### The Mizar article:

### Product of Families of Groups and Vector Spaces

**by****Anna Lango, and****Grzegorz Bancerek**

- Received December 29, 1992
Copyright (c) 1992 Association of Mizar Users

- MML identifier: PRVECT_1
- [ MML identifier index ]

environ vocabulary RLVECT_1, VECTSP_1, BINOP_1, FUNCT_1, FINSEQOP, ARYTM_1, FINSEQ_2, RELAT_1, CARD_3, CAT_1, SETWISEO, FUNCOP_1, PARTFUN1, FINSEQ_1, BOOLE, ZF_REFLE, FRAENKEL, FUNCT_3, FUNCT_6, GROUP_2, PRVECT_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NAT_1, CARD_3, RELAT_1, FUNCT_1, STRUCT_0, PARTFUN1, FUNCT_2, FUNCT_3, BINOP_1, FUNCOP_1, RLVECT_1, VECTSP_1, FINSEQOP, FINSEQ_1, FRAENKEL, FUNCT_6, SETWISEO, FINSEQ_2; constructors FUNCT_5, CARD_3, FUNCT_3, BINOP_1, VECTSP_1, FINSEQOP, FRAENKEL, FUNCT_6, SETWISEO, MEMBERED, PARTFUN1, XBOOLE_0; clusters FUNCT_1, VECTSP_1, FRAENKEL, STRUCT_0, FINSEQ_2, GROUP_1, RELSET_1, AMI_1, FUNCOP_1, FINSEQ_1, ARYTM_3, MEMBERED, ZFMISC_1, FUNCT_2, PARTFUN1, XBOOLE_0, NUMBERS, ORDINAL2; requirements NUMERALS, BOOLE, SUBSET; definitions TARSKI, BINOP_1, FINSEQOP, VECTSP_1, STRUCT_0, RELAT_1, RLVECT_1; theorems BINOP_1, FUNCT_1, FUNCT_2, FUNCT_3, FINSEQ_1, FINSEQ_2, FINSEQ_3, FINSEQOP, FUNCOP_1, VECTSP_1, ZFMISC_1, SETWISEO, TARSKI, CARD_3, FUNCT_6, RLVECT_1, RELAT_1, RELSET_1, XBOOLE_0, XBOOLE_1, FVSUM_1; schemes FUNCT_1, BINOP_1, FUNCT_2, FINSEQ_1; begin :: Abelian Groups and Fields reserve G for Abelian add-associative right_complementable right_zeroed (non empty LoopStr); deffunc car(1-sorted) = the carrier of $1; deffunc ad(LoopStr) = the add of $1; deffunc cc(non empty LoopStr) = comp $1; deffunc zr(LoopStr) = the Zero of $1; canceled 2; theorem Th3: the Zero of G is_a_unity_wrt the add of G proof now let x be Element of G; thus (the add of G).(the Zero of G,x)=(the Zero of G) + x by RLVECT_1:5 .=0.G + x by RLVECT_1:def 2 .= x by RLVECT_1:10; thus (the add of G).(x,the Zero of G)=x + (the Zero of G) by RLVECT_1:5 .= x + 0.G by RLVECT_1:def 2 .= x by RLVECT_1:10; end; hence thesis by BINOP_1:11; end; theorem Th4: for G being AbGroup holds comp G is_an_inverseOp_wrt the add of G proof let G be AbGroup; A1: the Zero of G is_a_unity_wrt the add of G by Th3; now let x be Element of G; thus (the add of G).(x,(comp G).x)=x + ((comp G).x) by RLVECT_1:5 .= x+(-x) by VECTSP_1:def 25 .= 0.G by RLVECT_1:16 .= the Zero of G by RLVECT_1:def 2 .= the_unity_wrt the add of G by A1,BINOP_1:def 8; thus (the add of G).((comp G).x,x)=((comp G).x)+x by RLVECT_1:5 .= x+(-x) by VECTSP_1:def 25 .= 0.G by RLVECT_1:16 .= the Zero of G by RLVECT_1:def 2 .= the_unity_wrt the add of G by A1,BINOP_1:def 8; end; hence thesis by FINSEQOP:def 1; end; Lm1: comp G is_an_inverseOp_wrt the add of G proof A1: the Zero of G is_a_unity_wrt the add of G by Th3; now let x be Element of G; thus (the add of G).(x,(comp G).x)=x + ((comp G).x) by RLVECT_1:5 .= x+(-x) by VECTSP_1:def 25 .= 0.G by RLVECT_1:16 .= the Zero of G by RLVECT_1:def 2 .= the_unity_wrt the add of G by A1,BINOP_1:def 8; thus (the add of G).((comp G).x,x)=((comp G).x)+x by RLVECT_1:5 .= x+(-x) by VECTSP_1:def 25 .= 0.G by RLVECT_1:16 .= the Zero of G by RLVECT_1:def 2 .= the_unity_wrt the add of G by A1,BINOP_1:def 8; end; hence thesis by FINSEQOP:def 1; end; reserve GS for non empty LoopStr; theorem the add of GS is commutative associative & the Zero of GS is_a_unity_wrt the add of GS & comp GS is_an_inverseOp_wrt the add of GS implies GS is AbGroup proof assume that A1: the add of GS is commutative and A2: the add of GS is associative and A3: the Zero of GS is_a_unity_wrt the add of GS and A4: comp GS is_an_inverseOp_wrt the add of GS; now let x,y,z be Element of GS; thus x+y = (the add of GS).(x,y) by RLVECT_1:5 .= (the add of GS).(y,x) by A1,BINOP_1:def 2 .= y+x by RLVECT_1:5; thus (x+y)+z = (the add of GS).(x+y,z) by RLVECT_1:5 .= (the add of GS).((the add of GS).(x,y),z) by RLVECT_1:5 .= (the add of GS).(x,(the add of GS).(y,z)) by A2,BINOP_1:def 3 .= (the add of GS).(x,y+z) by RLVECT_1:5 .= x+(y+z) by RLVECT_1:5; thus x+(0.GS) = (the add of GS).(x,0.GS) by RLVECT_1:5 .= (the add of GS).(x,the Zero of GS) by RLVECT_1:def 2 .= x by A3,BINOP_1:11; reconsider b = (comp GS).x as Element of GS; take b; thus x+b = (the add of GS).(x,(comp GS).x) by RLVECT_1:5 .= the_unity_wrt the add of GS by A4,FINSEQOP:def 1 .= the Zero of GS by A3,BINOP_1:def 8 .= (0.GS) by RLVECT_1:def 2; end; hence thesis by VECTSP_1:7; end; Lm2: the add of GS is commutative associative implies GS is Abelian add-associative proof assume that A1: the add of GS is commutative and A2: the add of GS is associative; thus GS is Abelian proof let x,y be Element of GS; thus x+y = (the add of GS).(x,y) by RLVECT_1:5 .= (the add of GS).(y,x) by A1,BINOP_1:def 2 .= y+x by RLVECT_1:5; end; let x,y,z be Element of GS; thus (x+y)+z = (the add of GS).(x+y,z) by RLVECT_1:5 .= (the add of GS).((the add of GS).(x,y),z) by RLVECT_1:5 .= (the add of GS).(x,(the add of GS).(y,z)) by A2,BINOP_1:def 3 .= (the add of GS).(x,y+z) by RLVECT_1:5 .= x+(y+z) by RLVECT_1:5; end; Lm3: the Zero of GS is_a_unity_wrt the add of GS implies GS is right_zeroed proof assume that A1: the Zero of GS is_a_unity_wrt the add of GS; let x be Element of GS; thus x+(0.GS) = (the add of GS).(x,0.GS) by RLVECT_1:5 .= (the add of GS).(x,the Zero of GS) by RLVECT_1:def 2 .= x by A1,BINOP_1:11; end; reserve F for Field; Lm4: the mult of F is associative proof let x,y,z be Element of F; thus (the mult of F).(x,(the mult of F).(y,z)) = (the mult of F).(x,y*z) by VECTSP_1:def 10 .= x*(y*z) by VECTSP_1:def 10 .= (x*y)*z by VECTSP_1:def 16 .= (the mult of F).(x*y,z) by VECTSP_1:def 10 .= (the mult of F).((the mult of F).(x,y),z) by VECTSP_1:def 10; end; canceled 4; theorem the Zero of F is_a_unity_wrt the add of F proof now let x be Element of F; thus (the add of F).(the Zero of F,x) = (the Zero of F)+x by RLVECT_1:5 .= x+(0.F) by RLVECT_1:def 2 .= x by RLVECT_1:10; thus (the add of F).(x,the Zero of F) = x+(the Zero of F) by RLVECT_1:5 .= x+(0.F) by RLVECT_1:def 2 .= x by RLVECT_1:10; end; hence thesis by BINOP_1:11; end; theorem Th11: the unity of F is_a_unity_wrt the mult of F proof now let x be Element of F; thus (the mult of F).(the unity of F,x) = (the unity of F)*x by VECTSP_1:def 10 .= x*(1_ F) by VECTSP_1:def 9 .= x by VECTSP_1:def 19; thus (the mult of F).(x,the unity of F) = x*(the unity of F) by VECTSP_1:def 10 .= x*(1_ F) by VECTSP_1:def 9 .= x by VECTSP_1:def 19; end; hence thesis by BINOP_1:11; end; Lm5: the mult of F is_distributive_wrt the add of F proof now let x,y,z be Element of F; thus (the mult of F).(x,(the add of F).(y,z)) = x*((the add of F).(y,z)) by VECTSP_1:def 10 .= x*(y + z) by RLVECT_1:5 .= x*y + x*z by VECTSP_1:def 18 .= (the add of F).(x*y,x*z) by RLVECT_1:5 .= (the add of F).((the mult of F).(x,y),x*z) by VECTSP_1:def 10 .= (the add of F).((the mult of F).(x,y),(the mult of F).(x,z)) by VECTSP_1:def 10; thus (the mult of F).((the add of F).(x,y),z) = ((the add of F).(x,y))*z by VECTSP_1:def 10 .= (x + y)*z by RLVECT_1:5 .= x*z + y*z by VECTSP_1:def 18 .= (the add of F).(x*z,y*z) by RLVECT_1:5 .= (the add of F).((the mult of F).(x,z),y*z) by VECTSP_1:def 10 .= (the add of F).((the mult of F).(x,z),(the mult of F).(y,z)) by VECTSP_1:def 10; end; hence thesis by BINOP_1:23; end; begin :: The $n$-Product of a Binary and Unary Operation reserve F for Field, n for Nat, D for non empty set, d for Element of D, B for BinOp of D, C for UnOp of D; definition let D,n; let F be BinOp of D; let x,y be Element of n-tuples_on D; redefine func F.:(x,y) -> Element of n-tuples_on D; coherence by FINSEQ_2:140; end; definition let D be non empty set; let F be BinOp of D; let n be Nat; func product(F,n) -> BinOp of n-tuples_on D means :Def1: for x,y being Element of n-tuples_on D holds it.(x,y)=F.:(x,y); existence proof defpred P[set,set,set] means ex x',y' be Element of n-tuples_on D st $1=x' & $2=y' & $3=F.:(x',y'); A1: for x,y being Element of (n-tuples_on D) qua non empty set ex z being Element of (n-tuples_on D) qua non empty set st P[x,y,z] proof let x,y be Element of (n-tuples_on D) qua non empty set; reconsider x'=x,y'=y as Element of n-tuples_on D; reconsider z = F.:(x',y') as Element of (n-tuples_on D) qua non empty set; take z; take x',y'; thus x=x' & y=y' & z=F.:(x',y'); end; consider G being BinOp of n-tuples_on D such that A2: for x,y being Element of (n-tuples_on D) qua non empty set holds P[x,y,G.(x,y)] from BinOpEx(A1); take G; let p1,p2 be Element of n-tuples_on D; reconsider x=p1,y=p2 as Element of (n-tuples_on D) qua non empty set; ex x',y' being Element of n-tuples_on D st x=x' & y=y' & G.(x,y)=F.:(x',y') by A2; hence thesis; end; uniqueness proof let G,H be BinOp of n-tuples_on D; assume that A3: (for x,y being Element of n-tuples_on D holds G.(x,y)=F.:(x,y)) and A4: (for x,y being Element of n-tuples_on D holds H.(x,y)=F.:(x,y)); now let x,y be Element of n-tuples_on D; G.(x,y)=F.:(x,y) by A3; hence G.(x,y)=H.(x,y) by A4; end; hence G=H by BINOP_1:2; end; end; definition let D; let F be UnOp of D; let n; func product(F,n) -> UnOp of n-tuples_on D means :Def2: for x being Element of n-tuples_on D holds it.x=F*x; existence proof defpred P[set,set] means ex x' being Element of n-tuples_on D st x'=$1 & $2=F*x'; A1: for x being Element of (n-tuples_on D) qua non empty set ex y being Element of (n-tuples_on D) qua non empty set st P[x,y] proof let x be Element of (n-tuples_on D) qua non empty set; reconsider x'=x as Element of n-tuples_on D; reconsider y=F*x' as Element of (n-tuples_on D) qua non empty set by FINSEQ_2:133; take y, x'; thus thesis; end; consider G being UnOp of n-tuples_on D such that A2: for x being Element of (n-tuples_on D) qua non empty set holds P[x,G.x] from FuncExD(A1); take G; let p1 be Element of n-tuples_on D; reconsider x=p1 as Element of (n-tuples_on D) qua non empty set; ex x' being Element of n-tuples_on D st x'=x & G.x=F*x' by A2; hence thesis; end; uniqueness proof let G,H be UnOp of n-tuples_on D; assume that A3: for x being Element of n-tuples_on D holds G.x=F*x and A4: for x being Element of n-tuples_on D holds H.x=F*x; now let x be Element of n-tuples_on D; G.x=F*x by A3; hence G.x=H.x by A4; end; hence thesis by FUNCT_2:113; end; end; definition let D be non empty set; let n; let x be Element of D; redefine func n |-> x -> Element of n-tuples_on D; correctness by FINSEQ_2:132; synonym n .--> x; end; canceled 2; theorem Th14: B is commutative implies product(B,n) is commutative proof assume A1: B is commutative; now let x,y be Element of n-tuples_on D; thus product(B,n).(x,y) = B.:(x,y) by Def1 .= B.:(y,x) by A1,FINSEQOP:34 .= product(B,n).(y,x) by Def1; end; hence thesis by BINOP_1:def 2; end; theorem Th15: B is associative implies product(B,n) is associative proof assume A1: B is associative; now let x,y,z be Element of n-tuples_on D; thus product(B,n).(product(B,n).(x,y),z) = product(B,n).(B.:(x,y),z) by Def1 .= B.:(B.:(x,y),z) by Def1 .= B.:(x,B.:(y,z)) by A1,FINSEQOP:29 .= product(B,n).(x,B.:(y,z)) by Def1 .= product(B,n).(x,product(B,n).(y,z)) by Def1; end; hence thesis by BINOP_1:def 3; end; theorem Th16: d is_a_unity_wrt B implies n .--> d is_a_unity_wrt product(B,n) proof assume d is_a_unity_wrt B; then A1: B has_a_unity & d = the_unity_wrt B by BINOP_1:def 8,SETWISEO:def 2 ; now let b be Element of (n-tuples_on D) qua non empty set; reconsider b' = b as Element of n-tuples_on D; thus product(B,n).(n .--> d,b) = B.:(n .--> d,b') by Def1 .= b by A1,FINSEQOP:57; thus product(B,n).(b,n .--> d) = B.:(b',n .--> d) by Def1 .= b by A1,FINSEQOP:57; end; hence thesis by BINOP_1:11; end; theorem Th17: B has_a_unity & B is associative & C is_an_inverseOp_wrt B implies product(C,n) is_an_inverseOp_wrt product(B,n) proof assume A1: B has_a_unity & B is associative & C is_an_inverseOp_wrt B; then A2: B has_an_inverseOp by FINSEQOP:def 2; then A3: C = the_inverseOp_wrt B by A1,FINSEQOP:def 3; ex d st d is_a_unity_wrt B by A1,SETWISEO:def 2; then the_unity_wrt B is_a_unity_wrt B by BINOP_1:def 8; then n .-->the_unity_wrt B is_a_unity_wrt product(B,n) by Th16; then A4: n |->the_unity_wrt B = the_unity_wrt product(B,n) by BINOP_1:def 8; now let f be Element of (n-tuples_on D) qua non empty set; reconsider f' = f as Element of n-tuples_on D; reconsider cf = C*f' as Element of n-tuples_on D by FINSEQ_2:133; thus product(B,n).(f,product(C,n).f) = product(B,n).(f',C*f') by Def2 .= B.:(f',cf) by Def1 .= n |->the_unity_wrt B by A1,A2,A3,FINSEQOP:77; thus product(B,n).(product(C,n).f,f) = product(B,n).(C*f',f') by Def2 .= B.:(cf,f') by Def1 .= n |->the_unity_wrt B by A1,A2,A3,FINSEQOP:77; end; hence thesis by A4,FINSEQOP:def 1; end; begin :: The $n$-Power of a Group and of a Field definition let F be non empty LoopStr, n; assume A1:F is Abelian add-associative right_zeroed right_complementable; func n -Group_over F -> strict AbGroup equals :Def3: LoopStr(# n-tuples_on the carrier of F, product(the add of F,n), (n .--> the Zero of F) qua Element of n-tuples_on the carrier of F#); coherence proof set G = LoopStr(# n-tuples_on the carrier of F,product(the add of F,n), (n .--> the Zero of F) qua Element of n-tuples_on the carrier of F#); reconsider G as non empty LoopStr; the add of F is commutative associative by A1,FVSUM_1:2,3; then A2: product(the add of F,n) is commutative associative by Th14,Th15; A3: the Zero of F is_a_unity_wrt the add of F by A1,Th3; then A4: (n .--> the Zero of F) qua Element of n-tuples_on the carrier of F is_a_unity_wrt product(the add of F,n) by Th16; G is right_complementable proof let x be Element of G; set C = comp F; set B = the add of F; A5: B is associative & C is_an_inverseOp_wrt B by A1,Lm1,FVSUM_1:3; B has_a_unity by A3,SETWISEO:def 2; then A6: product(C,n) is_an_inverseOp_wrt product(B,n) by A5,Th17; reconsider y = (product (comp F,n)).x as Element of G by FUNCT_2:7; the Zero of F is_a_unity_wrt the add of F by A1,Th3; then A7: the Zero of G is_a_unity_wrt the add of G by Th16; product(B,n).(x,product(C,n).x) = the_unity_wrt product(B,n) by A6,FINSEQOP:def 1; then x + y = the_unity_wrt product(B,n) by RLVECT_1:5; then x + y = the Zero of G by A7,BINOP_1:def 8; then x + y = 0.G by RLVECT_1:def 2; hence thesis; end; hence thesis by A2,A4,Lm2,Lm3; end; end; definition let F be AbGroup, n; cluster n-Group_over F -> non empty; coherence; end; reserve x,y for set; definition let F,n; func n -Mult_over F -> Function of [:the carrier of F,n-tuples_on the carrier of F:], n-tuples_on the carrier of F means :Def4: for x being Element of F, v being Element of n-tuples_on the carrier of F holds it.(x,v) = (the mult of F)[;](x,v); existence proof defpred P[set,set] means (ex x1 being Element of F, v being Element of n-tuples_on the carrier of F st $1=[x1,v] & $2=(the mult of F)[;](x1,v)); A1: for x st x in [:the carrier of F,n-tuples_on the carrier of F:] ex y st y in n-tuples_on the carrier of F & P[x,y] proof let x; assume x in [:the carrier of F,n-tuples_on the carrier of F:]; then consider x1,v being set such that A2: x1 in the carrier of F & v in n-tuples_on the carrier of F & [x1,v] = x by ZFMISC_1:103; reconsider x1 as Element of F by A2; reconsider v as Element of n-tuples_on the carrier of F by A2; take y = (the mult of F)[;](x1,v); y is Element of n-tuples_on the carrier of F by FINSEQ_2:141; hence y in n-tuples_on the carrier of F; take x1,v; thus thesis by A2; end; consider f be Function of [:the carrier of F,n-tuples_on the carrier of F:], n-tuples_on the carrier of F such that A3: for x st x in [:the carrier of F,n-tuples_on the carrier of F:] holds P[x,f.x] from FuncEx1(A1); take f;let x be Element of F, v be Element of n-tuples_on the carrier of F; [x,v] in [:the carrier of F,n-tuples_on the carrier of F:] by ZFMISC_1:106; then consider x1 being Element of F , v1 being Element of n-tuples_on the carrier of F such that A4: [x,v]=[x1,v1] & f.[x,v]=(the mult of F)[;](x1,v1) by A3; x1 = x & v1 = v by A4,ZFMISC_1:33; hence f.(x,v) = (the mult of F)[;](x,v) by A4,BINOP_1:def 1; end; uniqueness proof let f,g be Function of [:the carrier of F,n-tuples_on the carrier of F:], n-tuples_on the carrier of F; assume that A5: for x being Element of F,v being Element of n-tuples_on the carrier of F holds f.(x,v) = (the mult of F)[;](x,v) and A6: for x being Element of F,v being Element of n-tuples_on the carrier of F holds g.(x,v) = (the mult of F)[;](x,v); now let x be Element of F, v be Element of n-tuples_on the carrier of F; f.(x,v) = (the mult of F)[;](x,v) by A5; hence f.(x,v)=g.(x,v) by A6; end; hence thesis by BINOP_1:2; end; end; definition let F,n; func n -VectSp_over F -> strict VectSpStr over F means :Def5: the LoopStr of it = n -Group_over F & the lmult of it = n -Mult_over F; existence proof the carrier of n-Group_over F = n-tuples_on the carrier of F proof n-Group_over F = LoopStr(# n-tuples_on the carrier of F, product(the add of F,n), (n .--> the Zero of F) qua Element of n-tuples_on the carrier of F#) by Def3; hence thesis; end; then reconsider d = n-Mult_over F as Function of [:the carrier of F, the carrier of n-Group_over F:], the carrier of n-Group_over F; set G = n-Group_over F; take VectSpStr(#car(G), ad(G), zr(G), d#); thus thesis; end; uniqueness; end; definition let F,n; cluster n -VectSp_over F -> non empty; coherence proof A1: the carrier of n -Group_over F is non empty; the LoopStr of n -VectSp_over F = n -Group_over F by Def5; hence the carrier of n -VectSp_over F is non empty by A1; end; end; reserve D for non empty set, H,G for BinOp of D, d for Element of D, t1,t2 for Element of n-tuples_on D; theorem Th18: H is_distributive_wrt G implies H[;](d,G.:(t1,t2)) = G.: (H[;](d,t1),H[;](d,t2)) proof assume A1: H is_distributive_wrt G; A2: H[;](d,G.:(t1,t2)) = H[;](d,G*<:t1,t2:>) by FUNCOP_1:def 3 .= H*<:dom (G*<:t1,t2:>) --> d,G*<:t1,t2:>:> by FUNCOP_1:def 5; set A = H*<:dom (G*<:t1,t2:>) --> d,G*<:t1,t2:>:>; A3: G.:(H[;](d,t1),H[;](d,t2)) = G.:(H*<:dom t1 -->d,t1:>,H[;](d,t2)) by FUNCOP_1:def 5 .= G.:(H*<:dom t1 -->d,t1:>,H*<:dom t2 -->d,t2:>) by FUNCOP_1:def 5 .= G*<:H*<:dom t1 -->d,t1:>,H*<:dom t2 -->d,t2:>:> by FUNCOP_1:def 3; A4: dom A = Seg n proof A5: dom t1 = Seg (len t1) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; dom t2 = Seg (len t2) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; then A6: dom <:t1,t2:> = Seg n by A5,FUNCT_3:70; rng t1 c= D & rng t2 c= D by FINSEQ_1:def 4; then A7: [:rng t1,rng t2:] c= [:D,D:] by ZFMISC_1:119; rng <:t1,t2:> c= [:rng t1,rng t2:] by FUNCT_3:71; then rng <:t1,t2:> c= [:D,D:] by A7,XBOOLE_1:1; then rng <:t1,t2:> c= dom G by FUNCT_2:def 1; then A8: dom (G*<:t1,t2:>) = Seg n by A6,RELAT_1:46; then dom (dom (G*<:t1,t2:>) -->d) = Seg n by FUNCOP_1:19; then A9: dom <: dom (G*<:t1,t2:>) -->d,G*<:t1,t2:>:> = Seg n by A8,FUNCT_3:70; rng <: dom (G*<:t1,t2:>) -->d,G*<:t1,t2:>:> c= dom H proof A10: rng <: dom (G*<:t1,t2:>) -->d,G*<:t1,t2:>:> c= [:rng (dom (G*<:t1,t2:>) -->d),rng (G*<:t1,t2:>):] by FUNCT_3:71; rng (dom (G*<:t1,t2:>) -->d) c= {d} & {d} c= D by FUNCOP_1:19,ZFMISC_1:37; then A11: rng (dom (G*<:t1,t2:>) -->d) c= D by XBOOLE_1:1; A12: rng (G*<:t1,t2:>) c= rng G by RELAT_1:45; rng G c= D by RELSET_1:12; then rng (G*<:t1,t2:>) c= D by A12,XBOOLE_1:1; then [:rng (dom (G*<:t1,t2:>) -->d),rng (G*<:t1,t2:>):] c= [:D,D:] by A11,ZFMISC_1:119; then rng <: dom (G*<:t1,t2:>) -->d,G*<:t1,t2:>:> c= [:D,D:] & dom H = [:D,D:] by A10,FUNCT_2:def 1,XBOOLE_1:1; hence thesis; end; hence thesis by A9,RELAT_1:46; end; A13: dom (G*<:H*<:dom t1 -->d,t1:>,H*<:dom t2 -->d,t2:>:>) = Seg n proof A14: rng <:H*<:dom t1 -->d,t1:>,H*<:dom t2 -->d,t2:>:> c= dom G proof rng (H*<:dom t1 -->d,t1:>) c= rng H & rng (H*<:dom t2 -->d,t2:>) c= rng H by RELAT_1:45; then A15: rng <:H*<:dom t1 -->d,t1:>,H*<:dom t2 -->d,t2:>:> c= [:rng (H*<:dom t1 -->d,t1:>),rng (H*<:dom t2 -->d,t2:>):] & [:rng (H*<:dom t1 -->d,t1:>),rng (H*<:dom t2 -->d,t2:>):] c= [:rng H,rng H:] by FUNCT_3:71,ZFMISC_1:119; rng H c= D by RELSET_1:12; then rng <:H*<:dom t1 -->d,t1:>,H*<:dom t2 -->d,t2:>:> c= [:rng H,rng H:] & [:rng H,rng H:] c= [:D,D:] by A15,XBOOLE_1:1,ZFMISC_1:119; then rng <:H*<:dom t1 -->d,t1:>,H*<:dom t2 -->d,t2:>:> c= [:D,D:] & dom G = [:D,D:] by FUNCT_2:def 1,XBOOLE_1:1; hence thesis; end; dom (H*<:dom t1 -->d,t1:>) = Seg n & dom (H*<:dom t2 -->d,t2:>) = Seg n proof rng t1 c= D by FINSEQ_1:def 4; then A16: rng <:dom t1 -->d,t1:> c= [:rng (dom t1 -->d),rng t1:] & [:rng (dom t1 -->d),rng t1:] c= [:rng (dom t1 -->d),D:] by FUNCT_3:71,ZFMISC_1:119; rng (dom t1 -->d) c= {d} by FUNCOP_1:19; then A17: rng <:dom t1 -->d,t1:> c= [:rng (dom t1 -->d),D:] & [:rng (dom t1 -->d),D:] c= [:{d},D:] by A16,XBOOLE_1:1,ZFMISC_1:119; {d} c= D by ZFMISC_1:37; then A18: rng <:dom t1 -->d,t1:> c= [:{d},D:] & [:{d},D:] c= [:D,D:] by A17,XBOOLE_1:1,ZFMISC_1:119; dom H = [:D,D:] by FUNCT_2:def 1; then A19: rng <:dom t1 -->d,t1:> c= dom H by A18,XBOOLE_1:1; A20: dom (dom t1 -->d) = dom t1 by FUNCOP_1:19 .= Seg (len t1) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; A21: dom t1 = Seg (len t1) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; A22: dom (H*<:dom t1 -->d,t1:>) = dom <:dom t1 -->d,t1:> by A19,RELAT_1:46 .= Seg n by A20,A21,FUNCT_3:70; rng t2 c= D by FINSEQ_1:def 4; then A23: rng <:dom t2 -->d,t2:> c= [:rng (dom t2 -->d),rng t2:] & [:rng (dom t2 -->d),rng t2:] c= [:rng (dom t2 -->d),D:] by FUNCT_3:71,ZFMISC_1:119; rng (dom t2 -->d) c= {d} by FUNCOP_1:19; then A24: rng <:dom t2 -->d,t2:> c= [:rng (dom t2 -->d),D:] & [:rng (dom t2 -->d),D:] c= [:{d},D:] by A23,XBOOLE_1:1,ZFMISC_1:119; {d} c= D by ZFMISC_1:37; then A25: rng <:dom t2 -->d,t2:> c= [:{d},D:] & [:{d},D:] c= [:D,D:] by A24,XBOOLE_1:1,ZFMISC_1:119; dom H = [:D,D:] by FUNCT_2:def 1; then A26: rng <: dom t2 -->d,t2:> c= dom H by A25,XBOOLE_1:1; A27: dom (dom t2 -->d) = dom t2 by FUNCOP_1:19 .= Seg (len t2) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; A28: dom t2 = Seg (len t2) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; dom (H*<:dom t2-->d,t2:>) = dom <:dom t2 -->d,t2:> by A26,RELAT_1:46 .= Seg n by A27,A28,FUNCT_3:70; hence thesis by A22; end; then dom <:H*<:dom t1 -->d,t1:>,H*<:dom t2 -->d,t2:>:> = Seg n by FUNCT_3:70; hence thesis by A14,RELAT_1:46; end; for x st x in dom A holds H[;](d,G.:(t1,t2)).x = G.:(H[;](d,t1),H[;](d,t2)) . x proof let x;assume A29: x in dom A; then x in dom (H*<:dom (G.:(t1,t2)) -->d,G.: (t1,t2):>) by A2,FUNCOP_1:def 5; then A30: x in dom <:dom (G.:(t1,t2)) -->d,G.:(t1,t2):> by FUNCT_1:21; dom <:dom (G.:(t1,t2)) -->d,G.:(t1,t2):> = dom (dom (G.:(t1,t2)) -->d) /\ dom (G.:(t1,t2)) by FUNCT_3:def 8; then A31: x in dom (G.:(t1,t2)) by A30,XBOOLE_0:def 3; A32: t1.x in D & t2.x in D proof A33: dom t1 = Seg (len t1) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; A34: rng t1 c= D by FINSEQ_1:def 4; A35: t1.x in rng t1 by A4,A29,A33,FUNCT_1:def 5; A36: dom t2 = Seg (len t2) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; A37: rng t2 c= D by FINSEQ_1:def 4; t2.x in rng t2 by A4,A29,A36,FUNCT_1:def 5; hence thesis by A34,A35,A37; end; A38: x in dom (H[;](d,t1)) & x in dom (H[;](d,t2)) proof dom (H*<:dom t1 -->d,t1:>) = Seg n & dom (H*<:dom t2 -->d,t2:>) = Seg n proof rng t1 c= D by FINSEQ_1:def 4; then A39: rng <:dom t1 -->d,t1:> c= [:rng (dom t1 -->d),rng t1:] & [:rng (dom t1 -->d),rng t1:] c= [:rng (dom t1 -->d),D:] by FUNCT_3:71,ZFMISC_1:119; rng (dom t1 -->d) c= {d} by FUNCOP_1:19; then A40: rng <:dom t1 -->d,t1:> c= [:rng (dom t1 -->d),D:] & [:rng (dom t1 -->d),D:] c= [:{d},D:] by A39,XBOOLE_1:1,ZFMISC_1:119; {d} c= D by ZFMISC_1:37; then A41: rng <:dom t1 -->d,t1:> c= [:{d},D:] & [:{d},D:] c= [:D,D:] by A40,XBOOLE_1:1,ZFMISC_1:119; dom H = [:D,D:] by FUNCT_2:def 1; then A42: rng <:dom t1 -->d,t1:> c= dom H by A41,XBOOLE_1:1; A43: dom (dom t1 -->d) = dom t1 by FUNCOP_1:19 .= Seg (len t1) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; A44: dom t1 = Seg (len t1) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; A45: dom (H*<:dom t1 -->d,t1:>) = dom <:dom t1 -->d,t1:> by A42,RELAT_1:46 .= Seg n by A43,A44,FUNCT_3:70; rng t2 c= D by FINSEQ_1:def 4; then A46: rng <:dom t2 -->d,t2:> c= [:rng (dom t2 -->d),rng t2:] & [:rng (dom t2 -->d),rng t2:] c= [:rng (dom t2 -->d),D:] by FUNCT_3:71,ZFMISC_1:119; rng (dom t2 -->d) c= {d} by FUNCOP_1:19; then A47: rng <:dom t2 -->d,t2:> c= [:rng (dom t2 -->d),D:] & [:rng (dom t2 -->d),D:] c= [:{d},D:] by A46,XBOOLE_1:1,ZFMISC_1:119; {d} c= D by ZFMISC_1:37; then A48: rng <:dom t2 -->d,t2:> c= [:{d},D:] & [:{d},D:] c= [:D,D:] by A47,XBOOLE_1:1,ZFMISC_1:119; dom H = [:D,D:] by FUNCT_2:def 1; then A49: rng <: dom t2 -->d,t2:> c= dom H by A48,XBOOLE_1:1; A50: dom (dom t2 -->d) = dom t2 by FUNCOP_1:19 .= Seg (len t2) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; A51: dom t2 = Seg (len t2) by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:109; dom (H*<:dom t2-->d,t2:>) = dom <:dom t2 -->d,t2:> by A49,RELAT_1:46 .= Seg n by A50,A51,FUNCT_3:70; hence thesis by A45; end; hence thesis by A4,A29,FUNCOP_1:def 5; end; H[;](d,G.:(t1,t2)).x = H.(d,G.:(t1,t2).x) by A2,A29,FUNCOP_1:42 .= H.(d,G.(t1.x,t2.x)) by A31,FUNCOP_1:28 .= G.(H.(d,t1.x),H.(d,t2.x)) by A1,A32,BINOP_1:23 .= G.((H[;](d,t1)).x,H.(d,t2.x)) by A38,FUNCOP_1:42 .= G.((H[;](d,t1)).x,(H[;](d,t2)).x) by A38,FUNCOP_1:42 .= (G.: (H[;](d,t1),H[;](d,t2))).x by A3,A4,A13,A29,FUNCOP_1:28; hence thesis; end; hence thesis by A2,A3,A4,A13,FUNCT_1:9; end; definition let D be non empty set, n be Nat, F be BinOp of D, x be Element of D, v be Element of n-tuples_on D; redefine func F[;](x,v) -> Element of n-tuples_on D; coherence by FINSEQ_2:141; end; definition let F,n; cluster n -VectSp_over F -> VectSp-like; coherence proof A1: the unity of F is_a_unity_wrt the mult of F by Th11; then A2: the mult of F has_a_unity by SETWISEO:def 2; A3: the_unity_wrt the mult of F = the unity of F by A1,BINOP_1:def 8; A4: the mult of F is associative by Lm4; A5: the mult of F is_distributive_wrt the add of F by Lm5; A6: the LoopStr of n -VectSp_over F=n-Group_over F by Def5; A7: n-Group_over F=LoopStr(# n-tuples_on the carrier of F, product(the add of F,n), (n .--> the Zero of F) qua Element of n-tuples_on the carrier of F#) by Def3; n-VectSp_over F is VectSp-like proof let x,y be Element of F; let v,w be Element of n -VectSp_over F; reconsider v'=v,w'=w as Element of n-tuples_on the carrier of F by A6,A7; thus x*(v+w) = (the lmult of n-VectSp_over F).(x,v+w) by VECTSP_1:def 24 .= (the lmult of n-VectSp_over F).(x, (the add of n-Group_over F).(v,w)) by A6,RLVECT_1:5 .= (n-Mult_over F).(x,product(the add of F,n).(v,w)) by A7,Def5 .= (n-Mult_over F).(x,(the add of F).:(v',w')) by Def1 .= (the mult of F)[;](x,(the add of F).:(v',w')) by Def4 .= (the add of F).: ((the mult of F)[;](x,v'),(the mult of F)[;](x,w')) by A5,Th18 .= product(the add of F,n).((the mult of F)[;](x,v'), (the mult of F)[;](x,w')) by Def1 .= product(the add of F,n).((n-Mult_over F).(x,v'), (the mult of F)[;](x,w')) by Def4 .= product(the add of F,n).((n-Mult_over F).(x,v'), (n-Mult_over F).(x,w')) by Def4 .= product(the add of F,n).((the lmult of n-VectSp_over F).(x,v), (n-Mult_over F).(x,w')) by Def5 .= (the add of n-Group_over F).((the lmult of n-VectSp_over F). (x,v),(the lmult of n-VectSp_over F).(x,w)) by A7,Def5 .= (the add of n-Group_over F).(x*v,(the lmult of n-VectSp_over F). (x,w)) by VECTSP_1:def 24 .= (the add of n-Group_over F).(x*v,x*w) by VECTSP_1:def 24 .= x*v + x*w by A6,RLVECT_1:5; thus (x+y)*v = (the lmult of n-VectSp_over F).(x+y,v) by VECTSP_1:def 24 .= (n-Mult_over F).(x+y,v') by Def5 .= (the mult of F)[;](x+y,v') by Def4 .= (the mult of F)[;]((the add of F).(x,y),v') by RLVECT_1:5 .= (the add of F).: ((the mult of F)[;](x,v'),(the mult of F)[;](y,v')) by A5,FINSEQOP:47 .= product(the add of F,n).((the mult of F)[;](x,v'), (the mult of F)[;](y,v')) by Def1 .= product(the add of F,n).((n-Mult_over F).(x,v'), (the mult of F)[;](y,v')) by Def4 .= product(the add of F,n).((n-Mult_over F).(x,v'), (n-Mult_over F).(y,v')) by Def4 .= product(the add of F,n).((the lmult of n-VectSp_over F).(x,v), (n-Mult_over F).(y,v')) by Def5 .= (the add of n-Group_over F).((the lmult of n-VectSp_over F).(x,v), (the lmult of n-VectSp_over F).(y,v)) by A7,Def5 .= (the add of n-Group_over F).(x*v,(the lmult of n-VectSp_over F). (y,v)) by VECTSP_1:def 24 .= (the add of n-Group_over F).(x*v,y*v) by VECTSP_1:def 24 .= x*v + y*v by A6,RLVECT_1:5; thus (x*y)*v = (the lmult of n-VectSp_over F).(x*y,v) by VECTSP_1:def 24 .= (n-Mult_over F).(x*y,v') by Def5 .= (the mult of F)[;](x*y,v') by Def4 .= (the mult of F)[;]((the mult of F).(x,y),v') by VECTSP_1:def 10 .= (the mult of F)[;](x,(the mult of F)[;](y,v')) by A4,FINSEQOP:32 .= (n-Mult_over F).(x,(the mult of F)[;](y,v')) by Def4 .= (n-Mult_over F).(x,(n-Mult_over F).(y,v')) by Def4 .= (n-Mult_over F).(x,(the lmult of n-VectSp_over F).(y,v)) by Def5 .= (the lmult of n-VectSp_over F).(x,(the lmult of n-VectSp_over F). (y,v)) by Def5 .= (the lmult of n-VectSp_over F).(x,y*v) by VECTSP_1:def 24 .= x*(y*v) by VECTSP_1:def 24; thus (1_ F)*v = (the unity of F)*v by VECTSP_1:def 9 .= (the lmult of n-VectSp_over F).(the unity of F,v) by VECTSP_1:def 24 .= (n-Mult_over F).(the unity of F,v') by Def5 .= (the mult of F)[;](the unity of F,v') by Def4 .= v by A2,A3,FINSEQOP:58; thus thesis; end; hence thesis; end; end; begin :: Sequences of domains reserve x,y,y1,y2,z for set, i for Nat, A for AbGroup, D for non empty set; definition cluster non empty non-empty FinSequence; existence proof consider D; take <*D*>; A1: Seg 1 = dom <*D*> by FINSEQ_1:def 8; 1 in Seg 1 by FINSEQ_1:4,TARSKI:def 1; then [1,<*D*>.1] in <*D*> by A1,FUNCT_1:def 4; hence <*D*> is non empty; assume {} in rng <*D*>; then {} in {D} by FINSEQ_1:55; hence contradiction by TARSKI:def 1; end; end; definition let f be non-empty Function; cluster product f -> functional non empty; coherence proof product f is non empty set proof not {} in rng f by RELAT_1:def 9; hence thesis by CARD_3:37; end; hence thesis; end; end; definition mode Domain-Sequence is non empty non-empty FinSequence; end; definition let a be non empty Function; cluster dom a -> non empty; coherence proof consider x being Element of a; consider y,z such that A1: x = [y,z] by RELAT_1:def 1; thus thesis by A1,RELAT_1:def 4; end; end; scheme NEFinSeqLambda {w()-> non empty FinSequence, F(set)->set}: ex p being non empty FinSequence st len p = len w() & for i being Element of dom w() holds p.i = F(i) proof deffunc G(set) = F($1); consider p being FinSequence such that A1: len p = len w() & for i st i in Seg len w() holds p.i = G(i) from SeqLambda; consider x being Element of dom w(); dom p = dom w() by A1,FINSEQ_3:31; then [x,p.x] in p & p = p by FUNCT_1:def 4; then reconsider p'=p as non empty FinSequence; take p'; thus len p' = len w() by A1; let i be Element of dom w(); dom w() = Seg len w() by FINSEQ_1:def 3; hence p'.i = F(i) by A1; end; definition let a be non-empty non empty Function; let i be Element of dom a; cluster a.i -> non empty; coherence proof a.i in rng a by FUNCT_1:def 5; hence thesis by RELAT_1:def 9; end; end; definition let a be non-empty non empty Function; let f be Element of product a; let i be Element of dom a; redefine func f.i -> Element of a.i; coherence by CARD_3:18; end; begin :: The Product of Families of Operations reserve a for Domain-Sequence, i for Element of dom a, p for FinSequence; definition let a be non-empty non empty Function; canceled 2; mode BinOps of a -> Function means:Def8: dom it = dom a & for i being Element of dom a holds it.i is BinOp of a.i; existence proof deffunc F(set) = pr1(a.$1,a.$1); consider f being Function such that A1: dom f = dom a & for x st x in dom a holds f.x = F(x) from Lambda; take f; thus dom f = dom a by A1; let i be Element of dom a; f.i = pr1(a.i,a.i) by A1; hence thesis; end; mode UnOps of a -> Function means: Def9: dom it = dom a & for i being Element of dom a holds it.i is UnOp of a.i; existence proof deffunc F(set) = id (a.$1); consider f being Function such that A2: dom f = dom a & for x st x in dom a holds f.x = F(x) from Lambda; take f; thus dom f = dom a by A2; let i be Element of dom a; f.i = id (a.i) by A2; hence thesis; end; end; definition let a; cluster -> FinSequence-like BinOps of a; coherence proof let f be BinOps of a; dom a = dom f & dom a = Seg len a by Def8,FINSEQ_1:def 3; hence thesis by FINSEQ_1:def 2; end; cluster -> FinSequence-like UnOps of a; coherence proof let f be UnOps of a; dom a = dom f & dom a = Seg len a by Def9,FINSEQ_1:def 3; hence thesis by FINSEQ_1:def 2; end; end; theorem Th19: p is BinOps of a iff len p = len a & for i holds p.i is BinOp of a.i proof dom p = dom a iff len p = len a by FINSEQ_3:31; hence thesis by Def8; end; theorem Th20: p is UnOps of a iff len p = len a & for i holds p.i is UnOp of a.i proof dom p = dom a iff len p = len a by FINSEQ_3:31; hence thesis by Def9; end; definition let a; let b be BinOps of a; let i; redefine func b.i -> BinOp of a.i; coherence by Th19; end; definition let a; let u be UnOps of a; let i; redefine func u.i -> UnOp of a.i; coherence by Th20; end; definition let F be functional non empty set; let u be UnOp of F, f be Element of F; redefine func u.f -> Element of F; coherence proof u.f is Element of F; hence thesis; end; end; theorem for d,d' being UnOp of product a st for f being Element of product a, i being Element of dom a holds (d.f).i = (d'.f).i holds d = d' proof let d,d' be UnOp of product a such that A1: for f being Element of product a, i being Element of dom a holds (d.f).i = (d'.f).i; now let f be Element of product a; dom (d.f) = dom a & dom (d'.f) = dom a & for x st x in dom a holds (d.f).x = (d'.f).x by A1,CARD_3:18; hence d.f = d'.f by FUNCT_1:9; end; hence thesis by FUNCT_2:113; end; theorem Th22: for u being UnOps of a holds doms u = a & product rngs u c= product a proof let u be UnOps of a; A1: dom doms u = u"SubFuncs rng u & dom rngs u = u"SubFuncs rng u & dom a = Seg len a & dom u = Seg len u & len u = len a by Th20,FINSEQ_1:def 3,FUNCT_6:def 2,def 3; A2: dom u c= u"SubFuncs rng u proof let x; assume x in dom u; then reconsider x as Element of dom a by A1; u.x is UnOp of a.x; hence thesis by A1,FUNCT_6:28; end; u"SubFuncs rng u c= dom u by RELAT_1:167; then A3: dom doms u = dom u by A1,A2,XBOOLE_0:def 10; now let x; assume x in dom u; then reconsider i = x as Element of dom a by A1; (doms u).i = dom (u.i) by A1,FUNCT_6:31 .= a.i by FUNCT_2:def 1; hence (doms u).x = a.x; end; hence doms u = a by A1,A3,FUNCT_1:9; now let x; assume x in dom u; then reconsider i = x as Element of dom a by A1; (rngs u).i = rng (u.i) & rng (u.i) c= a.i by A1,FUNCT_6:31,RELSET_1:12; hence (rngs u).x c= a.x; end; hence thesis by A1,A3,CARD_3:38; end; definition let a; let u be UnOps of a; redefine func Frege u -> UnOp of product a; coherence proof dom Frege u = product doms u & doms u = a & product rngs u c= product a & rng Frege u = product rngs u by Th22,FUNCT_6:58,def 7; hence thesis by FUNCT_2:def 1,RELSET_1:11; end; end; theorem Th23: for u being UnOps of a for f being Element of product a, i being Element of dom a holds (Frege u).f.i = (u.i).(f.i) proof let u be UnOps of a, f be Element of product a; let i be Element of dom a; dom a = Seg len a & len a = len u & dom u = Seg len u & doms u = a by Th20,Th22,FINSEQ_1:def 3; hence thesis by FUNCT_6:57; end; definition let F be functional non empty set; let b be BinOp of F, f,g be Element of F; redefine func b.(f,g) -> Element of F; coherence proof b.(f,g) is Element of F; hence thesis; end; end; theorem Th24: for d,d' being BinOp of product a st for f,g being Element of product a, i being Element of dom a holds (d.(f,g)).i = (d'.(f,g)).i holds d = d' proof let d,d' be BinOp of product a such that A1: for f,g be Element of product a, i be Element of dom a holds (d.(f,g)).i = (d'.(f,g)).i; now let f,g be Element of product a; dom (d.(f,g)) = dom a & dom (d'.(f,g)) = dom a & for x st x in dom a holds (d.(f,g)).x = (d'.(f,g)).x by A1,CARD_3:18; hence d.(f,g) = d'.(f,g) by FUNCT_1:9; end; hence thesis by BINOP_1:2; end; reserve i for Element of dom a; definition let a; let b be BinOps of a; func [:b:] -> BinOp of product a means :Def10: for f,g being Element of product a, i being Element of dom a holds (it.(f,g)).i = (b.i).(f.i,g.i); existence proof defpred Q[Element of product a,Element of product a,Element of product a] means for i being Element of dom a holds $3.i = (b.i).($1.i,$2.i); A1: for f,g being Element of product a ex z being Element of product a st Q[f,g,z] proof let f,g be Element of product a; defpred P[set,set] means ex i st $1 =i & $2 = (b.i).(f.i,g.i); A2: for x,z1,z2 being set st x in dom a & P[x,z1] & P[x,z2] holds z1 = z2; A3: for x st x in dom a ex z st P[x,z] proof let x; assume x in dom a; then reconsider i = x as Element of dom a; take (b.i).(f.i,g.i); thus thesis; end; consider z being Function such that A4: dom z = dom a & for x being set st x in dom a holds P[x,z.x] from FuncEx(A2,A3); now let x; assume x in dom a; then ex i st x = i & z.x = (b.i).(f.i,g.i) by A4; hence z.x in a.x; end; then reconsider z' = z as Element of product a by A4,CARD_3:18; take z'; let i; ex j being Element of dom a st j = i & z.i = (b.j).(f.j,g.j) by A4; hence thesis; end; consider d being BinOp of product a such that A5: for f,g being Element of product a holds Q[f,g,d.(f,g)] from BinOpEx(A1); take d; thus thesis by A5; end; uniqueness proof let d,d' be BinOp of product a such that A6: for f,g being Element of product a,i being Element of dom a holds (d.(f,g)).i = (b.i).(f.i,g.i) and A7: for f,g being Element of product a,i being Element of dom a holds (d'.(f,g)).i = (b.i).(f.i,g.i); now let f,g be Element of product a; let i be Element of dom a; thus (d.(f,g)).i = (b.i).(f.i,g.i) by A6 .= (d'.(f,g)).i by A7; end; hence thesis by Th24; end; end; theorem Th25: for b being BinOps of a st for i holds b.i is commutative holds [:b:] is commutative proof let b be BinOps of a such that A1: for i holds b.i is commutative; let x,y be Element of product a; A2: dom ([:b:].(x,y)) = dom a & dom ([:b:].(y,x)) = dom a by CARD_3:18; now let z; assume z in dom a; then reconsider i = z as Element of dom a; b.i is commutative & [:b:].(x,y).i = (b.i).(x.i,y.i) & [:b:].(y,x).i = (b.i).(y.i,x.i) by A1,Def10; hence [:b:].(x,y).z = [:b:].(y,x).z by BINOP_1:def 2; end; hence thesis by A2,FUNCT_1:9; end; theorem Th26: for b being BinOps of a st for i holds b.i is associative holds [:b:] is associative proof let b be BinOps of a such that A1: for i holds b.i is associative; let x,y,z be Element of product a; A2: dom ([:b:].(x,[:b:].(y,z))) = dom a & dom ([:b:].([:b:].(x,y),z)) = dom a by CARD_3:18; now let v be set; assume v in dom a; then reconsider i = v as Element of dom a; set xy = [:b:].(x,y), yz = [:b:].(y,z); b.i is associative & [:b:].(x,y).i = (b.i).(x.i,y.i) & [:b:].(y,z).i = (b.i).(y.i,z.i) & [:b:].(x,yz).i = (b.i).(x.i,yz.i) & [:b:].(xy,z).i = (b.i).(xy.i,z.i) by A1,Def10; hence [:b:].(x,[:b:].(y,z)).v = [:b:].([:b:].(x,y),z).v by BINOP_1:def 3; end; hence thesis by A2,FUNCT_1:9; end; theorem Th27: for b being BinOps of a, f being Element of product a st for i holds f.i is_a_unity_wrt b.i holds f is_a_unity_wrt [:b:] proof let b be BinOps of a, f be Element of product a such that A1: for i holds f.i is_a_unity_wrt b.i; now let x be Element of product a; A2: dom ([:b:].(x,f)) = dom a & dom ([:b:].(f,x)) = dom a & dom x = dom a by CARD_3:18; now let y; assume y in dom a; then reconsider i = y as Element of dom a; [:b:].(f,x).i = (b.i).(f.i,x.i) & f.i is_a_unity_wrt b.i by A1,Def10; hence [:b:].(f,x).y = x.y by BINOP_1:11; end; hence [:b:].(f,x) = x by A2,FUNCT_1:9; now let y; assume y in dom a; then reconsider i = y as Element of dom a; [:b:].(x,f).i = (b.i).(x.i,f.i) & f.i is_a_unity_wrt b.i by A1,Def10; hence [:b:].(x,f).y = x.y by BINOP_1:11; end; hence [:b:].(x,f) = x by A2,FUNCT_1:9; end; hence thesis by BINOP_1:11; end; theorem Th28: for b being BinOps of a, u being UnOps of a st for i holds u.i is_an_inverseOp_wrt b.i & b.i has_a_unity holds Frege u is_an_inverseOp_wrt [:b:] proof let b be BinOps of a, u be UnOps of a such that A1: for i holds u.i is_an_inverseOp_wrt b.i & b.i has_a_unity; defpred P[set,set] means ex i st $1 = i & $2 = the_unity_wrt (b.i); A2: for x st x in dom a ex y st P[x,y] proof let x; assume x in dom a; then reconsider i = x as Element of dom a; take the_unity_wrt (b.i); thus thesis; end; A3: for x,y1,y2 st x in dom a & P[x,y1] & P[x,y2] holds y1 = y2; consider f being Function such that A4: dom f = dom a & for x st x in dom a holds P[x,f.x] from FuncEx(A3,A2); now let x; assume x in dom a; then ex i st x = i & f.x = the_unity_wrt (b.i) by A4; hence f.x in a.x; end; then reconsider f as Element of product a by A4,CARD_3:18; now let i; ex j being Element of dom a st i = j & f.i = the_unity_wrt (b.j) by A4; then f.i = the_unity_wrt (b.i) & b.i has_a_unity by A1; hence f.i is_a_unity_wrt b.i by SETWISEO:22; end; then f is_a_unity_wrt [:b:] by Th27; then A5: f = the_unity_wrt [:b:] by BINOP_1:def 8; let x be Element of product a; A6: dom ([:b:].(x,(Frege u).x)) = dom a & dom ([:b:].((Frege u).x,x)) = dom a & dom f = dom a by CARD_3:18; now let y; assume y in dom a; then reconsider i = y as Element of dom a; ex j being Element of dom a st i = j & f.i = the_unity_wrt (b.j) by A4; then [:b:].(x,(Frege u).x).i = (b.i).(x.i,(Frege u).x.i) & (Frege u).x.i = (u.i).(x.i) & f.i = the_unity_wrt (b.i) & u.i is_an_inverseOp_wrt b.i by A1,Def10,Th23; hence [:b:].(x,(Frege u).x).y = f.y by FINSEQOP:def 1; end; hence [:b:].(x,(Frege u).x) = the_unity_wrt [:b:] by A5,A6,FUNCT_1:9; now let y; assume y in dom a; then reconsider i = y as Element of dom a; ex j being Element of dom a st i = j & f.i = the_unity_wrt (b.j) by A4; then [:b:].((Frege u).x,x).i = (b.i).((Frege u).x.i,x.i) & (Frege u).x.i = (u.i).(x.i) & f.i = the_unity_wrt (b.i) & u.i is_an_inverseOp_wrt b.i by A1,Def10,Th23; hence [:b:].((Frege u).x,x).y = f.y by FINSEQOP:def 1; end; hence [:b:].((Frege u).x,x) = the_unity_wrt [:b:] by A5,A6,FUNCT_1:9; end; begin :: The Product of Groups definition let F be Function; attr F is AbGroup-yielding means:Def11: x in rng F implies x is AbGroup; end; definition cluster non empty AbGroup-yielding FinSequence; existence proof consider A; take <*A*>; A1: Seg 1 = dom <*A*> by FINSEQ_1:def 8; 1 in Seg 1 by FINSEQ_1:4,TARSKI:def 1; then [1,<*A*>.1] in <*A*> & <*A*> = <*A*> by A1,FUNCT_1:def 4; hence <*A*> is non empty; let x; assume A2: x in rng <*A*> & not x is AbGroup; then x in {A} by FINSEQ_1:55; hence contradiction by A2,TARSKI:def 1; end; end; definition mode Group-Sequence is non empty AbGroup-yielding FinSequence; end; definition let g be Group-Sequence; let i be Element of dom g; redefine func g.i -> AbGroup; coherence proof g.i in rng g by FUNCT_1:def 5; hence thesis by Def11; end; end; definition let g be Group-Sequence; func carr g -> Domain-Sequence means :Def12: len it = len g & for j be Element of dom g holds it.j = the carrier of g.j; existence proof defpred P[set,set] means ex j' being Element of dom g st j' = $1 & $2 = the carrier of g.j'; A1: for j be Nat,y1,y2 st j in Seg len g & P[j,y1] & P[j,y2] holds y1=y2; A2: for j being Nat st j in Seg len g ex x st P[j,x] proof let j be Nat; assume j in Seg len g; then reconsider j' = j as Element of dom g by FINSEQ_1:def 3; take the carrier of g.j'; thus thesis; end; consider p being FinSequence such that A3: dom p = Seg (len g) & for j being Nat st j in Seg len g holds P[j,p.j] from SeqEx(A1,A2); A4: dom g = Seg len g by FINSEQ_1:def 3; A5: p is non empty proof consider i being Element of dom g; dom p = dom g by A3,FINSEQ_1:def 3; then [i,p.i] in p & p = p by FUNCT_1:def 4; hence p is non empty; end; p is non-empty proof assume {} in rng p; then consider x such that A6: x in dom p & {} = p.x by FUNCT_1:def 5; reconsider x as Nat by A6; consider x' being Element of dom g such that A7: x' = x & p.x = the carrier of g.x' by A3,A6; thus contradiction by A6,A7; end; then reconsider p' = p as Domain-Sequence by A5; take p'; thus len p' = len g by A3,FINSEQ_1:def 3; let j be Element of dom g; reconsider k = j as Nat; ex j' being Element of dom g st j' = k & p'.k = the carrier of g.j' by A3,A4; hence thesis; end; uniqueness proof let f,h be Domain-Sequence such that A8: len f = len g & for j be Element of dom g holds f.j = the carrier of g.j and A9: len h = len g & for j be Element of dom g holds h.j = the carrier of g.j; reconsider f' = f ,h' = h as FinSequence; A10: dom f' = Seg len f' & dom h' = Seg len h' by FINSEQ_1:def 3; now let j be Nat; assume j in dom f'; then reconsider j' = j as Element of dom g by A8,FINSEQ_3:31; f'.j = the carrier of g.j' by A8; hence f'.j = h'.j by A9; end; hence f = h by A8,A9,A10,FINSEQ_1:17; end; end; reserve g for Group-Sequence, i for Element of dom carr g; definition let g,i; redefine func g.i -> AbGroup; coherence proof dom g = Seg len g & Seg len(carr g) = dom carr g by FINSEQ_1:def 3; then reconsider i' = i as Element of dom g by Def12; g.i' is AbGroup; hence thesis; end; end; definition let g; func addop g -> BinOps of carr g means :Def13: len it = len carr g & for i holds it.i = the add of g.i; existence proof deffunc F(Element of dom carr g) = the add of g.$1; consider p being non empty FinSequence such that A1: len p = len carr g & for i holds p.i = F(i) from NEFinSeqLambda; now let i; len g = len (carr g) by Def12; then reconsider j = i as Element of dom g by FINSEQ_3:31; p.i = the add of g.i & the carrier of g.j = (carr g).j by A1,Def12; hence p.i is BinOp of (carr g).i; end; then reconsider p' = p as BinOps of carr g by A1,Th19; take p'; thus thesis by A1; end; uniqueness proof let f,h be BinOps of carr g; assume that A2: (len f = len carr g & for i holds f.i = the add of g.i) and A3: (len h = len carr g & for i holds h.i = the add of g.i); reconsider f' = f,h' = h as FinSequence; A4: dom f' = Seg len f' & dom h' = Seg len h' by FINSEQ_1:def 3; now let i be Nat; assume i in dom f'; then reconsider i' = i as Element of dom carr g by A2,FINSEQ_3:31; f'.i = the add of g.i' by A2; hence f'.i = h'.i by A3; end; hence f = h by A2,A3,A4,FINSEQ_1:17; end; func complop g -> UnOps of carr g means:Def14: len it = len carr g & for i holds it.i = comp g.i; existence proof deffunc F(Element of dom carr g) = comp g.$1; consider p being non empty FinSequence such that A5: len p = len carr g & for i holds p.i = F(i) from NEFinSeqLambda; now let i; len g = len (carr g) by Def12; then reconsider j = i as Element of dom g by FINSEQ_3:31; p.i = comp g.i & the carrier of g.j = (carr g).j by A5,Def12; hence p.i is UnOp of (carr g).i; end; then reconsider p' = p as UnOps of carr g by A5,Th20; take p'; thus thesis by A5; end; uniqueness proof let f,h be UnOps of carr g; assume that A6: len f = len carr g & for i holds f.i = comp g.i and A7: len h = len carr g & for i holds h.i = comp g.i; reconsider f' = f,h' = h as FinSequence; A8: dom f' = Seg len f' & dom h' = Seg len h' by FINSEQ_1:def 3; now let i be Nat; assume i in dom f'; then reconsider i' = i as Element of dom carr g by A6,FINSEQ_3:31; f.i = comp g.i' by A6; hence f.i = h.i by A7; end; hence f = h by A6,A7,A8,FINSEQ_1:17; end; func zeros g -> Element of product carr g means :Def15: for i holds it.i = the Zero of g.i; existence proof deffunc F(Element of dom carr g) = the Zero of g.$1; consider p being non empty FinSequence such that A9: len p = len carr g & for i holds p.i = F(i) from NEFinSeqLambda; A10:dom g = Seg len g & dom (carr g) = Seg len (carr g) & len g = len carr g & dom p = Seg len p by Def12,FINSEQ_1:def 3; now let i be set; assume i in dom (carr g); then reconsider x = i as Element of dom carr g; reconsider x' = x as Element of dom g by A10; p.x = the Zero of g.x & (carr g).x' = the carrier of g.x' by A9,Def12; hence p.i in (carr g).i; end; then reconsider t = p as Element of product (carr g) by A9,A10,CARD_3:18; take t; thus thesis by A9; end; uniqueness proof let f,h be Element of product carr g; assume that A11: (for i holds f.i = the Zero of g.i) and A12: (for i holds h.i = the Zero of g.i); reconsider f' = f , h' = h as Function; A13: dom f' = dom carr g & dom h' = dom carr g by CARD_3:18; now let x; assume x in dom carr g; then reconsider i = x as Element of dom carr g; thus f'.x = the Zero of g.i by A11 .= h'.x by A12; end; hence thesis by A13,FUNCT_1:9; end; end; definition let G be Group-Sequence; func product G -> strict AbGroup equals LoopStr(# product carr G, [:addop G:], zeros G #); coherence proof reconsider GS = LoopStr(# product carr G, [:addop G:], zeros G #) as non empty LoopStr; A1: now let i be Element of dom carr G; dom G = Seg len G by FINSEQ_1:def 3 .= Seg len carr G by Def12 .= dom carr G by FINSEQ_1:def 3; hence (carr G).i = car(G.i) by Def12; end; now let i be Element of dom carr G; (addop G).i = ad(G.i) & (carr G).i = car(G.i) by A1,Def13; hence (addop G).i is commutative by FVSUM_1:2; end; then A2: [:addop G:] is commutative by Th25; now let i be Element of dom carr G; (addop G).i = ad(G.i) & (carr G).i = car(G.i) by A1,Def13; hence (addop G).i is associative by FVSUM_1:3; end; then A3: [:addop G:] is associative by Th26; now let i be Element of dom carr G; (addop G).i = ad(G.i) & (zeros G).i = zr(G.i) & (carr G).i = car(G.i) by A1,Def13,Def15; hence (zeros G).i is_a_unity_wrt (addop G).i by Th3; end; then A4: zeros G is_a_unity_wrt [:addop G:] by Th27; GS is right_complementable proof let x be Element of GS; reconsider y = (Frege complop G).x as Element of GS by FUNCT_2:7; take y; now let i be Element of dom carr G; zr(G.i) is_a_unity_wrt ad(G.i) & (carr G).i = car(G.i) & (addop G).i = ad(G.i) & (complop G).i = cc(G.i) by A1,Def13,Def14,Th3; hence (complop G).i is_an_inverseOp_wrt (addop G).i & (addop G).i has_a_unity by Th4,SETWISEO:def 2; end; then Frege complop G is_an_inverseOp_wrt [:addop G:] by Th28; then [:addop G:].(x,y) = the_unity_wrt [:addop G:] by FINSEQOP:def 1; then x + y = the_unity_wrt [:addop G:] by RLVECT_1:5; then x + y = the Zero of GS by A4,BINOP_1:def 8; hence thesis by RLVECT_1:def 2; end; hence thesis by A2,A3,A4,Lm2,Lm3; end; end;

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