Volume 12, 2000

University of Bialystok

Copyright (c) 2000 Association of Mizar Users

### The abstract of the Mizar article:

### The Binomial Theorem for Algebraic Structures

**by****Christoph Schwarzweller**- Received November 20, 2000
- MML identifier: BINOM

- [ Mizar article, MML identifier index ]

environ vocabulary RLVECT_1, ALGSTR_1, VECTSP_2, ARYTM_1, BINOP_1, LATTICES, GROUP_1, VECTSP_1, ALGSTR_2, FUNCT_1, FUNCT_2, MCART_1, RELAT_1, FINSEQ_1, FINSEQ_4, NEWTON, FINSEQ_2, BOOLE, BINOM; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, XCMPLX_0, XREAL_0, STRUCT_0, FINSEQ_1, FUNCT_1, FUNCT_2, NAT_1, FINSEQ_4, RELSET_1, BINOP_1, ALGSTR_1, FINSEQ_2, VECTSP_1, VECTSP_2, GROUP_1, NEWTON, RLVECT_1, MCART_1, POLYNOM1; constructors BINOP_1, REAL_1, ALGSTR_2, NEWTON, DOMAIN_1, POLYNOM1, MONOID_0, MEMBERED; clusters STRUCT_0, ALGSTR_1, RELSET_1, FUNCT_2, MONOID_0, FINSEQ_2, VECTSP_2, INT_1, VECTSP_1, MEMBERED, ZFMISC_1, ORDINAL2, NUMBERS; requirements NUMERALS, SUBSET, REAL, BOOLE, ARITHM; begin :: Preliminaries :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: definition let L be non empty LoopStr; canceled 2; attr L is add-cancelable means :: BINOM:def 3 for a,b,c being Element of L holds (a + b = a + c implies b = c) & (b + a = c + a implies b = c); end; definition cluster add-left-cancelable (non empty LoopStr); cluster add-right-cancelable (non empty LoopStr); cluster add-cancelable (non empty LoopStr); end; definition cluster add-left-cancelable add-right-cancelable -> add-cancelable (non empty LoopStr); cluster add-cancelable -> add-left-cancelable add-right-cancelable (non empty LoopStr); end; definition cluster Abelian add-right-cancelable -> add-left-cancelable (non empty LoopStr); cluster Abelian add-left-cancelable -> add-right-cancelable (non empty LoopStr); end; definition cluster right_zeroed right_complementable add-associative -> add-right-cancelable (non empty LoopStr); end; definition cluster Abelian add-associative left_zeroed right_zeroed commutative associative add-cancelable distributive unital (non empty doubleLoopStr); end; theorem :: BINOM:1 for R being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), a being Element of R holds 0.R * a = 0.R; theorem :: BINOM:2 for R being left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), a being Element of R holds a * 0.R = 0.R; scheme Ind2{M() -> Nat, P[Nat]} : for i being Nat st M() <= i holds P[i] provided P[M()] and for j being Nat st M() <= j holds P[j] implies P[j+1]; scheme RecDef1 {C,D() -> non empty set, b() -> Element of D(), F() -> Function of [:C(),D():],D()} : ex g being Function of [:NAT,C():],D() st for a being Element of C() holds g.(0,a) = b() & for n being Nat holds g.(n+1,a) = F().(a,g.(n,a)); scheme RecDef2 {C,D() -> non empty set, b() -> Element of D(), F() -> Function of [:D(),C():],D()} : ex g being Function of [:C(),NAT:],D() st for a being Element of C() holds g.(a,0) = b() & for n being Nat holds g.(a,n+1) = F().(g.(a,n),a); begin :: On Finite Sequences :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: BINOM:3 for L being left_zeroed (non empty LoopStr), a being Element of L holds Sum <* a *> = a; theorem :: BINOM:4 for R being left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), a being Element of R, p being FinSequence of the carrier of R holds Sum(a * p) = a * Sum p; theorem :: BINOM:5 for R being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), a being Element of R, p being FinSequence of the carrier of R holds Sum(p * a) = Sum p * a; theorem :: BINOM:6 for R being commutative (non empty doubleLoopStr), a being Element of R, p being FinSequence of the carrier of R holds Sum(p * a) = Sum(a * p); definition let R be non empty LoopStr, p,q be FinSequence of the carrier of R such that dom p = dom q; func p + q -> FinSequence of the carrier of R means :: BINOM:def 4 dom it = dom p & for i being Nat st 1 <= i & i <= len it holds it/.i = p/.i + q/.i; end; theorem :: BINOM:7 for R being Abelian right_zeroed add-associative (non empty LoopStr), p,q being FinSequence of the carrier of R st dom p = dom q holds Sum(p + q) = Sum p + Sum q; begin :: On Powers in Rings :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: definition let R be unital (non empty HGrStr), a be Element of R, n be Nat; func a|^n -> Element of R equals :: BINOM:def 5 power(R).(a,n); end; theorem :: BINOM:8 for R being unital (non empty HGrStr), a being Element of R holds a|^0 = 1.R & a|^1 = a; theorem :: BINOM:9 for R being unital (non empty HGrStr), a being Element of R, n being Nat holds a|^(n+1) = (a|^n) * a; theorem :: BINOM:10 for R being unital associative commutative (non empty HGrStr), a,b being Element of R, n being Nat holds (a * b)|^n = (a|^n) * (b|^n); theorem :: BINOM:11 for R being unital associative (non empty HGrStr), a being Element of R, n,m being Nat holds a|^(n+m) = (a|^n) * (a|^m); theorem :: BINOM:12 for R being unital associative (non empty HGrStr), a being Element of R, n,m being Nat holds (a|^n)|^m = a|^(n * m); begin :: On Natural Products in Rings ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: definition let R be non empty LoopStr; func Nat-mult-left(R) -> Function of [:NAT,the carrier of R:],the carrier of R means :: BINOM:def 6 for a being Element of R holds it.(0,a) = 0.R & for n being Nat holds it.(n+1,a) = a + it.(n,a); func Nat-mult-right(R) -> Function of [:the carrier of R,NAT:],the carrier of R means :: BINOM:def 7 for a being Element of R holds it.(a,0) = 0.R & for n being Nat holds it.(a,n+1) = it.(a,n) + a; end; definition let R be non empty LoopStr, a be Element of R, n be Nat; func n * a -> Element of R equals :: BINOM:def 8 (Nat-mult-left(R)).(n,a); func a * n -> Element of R equals :: BINOM:def 9 (Nat-mult-right(R)).(a,n); end; theorem :: BINOM:13 for R being non empty LoopStr, a being Element of R holds 0 * a = 0.R & a * 0 = 0.R; theorem :: BINOM:14 for R being right_zeroed (non empty LoopStr), a being Element of R holds 1 * a = a; theorem :: BINOM:15 for R being left_zeroed (non empty LoopStr), a being Element of R holds a * 1 = a; theorem :: BINOM:16 for R being left_zeroed add-associative (non empty LoopStr), a being Element of R, n,m being Nat holds (n + m) * a = n * a + m * a; theorem :: BINOM:17 for R being right_zeroed add-associative (non empty LoopStr), a being Element of R, n,m being Nat holds a * (n + m) = a * n + a * m; theorem :: BINOM:18 for R being left_zeroed right_zeroed add-associative (non empty LoopStr), a being Element of R, n being Nat holds n * a = a * n; theorem :: BINOM:19 for R being Abelian (non empty LoopStr), a being Element of R, n being Nat holds n * a = a * n; theorem :: BINOM:20 for R being left_zeroed right_zeroed add-left-cancelable add-associative left-distributive (non empty doubleLoopStr), a,b being Element of R, n being Nat holds (n * a) * b = n * (a * b); theorem :: BINOM:21 for R being left_zeroed right_zeroed add-right-cancelable add-associative distributive (non empty doubleLoopStr), a,b being Element of R, n being Nat holds b * (n * a) = (b * a) * n; theorem :: BINOM:22 for R being left_zeroed right_zeroed add-associative add-cancelable distributive (non empty doubleLoopStr), a,b being Element of R, n being Nat holds (a * n) * b = a * (n * b); begin :: The Binomial Theorem ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: definition let k,n be Nat; redefine func n choose k -> Nat; end; definition let R be unital (non empty doubleLoopStr), a,b be Element of R, n be Nat; func (a,b) In_Power n -> FinSequence of the carrier of R means :: BINOM:def 10 len it = n + 1 & for i,l,m being Nat st i in dom it & m = i - 1 & l = n - m holds it/.i = (n choose m) * a|^l * b|^m; end; theorem :: BINOM:23 for R being right_zeroed unital (non empty doubleLoopStr), a,b being Element of R holds (a,b) In_Power 0 = <*1.R*>; theorem :: BINOM:24 for R being right_zeroed unital (non empty doubleLoopStr), a,b being Element of R, n being Nat holds ((a,b) In_Power n).1 = a|^n; theorem :: BINOM:25 for R being right_zeroed unital (non empty doubleLoopStr), a,b being Element of R, n being Nat holds ((a,b) In_Power n).(n+1) = b|^n; theorem :: BINOM:26 for R being Abelian add-associative left_zeroed right_zeroed commutative associative add-cancelable distributive unital (non empty doubleLoopStr), a,b being Element of R, n being Nat holds (a+b)|^n = Sum((a,b) In_Power n);

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