (dom g -->x)|A by Th18; thus (F[:](f,x))|A = (F.:(f, dom f --> x))|A by Th34 .= (F.:(g, dom f --> x))|A by A1,Th29 .= (F.:(g, dom g --> x))|A by A2,Th30 .= (F[:](g,x))|A by Th34; end; theorem Th37: F[:](f,x)*h = F[:](f*h,x) proof A1: dom (dom f -->x) = dom f by Th19; then A2: dom ((dom f --> x)*h) = dom (f*h) by Th3; A3: now let z; assume A4: z in dom ((dom f --> x)*h); then A5: h.z in dom(dom f -->x) by FUNCT_1:21; thus ((dom f --> x)*h).z = (dom f --> x).(h.z) by A4,FUNCT_1:22 .= x by A1,A5,Th13; end; thus F[:](f,x)*h = F.:(f, dom f --> x)*h by Th34 .= F.:(f*h, (dom f --> x)*h) by Th31 .= F.:(f*h, dom (f*h) --> x) by A2,A3,Th17 .= F[:](f*h,x) by Th34; end; theorem h*F[:](f,x) = (h*F)[:](f,x) proof thus h*F[:](f,x) = h*(F*<:f, dom f -->x:>) by Def4 .= h*F*<:f, dom f-->x:> by RELAT_1:55 .= (h*F)[:](f,x) by Def4; end; theorem F[:](f,x)*id A = F[:](f|A,x) proof thus F[:](f,x)*id A = F[:](f*id A, x) by Th37 .= F[:](f|A, x) by RELAT_1:94; end; definition let F,x,g; func F[;](x,g) -> set equals :Def5: F * <:dom g --> x, g:>; correctness; end; definition let F,x,g; cluster F[;](x,g) -> Function-like Relation-like; coherence proof F * <:dom g --> x, g:> = F[;](x,g) by Def5; hence thesis; end; end; canceled; theorem Th41: F[;](x,g) = F.:(dom g --> x, g) proof thus F[;](x,g) = F * <:dom g --> x, g:> by Def5 .= F.:(dom g --> x, g) by Def3; end; theorem Th42: x in dom (F[;](z,f)) implies (F[;](z,f)).x = F.(z,f.x) proof assume x in dom (F[;](z,f)); then A1: x in dom (F * <:dom f --> z, f:>) by Def5; then A2: x in dom <:dom f --> z, f:> by FUNCT_1:21; dom <:dom f --> z, f:> = dom (dom f --> z) /\ dom f by FUNCT_3:def 8; then A3: x in dom f by A2,XBOOLE_0:def 3; thus (F[;](z,f)).x = (F * <:dom f --> z, f:>).x by Def5 .= F.((dom f --> z).x, f.x) by A1,Lm2 .= F.(z, f.x) by A3,Th13; end; theorem f|A = g|A implies (F[;](x,f))|A = (F[;](x,g))|A proof assume A1: f|A = g|A; dom f /\ A = dom (f|A) by RELAT_1:90 .= dom g /\ A by A1,RELAT_1:90; then A2: (dom f --> x)|A = dom g /\ A --> x by Th18 .= (dom g -->x)|A by Th18; thus (F[;](x,f))|A = (F.:(dom f --> x, f))|A by Th41 .= (F.:(dom f --> x, g))|A by A1,Th30 .= (F.:(dom g --> x, g))|A by A2,Th29 .= (F[;](x,g))|A by Th41; end; theorem Th44: F[;](x,f)*h = F[;](x,f*h) proof A1: dom (dom f -->x) = dom f by Th19; then A2: dom ((dom f --> x)*h) = dom (f*h) by Th3; A3: now let z; assume A4: z in dom ((dom f --> x)*h); then A5: h.z in dom(dom f -->x) by FUNCT_1:21; thus ((dom f --> x)*h).z = (dom f --> x).(h.z) by A4,FUNCT_1:22 .= x by A1,A5,Th13; end; thus F[;](x,f)*h = F.:(dom f --> x, f)*h by Th41 .= F.:((dom f --> x)*h, f*h) by Th31 .= F.:(dom (f*h) --> x, f*h) by A2,A3,Th17 .= F[;](x,f*h) by Th41; end; theorem h*F[;](x,f) = (h*F)[;](x,f) proof thus h*F[;](x,f) = h*(F*<:dom f -->x, f:>) by Def5 .= h*F*<:dom f-->x, f:> by RELAT_1:55 .= (h*F)[;](x,f) by Def5; end; theorem F[;](x,f)*id A = F[;](x,f|A) proof thus F[;](x,f)*id A = F[;](x,f*id A) by Th44 .= F[;](x,f|A) by RELAT_1:94; end; 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 Th47: F.:(f,g) is Function of Y,X proof F*<:f,g:> is Function of Y,X; hence thesis by Def3; end; 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; coherence by Th47; end; theorem Th48: for z being Element of Y holds (F.:(f,g)).z = F.(f.z,g.z) proof let z be Element of Y; dom (F.:(f,g)) = Y by FUNCT_2:def 1; hence (F.:(f,g)).z = F.(f.z,g.z) by Lm3; end; theorem Th49: 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) proof let h be Function of Y,X; assume A1: for z being Element of Y holds h.z = F.(f.z,g.z); now let z be Element of Y; thus h.z = F.(f.z,g.z) by A1 .= (F.:(f,g)).z by Th48; end; hence h = F.:(f,g) by FUNCT_2:113; end; canceled; theorem for g being Function of X,X holds F.:(id X, g)*f = F.:(f,g*f) proof let g be Function of X,X; thus F.:(id X, g)*f = F.:(id X*f, g*f) by Th31 .= F.:(f,g*f) by FUNCT_2:23; end; theorem for g being Function of X,X holds F.:(g, id X)*f = F.:(g*f,f) proof let g be Function of X,X; thus F.:(g, id X)*f = F.:(g*f, id X*f) by Th31 .= F.:(g*f, f) by FUNCT_2:23; end; theorem F.:(id X, id X)*f = F.:(f,f) proof thus F.:(id X, id X)*f = F.:(id X*f, id X*f) by Th31 .= F.:(id X*f, f) by FUNCT_2:23 .= F.:(f,f) by FUNCT_2:23; end; theorem for g being Function of X,X holds F.:(id X, g).x = F.(x,g.x) proof let g be Function of X,X; thus F.:(id X, g).x = F.(id X.x, g.x) by Th48 .= F.(x,g.x) by FUNCT_1:35; end; theorem for g being Function of X,X holds F.:(g, id X).x = F.(g.x,x) proof let g be Function of X,X; thus F.:(g, id X).x = F.(g.x, id X.x) by Th48 .= F.(g.x, x) by FUNCT_1:35; end; theorem F.:(id X, id X).x = F.(x,x) proof thus F.:(id X, id X).x = F.(id X.x, id X.x) by Th48 .= F.(id X.x, x) by FUNCT_1:35 .= F.(x,x) by FUNCT_1:35; end; theorem Th57: for A,B for x being set st x in B holds A --> x is Function of A, B proof let A,B; let x be set; assume A1: x in B; then A2: {x} c= B by ZFMISC_1:37; dom (A --> x) = A & rng (A --> x) c= {x} by Th19; then dom (A --> x) = A & rng (A --> x) c= B by A2,XBOOLE_1:1; hence thesis by A1,FUNCT_2:def 1,RELSET_1:11; end; theorem for A,X,x holds A --> x is Function of A, X by Th57; theorem Th59: F[:](f,x) is Function of Y,X proof dom f = Y by FUNCT_2:def 1; then reconsider g = dom f --> x as Function of Y,X by Th57; F*<:f,g:> is Function of Y,X; hence thesis by Def4; end; 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; coherence by Th59; end; theorem Th60: for y being Element of Y holds (F[:](f,x)).y = F.(f.y,x) proof let y be Element of Y; dom (F[:](f,x)) = Y by FUNCT_2:def 1; hence (F[:](f,x)).y = F.(f.y,x) by Th35; end; theorem Th61: (for y being Element of Y holds g.y = F.(f.y,x)) implies g = F[:](f,x) proof assume A1: for y being Element of Y holds g.y = F.(f.y,x); now let y be Element of Y; thus g.y = F.(f.y,x) by A1 .= (F[:](f,x)).y by Th60; end; hence g = F[:](f,x) by FUNCT_2:113; end; canceled; theorem F[:](id X, x)*f = F[:](f,x) proof thus F[:](id X, x)*f = F[:](id X*f, x) by Th37.= F[:](f,x) by FUNCT_2:23; end; theorem F[:](id X, x).x = F.(x,x) proof thus F[:](id X, x).x = F.(id X.x, x) by Th60 .= F.(x,x) by FUNCT_1:35; end; theorem Th65: F[;](x,g) is Function of Y,X proof dom g = Y by FUNCT_2:def 1; then reconsider f = dom g --> x as Function of Y,X by Th57; F*<:f,g:> is Function of Y,X; hence thesis by Def5; end; 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; coherence by Th65; end; theorem Th66: for y being Element of Y holds (F[;](x,f)).y = F.(x,f.y) proof let y be Element of Y; dom (F[;](x,f)) = Y by FUNCT_2:def 1; hence (F[;](x,f)).y = F.(x,f.y) by Th42; end; theorem Th67: (for y being Element of Y holds g.y = F.(x,f.y)) implies g = F[;](x,f) proof assume A1: for y being Element of Y holds g.y = F.(x,f.y); now let y be Element of Y; thus g.y = F.(x,f.y) by A1 .= (F[;](x,f)).y by Th66; end; hence g = F[;](x,f) by FUNCT_2:113; end; canceled; theorem F[;](x, id X)*f = F[;](x,f) proof thus F[;](x, id X)*f = F[;](x, id X*f) by Th44 .= F[;](x,f) by FUNCT_2:23; end; theorem F[;](x, id X).x = F.(x,x) proof thus F[;](x, id X).x = F.(x, id X.x) by Th66 .= F.(x,x) by FUNCT_1:35; end; theorem 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] proof let X,Y,Z be non empty set; let f be Function of X, [:Y,Z:]; let x be Element of X; x in X; then A1:x in dom f by FUNCT_2:def 1; f.x = [(f.x)`1, (f.x)`2] by MCART_1:24; hence f~.x =[(f.x)`2,(f.x)`1] by A1,Def1; end; theorem Th72: for X,Y,Z being non empty set for f being Function of X, [:Y,Z:] holds rng f is Relation of Y,Z proof let X,Y,Z be non empty set; let f be Function of X, [:Y,Z:]; rng f c= [:Y,Z:] by RELSET_1:12; hence rng f is Relation of Y,Z by RELSET_1:def 1; end; 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; coherence by Th72; 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:]; coherence proof A1: X = dom f by FUNCT_2:def 1 .= dom (f~) by Def1; rng (f~) c= [:Z,Y:] proof let x be set; assume x in rng (f~); then consider y being set such that A2: y in dom (f~) and A3: x = f~.y by FUNCT_1:def 5; A4: y in dom f by A2,Def1; then reconsider y as Element of X by FUNCT_2:def 1; A5: f.y = [(f.y)`1,(f.y)`2] by MCART_1:23; then f~.y = [(f.y)`2,(f.y)`1] by A4,Def1; hence x in [:Z,Y:] by A3,A5,ZFMISC_1:107; end; hence thesis by A1,FUNCT_2:def 1,RELSET_1:11; end; end; theorem for X,Y,Z being non empty set for f being Function of X, [:Y,Z:] holds rng (f~) = (rng f)~ proof let X,Y,Z be non empty set; let f be Function of X, [:Y,Z:]; let x,y be set; thus [x,y] in rng (f~) implies [x,y] in (rng f)~ proof assume [x,y] in rng (f~); then consider z being set such that A1: z in dom (f~) and A2: [x,y] = f~.z by FUNCT_1:def 5; A3: z in dom f by A1,Def1; f.z = f~~.z by Th8 .= [y,x] by A1,A2,Def1; then [y,x] in rng f by A3,FUNCT_1:def 5; hence [x,y] in (rng f)~ by RELAT_1:def 7; end; assume [x,y] in (rng f)~; then [y,x] in rng f by RELAT_1:def 7; then consider z being set such that A4: z in dom f and A5: [y,x] = f.z by FUNCT_1:def 5; A6: z in dom (f~) by A4,Def1; f~.z = [x,y] by A4,A5,Def1; hence [x,y] in rng (f~) by A6,FUNCT_1:def 5; end; reserve y for Element of Y; theorem F is associative implies F[:](F[;](x1,f),x2) = F[;](x1,F[:](f,x2)) proof assume A1: F is associative; now let y; thus (F[:](F[;](x1,f),x2)).y = F.((F[;](x1,f)).y,x2) by Th60 .= F.(F.(x1,f.y),x2) by Th66 .= F.(x1,F.(f.y,x2)) by A1,BINOP_1:def 3 .= F.(x1,(F[:](f,x2)).y) by Th60; end; hence F[:](F[;](x1,f),x2) = F[;](x1,F[:](f,x2)) by Th67; end; theorem F is associative implies F.:(F[:](f,x),g) = F.:(f,F[;](x,g)) proof assume A1: F is associative; now let y; thus (F.:(F[:](f,x),g)).y = F.((F[:](f,x)).y,g.y) by Th48 .= F.(F.(f.y,x),g.y) by Th60 .= F.(f.y,F.(x,g.y)) by A1,BINOP_1:def 3 .= F.(f.y,(F[;](x,g)).y) by Th66; end; hence F.:(F[:](f,x),g) = F.:(f,F[;](x,g)) by Th49; end; theorem F is associative implies F.:(F.:(f,g),h) = F.:(f,F.:(g,h)) proof assume A1: F is associative; now let y; thus (F.:(F.:(f,g),h)).y = F.((F.:(f,g)).y,h.y) by Th48 .= F.(F.(f.y,g.y),h.y) by Th48 .= F.(f.y,F.(g.y,h.y)) by A1,BINOP_1:def 3 .= F.(f.y,(F.:(g,h)).y) by Th48; end; hence F.:(F.:(f,g),h) = F.:(f,F.:(g,h)) by Th49; end; theorem F is associative implies F[;](F.(x1,x2),f) = F[;](x1,F[;](x2,f)) proof assume A1: F is associative; now let y; thus (F[;](F.(x1,x2),f)).y = F.(F.(x1,x2),f.y) by Th66 .= F.(x1,F.(x2,f.y)) by A1,BINOP_1:def 3 .= F.(x1,(F[;](x2,f)).y) by Th66; end; hence F[;](F.(x1,x2),f) = F[;](x1,F[;](x2,f)) by Th67; end; theorem F is associative implies F[:](f, F.(x1,x2)) = F[:](F[:](f,x1),x2) proof assume A1: F is associative; now let y; thus (F[:](f, F.(x1,x2))).y = F.(f.y, F.(x1,x2)) by Th60 .= F.(F.(f.y,x1),x2) by A1,BINOP_1:def 3 .= F.(F[:](f,x1).y,x2) by Th60; end; hence F[:](f,F.(x1,x2)) = F[:](F[:](f,x1),x2) by Th61; end; theorem F is commutative implies F[;](x,f) = F[:](f,x) proof assume A1: F is commutative; now let y; thus (F[;](x,f)).y = F.(x,f.y) by Th66 .= F.(f.y,x) by A1,BINOP_1:def 2; end; hence F[;](x,f) = F[:](f,x) by Th61; end; theorem F is commutative implies F.:(f,g) = F.:(g,f) proof assume A1: F is commutative; now let y; thus (F.:(f,g)).y = F.(f.y,g.y) by Th48 .= F.(g.y,f.y) by A1,BINOP_1:def 2; end; hence F.:(f,g) = F.:(g,f) by Th49; end; theorem F is idempotent implies F.:(f,f) = f proof assume F is idempotent; then for y holds f.y = F.(f.y,f.y) by BINOP_1:def 4; hence F.:(f,f) = f by Th49; end; theorem F is idempotent implies F[;](f.y,f).y = f.y proof assume A1:F is idempotent; thus F[;](f.y,f).y = F.(f.y,f.y) by Th66 .= f.y by A1,BINOP_1:def 4; end; theorem F is idempotent implies F[:](f,f.y).y = f.y proof assume A1:F is idempotent; thus F[:](f,f.y).y = F.(f.y,f.y) by Th60 .= f.y by A1,BINOP_1:def 4; end; :: Addendum, 2002.07.08 theorem for F,f,g being Function st [:rng f, rng g:] c= dom F holds dom(F.:(f,g)) = dom f /\ dom g proof let F,f,g be Function such that A1: [:rng f, rng g:] c= dom F; rng<:f,g:> c= [:rng f, rng g:] by FUNCT_3:71; then A2: rng<:f,g:> c= dom F by A1,XBOOLE_1:1; thus dom(F.:(f,g)) = dom(F*<:f,g:>) by Def3 .= dom<:f,g:> by A2,RELAT_1:46 .= dom f /\ dom g by FUNCT_3:def 8; end;

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