Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

### The abstract of the Mizar article:

### Sum and Product of Finite Sequences of Elements of a Field

**by****Katarzyna Zawadzka**- Received December 29, 1992
- MML identifier: FVSUM_1

- [ Mizar article, MML identifier index ]

environ vocabulary RLVECT_1, BINOP_1, FUNCT_1, VECTSP_1, ALGSTR_1, SETWISEO, LATTICES, FUNCOP_1, ARYTM_1, FINSEQOP, FINSEQ_1, RELAT_1, FINSEQ_2, RVSUM_1, SQUARE_1, FINSUB_1, BOOLE, SUBSET_1, FINSEQ_4, RLSUB_2, FINSET_1, CARD_1, CAT_3, FVSUM_1, CARD_3; notation TARSKI, XBOOLE_0, SUBSET_1, NUMBERS, XREAL_0, NAT_1, SQUARE_1, RELAT_1, FUNCT_1, STRUCT_0, PARTFUN1, FUNCT_2, FINSEQ_1, FINSEQ_4, BINOP_1, FUNCOP_1, RLVECT_1, SETWISEO, FINSEQ_2, FINSEQOP, SETWOP_2, TOPREAL1, ALGSTR_1, GROUP_4, VECTSP_1, CARD_1, FINSET_1, FINSUB_1, MATRIX_2; constructors DOMAIN_1, NAT_1, SQUARE_1, SETWISEO, FINSEQOP, SETWOP_2, TOPREAL1, ALGSTR_1, GROUP_4, MATRIX_2, FINSOP_1, FINSEQ_4, MEMBERED, PARTFUN1, XBOOLE_0; clusters FUNCT_1, FINSEQ_1, VECTSP_1, RELSET_1, STRUCT_0, FINSEQ_2, ALGSTR_1, FINSET_1, FINSUB_1, NAT_1, MEMBERED, ZFMISC_1, FUNCT_2, PARTFUN1, XBOOLE_0, NUMBERS, ORDINAL2; requirements NUMERALS, REAL, BOOLE, SUBSET; begin :: Auxiliary theorems reserve i,j,k for Nat; canceled; theorem :: FVSUM_1:2 for K being Abelian (non empty LoopStr) holds the add of K is commutative; theorem :: FVSUM_1:3 for K being add-associative (non empty LoopStr) holds the add of K is associative; theorem :: FVSUM_1:4 for K being commutative (non empty HGrStr) holds the mult of K is commutative; canceled; theorem :: FVSUM_1:6 :::multLoopStr for K being commutative left_unital (non empty doubleLoopStr) holds 1_ K is_a_unity_wrt the mult of K; theorem :: FVSUM_1:7 :::multLoopStr for K being commutative left_unital (non empty doubleLoopStr) holds the_unity_wrt the mult of K = 1_ K; theorem :: FVSUM_1:8 for K being left_zeroed right_zeroed (non empty LoopStr) holds 0.K is_a_unity_wrt the add of K; theorem :: FVSUM_1:9 for K being left_zeroed right_zeroed (non empty LoopStr) holds the_unity_wrt the add of K = 0.K; theorem :: FVSUM_1:10 for K being left_zeroed right_zeroed (non empty LoopStr) holds the add of K has_a_unity; theorem :: FVSUM_1:11 :::multLoopStr for K being commutative left_unital (non empty doubleLoopStr) holds the mult of K has_a_unity; theorem :: FVSUM_1:12 for K being distributive (non empty doubleLoopStr) holds the mult of K is_distributive_wrt the add of K; definition let K be non empty HGrStr;let a be Element of K; func a multfield -> UnOp of the carrier of K equals :: FVSUM_1:def 1 (the mult of K)[;](a,id (the carrier of K)); end; definition let K be non empty LoopStr; func diffield(K) -> BinOp of the carrier of K equals :: FVSUM_1:def 2 (the add of K)*(id the carrier of K,comp K); end; canceled; theorem :: FVSUM_1:14 for K being non empty LoopStr, a1,a2 being Element of K holds (diffield(K)).(a1,a2) = a1 - a2; theorem :: FVSUM_1:15 for K being distributive (non empty doubleLoopStr), a being Element of K holds a multfield is_distributive_wrt the add of K; theorem :: FVSUM_1:16 for K being left_zeroed right_zeroed add-associative right_complementable (non empty LoopStr) holds comp K is_an_inverseOp_wrt the add of K; theorem :: FVSUM_1:17 for K being left_zeroed right_zeroed add-associative right_complementable (non empty LoopStr) holds the add of K has_an_inverseOp; theorem :: FVSUM_1:18 for K being left_zeroed right_zeroed add-associative right_complementable (non empty LoopStr) holds the_inverseOp_wrt the add of K = comp K; theorem :: FVSUM_1:19 for K being right_zeroed add-associative right_complementable Abelian (non empty LoopStr) holds comp K is_distributive_wrt the add of K; begin :: :: Some Operations on the i-tuples on Element of K :: definition let K be non empty LoopStr; let p1,p2 be FinSequence of the carrier of K; func p1 + p2 -> FinSequence of the carrier of K equals :: FVSUM_1:def 3 (the add of K).:(p1,p2); end; canceled; theorem :: FVSUM_1:21 for K being non empty LoopStr, p1, p2 be FinSequence of the carrier of K, a1, a2 being Element of K, i being Nat st i in dom (p1+p2) & a1 = p1.i & a2 = p2.i holds (p1+p2).i = a1 + a2; definition let i; let K be non empty LoopStr; let R1,R2 be Element of i-tuples_on the carrier of K; redefine func R1 + R2 -> Element of i-tuples_on the carrier of K; end; theorem :: FVSUM_1:22 for K being non empty LoopStr, a1,a2 being Element of K, R1,R2 being Element of i-tuples_on the carrier of K holds j in Seg i & a1 = R1.j & a2 = R2.j implies (R1+R2).j = a1 + a2; theorem :: FVSUM_1:23 for K being non empty LoopStr, p being FinSequence of the carrier of K holds <*>(the carrier of K) + p = <*>(the carrier of K) & p + <*>(the carrier of K) = <*>(the carrier of K); theorem :: FVSUM_1:24 for K being non empty LoopStr, a1,a2 being Element of K holds <*a1*> + <*a2*> = <*a1+a2*>; theorem :: FVSUM_1:25 for K being non empty LoopStr, a1,a2 being Element of K holds (i|->a1) + (i|->a2) = i|->(a1+a2); theorem :: FVSUM_1:26 for K being Abelian (non empty LoopStr), R1,R2 being Element of i-tuples_on the carrier of K holds R1 + R2 = R2 + R1; theorem :: FVSUM_1:27 for K be add-associative (non empty LoopStr), R1,R2,R3 being Element of i-tuples_on the carrier of K holds R1 + (R2 + R3) = R1 + R2 + R3; theorem :: FVSUM_1:28 for K being Abelian left_zeroed right_zeroed (non empty LoopStr), R being Element of i-tuples_on the carrier of K holds R + (i|->(0.K)) = R & R = (i|->(0.K)) + R; definition let K be non empty LoopStr; let p be FinSequence of the carrier of K; func -p -> FinSequence of the carrier of K equals :: FVSUM_1:def 4 (comp K)*p; end; reserve K for non empty LoopStr, a for Element of K, p for FinSequence of the carrier of K, R for Element of i-tuples_on the carrier of K; canceled; theorem :: FVSUM_1:30 i in dom -p & a = p.i implies (-p).i = -a; definition let i;let K be non empty LoopStr; let R be Element of i-tuples_on the carrier of K; redefine func -R -> Element of i-tuples_on the carrier of K; end; theorem :: FVSUM_1:31 j in Seg i & a = R.j implies (-R).j = -a; theorem :: FVSUM_1:32 -(<*>(the carrier of K)) = <*>(the carrier of K); theorem :: FVSUM_1:33 -<*a*> = <*-a*>; theorem :: FVSUM_1:34 -(i|->a) = i|->-a; theorem :: FVSUM_1:35 for K being Abelian right_zeroed add-associative right_complementable (non empty LoopStr), R being Element of i-tuples_on the carrier of K holds R + -R = (i|->0.K) & -R + R = (i|->0.K); reserve K for left_zeroed right_zeroed add-associative right_complementable (non empty LoopStr), R,R1,R2 for Element of i-tuples_on the carrier of K; theorem :: FVSUM_1:36 R1 + R2 = (i|->0.K) implies R1 = -R2 & R2 = -R1; theorem :: FVSUM_1:37 --R = R; theorem :: FVSUM_1:38 -R1 = -R2 implies R1 = R2; theorem :: FVSUM_1:39 for K being Abelian right_zeroed add-associative right_complementable (non empty LoopStr), R,R1,R2 being Element of i-tuples_on the carrier of K holds R1 + R = R2 + R or R1 + R = R + R2 implies R1 = R2; theorem :: FVSUM_1:40 for K being Abelian right_zeroed add-associative right_complementable (non empty LoopStr), R1,R2 being Element of i-tuples_on the carrier of K holds -(R1 + R2) = -R1 + -R2; definition let K be non empty LoopStr; let p1,p2 be FinSequence of the carrier of K; func p1 - p2 -> FinSequence of the carrier of K equals :: FVSUM_1:def 5 (diffield(K)).:(p1,p2); end; reserve K for non empty LoopStr, a1,a2 for Element of K, p1,p2 for FinSequence of the carrier of K, R1,R2 for Element of i-tuples_on the carrier of K; canceled; theorem :: FVSUM_1:42 i in dom (p1-p2) & a1 = p1.i & a2 = p2.i implies (p1-p2).i = a1 - a2; definition let i;let K be non empty LoopStr; let R1,R2 be Element of i-tuples_on the carrier of K; redefine func R1 - R2 -> Element of i-tuples_on the carrier of K; end; theorem :: FVSUM_1:43 j in Seg i & a1 = R1.j & a2 = R2.j implies (R1-R2).j = a1 - a2; theorem :: FVSUM_1:44 <*>(the carrier of K) - p1 = <*>(the carrier of K) & p1 - <*>(the carrier of K) = <*>(the carrier of K); theorem :: FVSUM_1:45 <*a1*> - <*a2*> = <*a1-a2*>; theorem :: FVSUM_1:46 (i|->a1) - (i|->a2) = i|->(a1-a2); theorem :: FVSUM_1:47 R1 - R2 = R1 + - R2; theorem :: FVSUM_1:48 for K being add-associative right_complementable left_zeroed right_zeroed (non empty LoopStr), R being Element of i-tuples_on the carrier of K holds R - (i|->(0.K)) = R; theorem :: FVSUM_1:49 for K being Abelian left_zeroed right_zeroed (non empty LoopStr), R being Element of i-tuples_on the carrier of K holds (i|->(0.K)) - R = -R; theorem :: FVSUM_1:50 for K being left_zeroed right_zeroed add-associative right_complementable (non empty LoopStr), R1,R2 being Element of i-tuples_on the carrier of K holds R1 - -R2 = R1 + R2; reserve K for Abelian right_zeroed add-associative right_complementable (non empty LoopStr), R,R1,R2,R3 for Element of i-tuples_on the carrier of K; theorem :: FVSUM_1:51 -(R1 - R2) = R2 - R1; theorem :: FVSUM_1:52 -(R1 - R2) = -R1 + R2; theorem :: FVSUM_1:53 R - R = (i|->0.K); theorem :: FVSUM_1:54 R1 - R2 = (i|->0.K) implies R1 = R2; theorem :: FVSUM_1:55 R1 - R2 - R3 = R1 - (R2 + R3); theorem :: FVSUM_1:56 R1 + (R2 - R3) = R1 + R2 - R3; theorem :: FVSUM_1:57 R1 - (R2 - R3) = R1 - R2 + R3; theorem :: FVSUM_1:58 R1 = R1 + R - R; theorem :: FVSUM_1:59 R1 = R1 - R + R; reserve K for non empty HGrStr, a,a',a1,a2 for Element of K, p for FinSequence of the carrier of K, R for Element of i-tuples_on the carrier of K; theorem :: FVSUM_1:60 for a,b being Element of K holds ((the mult of K)[;](a,id the carrier of K)).b = a*b; theorem :: FVSUM_1:61 for a,b being Element of K holds (a multfield).b = a*b; definition let K be non empty HGrStr; let p be FinSequence of the carrier of K; let a be Element of K; func a*p -> FinSequence of the carrier of K equals :: FVSUM_1:def 6 (a multfield)*p; end; theorem :: FVSUM_1:62 i in dom (a*p) & a' = p.i implies (a*p).i = a*a'; definition let i;let K be non empty HGrStr; let R be Element of i-tuples_on the carrier of K; let a be Element of K; redefine func a*R -> Element of i-tuples_on the carrier of K; end; theorem :: FVSUM_1:63 j in Seg i & a' = R.j implies (a*R).j = a*a'; theorem :: FVSUM_1:64 a*(<*>(the carrier of K)) = <*>(the carrier of K); theorem :: FVSUM_1:65 a*<*a1*> = <*a*a1*>; theorem :: FVSUM_1:66 a1*(i|->a2) = i|->(a1*a2); theorem :: FVSUM_1:67 for K being associative (non empty HGrStr), a1,a2 being Element of K, R being Element of i-tuples_on the carrier of K holds (a1*a2)*R = a1*(a2*R); reserve K for distributive (non empty doubleLoopStr), a,a1,a2 for Element of K, R,R1,R2 for Element of i-tuples_on the carrier of K; theorem :: FVSUM_1:68 (a1 + a2)*R = a1*R + a2*R; theorem :: FVSUM_1:69 a*(R1+R2) = a*R1 + a*R2; theorem :: FVSUM_1:70 for K being distributive commutative left_unital (non empty doubleLoopStr), R being Element of i-tuples_on the carrier of K holds 1_ K * R = R; theorem :: FVSUM_1:71 for K being add-associative right_zeroed right_complementable distributive (non empty doubleLoopStr), R being Element of i-tuples_on the carrier of K holds 0.K*R = i|->0.K; theorem :: FVSUM_1:72 for K being add-associative right_zeroed right_complementable commutative left_unital distributive (non empty doubleLoopStr), R being Element of i-tuples_on the carrier of K holds (-1_ K) * R = -R; definition let M be non empty HGrStr, p1, p2 be FinSequence of the carrier of M; func mlt(p1,p2) -> FinSequence of the carrier of M equals :: FVSUM_1:def 7 (the mult of M).:(p1,p2); end; reserve K for non empty HGrStr, a1,a2,b1,b2 for Element of K, p,p1,p2 for FinSequence of the carrier of K, R1,R2 for Element of i-tuples_on the carrier of K; theorem :: FVSUM_1:73 i in dom mlt(p1,p2) & a1 = p1.i & a2 = p2.i implies mlt(p1,p2).i = a1 * a2; definition let i;let K be non empty HGrStr; let R1,R2 be Element of i-tuples_on the carrier of K; redefine func mlt(R1,R2) -> Element of i-tuples_on the carrier of K; end; theorem :: FVSUM_1:74 j in Seg i & a1 = R1.j & a2 = R2.j implies mlt(R1,R2).j = a1 * a2; theorem :: FVSUM_1:75 mlt(<*>(the carrier of K),p) = <*>(the carrier of K) & mlt(p,<*>(the carrier of K)) = <*>(the carrier of K); theorem :: FVSUM_1:76 mlt(<*a1*>,<*a2*>) = <*a1*a2*>; reserve K for commutative (non empty HGrStr), p,q for FinSequence of the carrier of K, R1,R2 for Element of i-tuples_on the carrier of K; theorem :: FVSUM_1:77 mlt(R1,R2) = mlt(R2,R1); theorem :: FVSUM_1:78 mlt(p,q)=mlt(q,p); theorem :: FVSUM_1:79 for K being associative (non empty HGrStr), R1,R2,R3 being Element of i-tuples_on the carrier of K holds mlt(R1,mlt(R2,R3)) = mlt(mlt(R1,R2),R3); reserve K for commutative associative (non empty HGrStr), a,a1,a2 for Element of K, R for Element of i-tuples_on the carrier of K; theorem :: FVSUM_1:80 mlt(i|->a,R) = a*R & mlt(R,i|->a) = a*R; theorem :: FVSUM_1:81 mlt(i|->a1,i|->a2) = i|->(a1*a2); theorem :: FVSUM_1:82 for K being associative (non empty HGrStr), a being Element of K, R1,R2 being Element of i-tuples_on the carrier of K holds a*mlt(R1,R2) = mlt(a*R1,R2); reserve K for commutative associative (non empty HGrStr), a for Element of K, R,R1,R2 for Element of i-tuples_on the carrier of K; theorem :: FVSUM_1:83 a*mlt(R1,R2) = mlt(a*R1,R2) & a*mlt(R1,R2) = mlt(R1,a*R2); theorem :: FVSUM_1:84 a*R = mlt(i|->a,R); begin :: ::The Sum of Finite Number of Elements :: definition cluster Abelian right_zeroed -> left_zeroed (non empty LoopStr); end; definition let K be Abelian add-associative right_zeroed right_complementable (non empty LoopStr); let p be FinSequence of the carrier of K; redefine func Sum(p) equals :: FVSUM_1:def 8 (the add of K) $$ p; end; reserve K for add-associative right_zeroed right_complementable (non empty LoopStr), a for Element of K, p for FinSequence of the carrier of K; canceled 2; theorem :: FVSUM_1:87 Sum(p^<*a*>) = Sum p + a; canceled; theorem :: FVSUM_1:89 Sum(<*a*>^p) = a + Sum p; canceled 2; theorem :: FVSUM_1:92 for K being Abelian add-associative right_zeroed right_complementable distributive(non empty doubleLoopStr), a being Element of K, p being FinSequence of the carrier of K holds Sum(a*p) = a*(Sum p); theorem :: FVSUM_1:93 for K being non empty LoopStr for R being Element of 0-tuples_on the carrier of K holds Sum R = 0.K; reserve K for Abelian add-associative right_zeroed right_complementable (non empty LoopStr), p for FinSequence of the carrier of K, R1,R2 for Element of i-tuples_on the carrier of K; theorem :: FVSUM_1:94 Sum -p = -(Sum p); theorem :: FVSUM_1:95 Sum(R1 + R2) = Sum R1 + Sum R2; theorem :: FVSUM_1:96 Sum(R1 - R2) = Sum R1 - Sum R2; begin :: ::The Product of Finite Number of Elements :: definition let K be non empty HGrStr;let p be FinSequence of the carrier of K; func Product(p) -> Element of K equals :: FVSUM_1:def 9 (the mult of K) $$ p; end; canceled; theorem :: FVSUM_1:98 :::multLoopStr for K being commutative left_unital (non empty doubleLoopStr) holds Product(<*> (the carrier of K)) = 1_ K; theorem :: FVSUM_1:99 for K being non empty HGrStr, a being Element of K holds Product <*a*> = a; theorem :: FVSUM_1:100 for K being commutative left_unital (non empty doubleLoopStr), a being Element of K, p being FinSequence of the carrier of K holds Product(p^<*a*>) = Product p * a; reserve K for commutative associative left_unital (non empty doubleLoopStr), a,a1,a2,a3 for Element of K, p1,p2 for FinSequence of the carrier of K, R1,R2 for Element of i-tuples_on the carrier of K; theorem :: FVSUM_1:101 Product(p1^p2) = Product p1 * Product p2; theorem :: FVSUM_1:102 Product(<*a*>^p1) = a * Product p1; theorem :: FVSUM_1:103 Product<*a1,a2*> = a1 * a2; theorem :: FVSUM_1:104 Product<*a1,a2,a3*> = a1 * a2 * a3; theorem :: FVSUM_1:105 for R being Element of 0-tuples_on the carrier of K holds Product R =1_ K; theorem :: FVSUM_1:106 Product(i|->(1_ K)) = 1_ K; theorem :: FVSUM_1:107 for K being add-associative right_zeroed right_complementable Abelian commutative associative left_unital distributive Field-like non degenerated (non empty doubleLoopStr) for p being FinSequence of the carrier of K holds (ex k st k in dom p & p.k = 0.K) iff Product p = 0.K; theorem :: FVSUM_1:108 Product((i+j)|->a) = (Product(i|->a))*(Product(j|->a)); theorem :: FVSUM_1:109 Product((i*j)|->a) = Product(j|->(Product(i|->a))); theorem :: FVSUM_1:110 Product(i|->(a1*a2)) = (Product(i|->a1))*(Product(i|->a2)); theorem :: FVSUM_1:111 Product mlt(R1,R2) = Product R1 * Product R2; theorem :: FVSUM_1:112 Product(a*R1) = Product (i|->a) * Product R1; begin :: ::The Product of Vectors :: definition let K be non empty doubleLoopStr; let p,q be FinSequence of the carrier of K; func p "*" q -> Element of K equals :: FVSUM_1:def 10 Sum(mlt(p,q)); end; theorem :: FVSUM_1:113 for K being commutative associative left_unital Abelian add-associative right_zeroed right_complementable (non empty doubleLoopStr) for a,b being Element of K holds <*a*> "*" <*b*>= a * b; theorem :: FVSUM_1:114 for K being commutative associative left_unital Abelian add-associative right_zeroed right_complementable (non empty doubleLoopStr) for a1,a2,b1,b2 being Element of K holds <*a1,a2*> "*" <*b1,b2*>= a1*b1 + a2*b2; theorem :: FVSUM_1:115 for p,q be FinSequence of the carrier of K holds p "*" q = q "*" p;

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