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Theorem List for Intuitionistic Logic Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclimre 11301* Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( Re
 `  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( Re `  A ) )
 
Theoremclimim 11302* Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( Im
 `  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( Im `  A ) )
 
Theoremclimrecl 11303* The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremclimge0 11304* A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  ( F `
  k ) )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremclimadd 11305* Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  +  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  +  B ) )
 
Theoremclimmul 11306* Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  x.  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
 
Theoremclimsub 11307* Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  -  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  -  B ) )
 
Theoremclimaddc1 11308* Limit of a constant  C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `  k )  +  C ) )   =>    |-  ( ph  ->  G  ~~>  ( A  +  C ) )
 
Theoremclimaddc2 11309* Limit of a constant  C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  +  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  +  A ) )
 
Theoremclimmulc2 11310* Limit of a sequence multiplied by a constant  C. Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  x.  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  x.  A ) )
 
Theoremclimsubc1 11311* Limit of a constant  C subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `  k )  -  C ) )   =>    |-  ( ph  ->  G  ~~>  ( A  -  C ) )
 
Theoremclimsubc2 11312* Limit of a constant  C minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  -  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  -  A ) )
 
Theoremclimle 11313* Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremclimsqz 11314* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  <_  A )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremclimsqz2 11315* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  <_  ( F `  k ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  <_  ( G `  k ) )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremclim2ser 11316* The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ph  ->  seq
 M (  +  ,  F )  ~~>  A )   =>    |-  ( ph  ->  seq ( N  +  1 ) (  +  ,  F )  ~~>  ( A  -  (  seq M (  +  ,  F ) `  N ) ) )
 
Theoremclim2ser2 11317* The limit of an infinite series with an initial segment added. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ph  ->  seq ( N  +  1 ) (  +  ,  F )  ~~>  A )   =>    |-  ( ph  ->  seq
 M (  +  ,  F )  ~~>  ( A  +  (  seq M (  +  ,  F ) `  N ) ) )
 
Theoremiserex 11318* An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) )
 
Theoremisermulc2 11319* Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Jim Kingdon, 8-Apr-2023.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  seq M (  +  ,  F ) 
 ~~>  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  x.  ( F `  k ) ) )   =>    |-  ( ph  ->  seq M (  +  ,  G ) 
 ~~>  ( C  x.  A ) )
 
Theoremclimlec2 11320* Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  F  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  A  <_  ( F `  k
 ) )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremiserle 11321* Comparison of the limits of two infinite series. (Contributed by Paul Chapman, 12-Nov-2007.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  seq M (  +  ,  F )  ~~>  A )   &    |-  ( ph  ->  seq M (  +  ,  G )  ~~>  B )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremiserge0 11322* The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  seq M (  +  ,  F )  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  ( F `
  k ) )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremclimub 11323* The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   =>    |-  ( ph  ->  ( F `  N ) 
 <_  A )
 
Theoremclimserle 11324* The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  seq M (  +  ,  F )  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  ( F `
  k ) )   =>    |-  ( ph  ->  (  seq M (  +  ,  F ) `  N )  <_  A )
 
Theoremiser3shft 11325* Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Jim Kingdon, 17-Oct-2022.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F )  ~~>  A  <->  seq ( M  +  N ) (  .+  ,  ( F  shift  N ) )  ~~>  A ) )
 
Theoremclimcau 11326* A converging sequence of complex numbers is a Cauchy sequence. The converse would require excluded middle or a different definition of Cauchy sequence (for example, fixing a rate of convergence as in climcvg1n 11329). Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  dom  ~~>  ) 
 ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x )
 
Theoremclimrecvg1n 11327* A Cauchy sequence of real numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within  C  /  n of the nth term, where  C is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( abs `  ( ( F `  k )  -  ( F `  n ) ) )  <  ( C  /  n ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremclimcvg1nlem 11328* Lemma for climcvg1n 11329. We construct sequences of the real and imaginary parts of each term of  F, show those converge, and use that to show that  F converges. (Contributed by Jim Kingdon, 24-Aug-2021.)
 |-  ( ph  ->  F : NN --> CC )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( abs `  ( ( F `  k )  -  ( F `  n ) ) )  <  ( C  /  n ) )   &    |-  G  =  ( x  e.  NN  |->  ( Re `  ( F `  x ) ) )   &    |-  H  =  ( x  e.  NN  |->  ( Im `  ( F `
  x ) ) )   &    |-  J  =  ( x  e.  NN  |->  ( _i  x.  ( H `
  x ) ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremclimcvg1n 11329* A Cauchy sequence of complex numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within  C  /  n of the nth term, where  C is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
 |-  ( ph  ->  F : NN --> CC )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( abs `  ( ( F `  k )  -  ( F `  n ) ) )  <  ( C  /  n ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremclimcaucn 11330* A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 11326 but adds the part that  ( F `  k ) is complex. (Contributed by Jim Kingdon, 24-Aug-2021.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  dom  ~~>  ) 
 ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  ( abs `  ( ( F `
  k )  -  ( F `  j ) ) )  <  x ) )
 
Theoremserf0 11331* If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  seq M (  +  ,  F )  e.  dom  ~~>  )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  F  ~~>  0 )
 
4.8.2  Finite and infinite sums
 
Syntaxcsu 11332 Extend class notation to include finite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is a commonly used word in comments.)
 class  sum_ k  e.  A  B
 
Definitiondf-sumdc 11333* Define the sum of a series with an index set of integers  A. The variable  k is normally a free variable in  B, i.e.,  B can be thought of as  B ( k ). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. In both cases we have an  if expression so that we only need  B to be defined where  k  e.  A. In the infinite case, we also require that the indexing set be a decidable subset of an upperset of integers (that is, membership of integers in it is decidable). These two methods of summation produce the same result on their common region of definition (i.e., finite sets of integers). Examples:  sum_ k  e. 
{ 1 ,  2 ,  4 } k means  1  +  2  +  4  =  7, and  sum_ k  e.  NN ( 1  / 
( 2 ^ k
) )  =  1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 11501). (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 21-May-2023.)
 |- 
 sum_ k  e.  A  B  =  ( iota x ( E. m  e. 
 ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>=
 `  m )DECID  j  e.  A  /\  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
 ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m )
 -1-1-onto-> A  /\  x  =  ( 
 seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ) `  m ) ) ) )
 
Theoremsumeq1 11334 Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  ( A  =  B  -> 
 sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremnfsum1 11335 Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  F/_ k A   =>    |-  F/_ k sum_ k  e.  A  B
 
Theoremnfsum 11336 Bound-variable hypothesis builder for sum: if  x is (effectively) not free in  A and  B, it is not free in  sum_ k  e.  A B. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x sum_ k  e.  A  B
 
Theoremsumdc 11337* Decidability of a subset of upper integers. (Contributed by Jim Kingdon, 1-Jan-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A 
 C_  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A. x  e.  ( ZZ>= `  M )DECID  x  e.  A )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  -> DECID  N  e.  A )
 
Theoremsumeq2 11338* Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( A. k  e.  A  B  =  C  -> 
 sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremcbvsum 11339 Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  ( j  =  k 
 ->  B  =  C )   &    |-  F/_ k A   &    |-  F/_ j A   &    |-  F/_ k B   &    |-  F/_ j C   =>    |- 
 sum_ j  e.  A  B  =  sum_ k  e.  A  C
 
Theoremcbvsumv 11340* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( j  =  k 
 ->  B  =  C )   =>    |-  sum_
 j  e.  A  B  =  sum_ k  e.  A  C
 
Theoremcbvsumi 11341* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
 |-  F/_ k B   &    |-  F/_ j C   &    |-  (
 j  =  k  ->  B  =  C )   =>    |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
 
Theoremsumeq1i 11342* Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
 |-  A  =  B   =>    |-  sum_ k  e.  A  C  =  sum_ k  e.  B  C
 
Theoremsumeq2i 11343* Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
 |-  ( k  e.  A  ->  B  =  C )   =>    |-  sum_
 k  e.  A  B  =  sum_ k  e.  A  C
 
Theoremsumeq12i 11344* Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
 |-  A  =  B   &    |-  (
 k  e.  A  ->  C  =  D )   =>    |-  sum_ k  e.  A  C  =  sum_ k  e.  B  D
 
Theoremsumeq1d 11345* Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremsumeq2d 11346* Equality deduction for sum. Note that unlike sumeq2dv 11347, 
k may occur in  ph. (Contributed by NM, 1-Nov-2005.)
 |-  ( ph  ->  A. k  e.  A  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2dv 11347* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2ad 11348* Equality deduction for sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2sdv 11349* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theorem2sumeq2dv 11350* Equality deduction for double sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( ph  /\  j  e.  A  /\  k  e.  B )  ->  C  =  D )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_ k  e.  B  C  =  sum_ j  e.  A  sum_ k  e.  B  D )
 
Theoremsumeq12dv 11351* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  =  D )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  D )
 
Theoremsumeq12rdv 11352* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  B ) 
 ->  C  =  D )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  D )
 
Theoremsumfct 11353* A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 18-Sep-2022.)
 |-  ( A. k  e.  A  B  e.  CC  -> 
 sum_ j  e.  A  ( ( k  e.  A  |->  B ) `  j )  =  sum_ k  e.  A  B )
 
Theoremfz1f1o 11354* A lemma for working with finite sums. (Contributed by Mario Carneiro, 22-Apr-2014.)
 |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
 ( `  A )  e. 
 NN  /\  E. f  f : ( 1 ... ( `  A )
 )
 -1-1-onto-> A ) ) )
 
Theoremnnf1o 11355 Lemma for sum and product theorems. (Contributed by Jim Kingdon, 15-Aug-2022.)
 |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN ) )   &    |-  ( ph  ->  F : ( 1 ... M ) -1-1-onto-> A )   &    |-  ( ph  ->  G : ( 1 ...
 N ) -1-1-onto-> A )   =>    |-  ( ph  ->  N  =  M )
 
Theoremsumrbdclem 11356* Lemma for sumrbdc 11358. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> DECID  k  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   =>    |-  ( ( ph  /\  A  C_  ( ZZ>= `  N )
 )  ->  (  seq M (  +  ,  F )  |`  ( ZZ>= `  N ) )  =  seq N (  +  ,  F ) )
 
Theoremfsum3cvg 11357* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 12-Nov-2022.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> DECID  k  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A 
 C_  ( M ... N ) )   =>    |-  ( ph  ->  seq M (  +  ,  F ) 
 ~~>  (  seq M (  +  ,  F ) `
  N ) )
 
Theoremsumrbdc 11358* Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  A  C_  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  A  C_  ( ZZ>= `  N )
 )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  -> DECID  k  e.  A )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  N )
 )  -> DECID  k  e.  A )   =>    |-  ( ph  ->  (  seq M (  +  ,  F )  ~~>  C  <->  seq N (  +  ,  F )  ~~>  C )
 )
 
Theoremsummodclem3 11359* Lemma for summodc 11362. (Contributed by Mario Carneiro, 29-Mar-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN )
 )   &    |-  ( ph  ->  f : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ph  ->  K : ( 1 ...
 N ) -1-1-onto-> A )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  M ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   &    |-  H  =  ( n  e.  NN  |->  if ( n  <_  N ,  [_ ( K `  n )  /  k ]_ B ,  0 ) )   =>    |-  ( ph  ->  (  seq 1 (  +  ,  G ) `  M )  =  (  seq 1 (  +  ,  H ) `  N ) )
 
Theoremsummodclem2a 11360* Lemma for summodc 11362. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> DECID  k  e.  A )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   &    |-  H  =  ( n  e.  NN  |->  if ( n  <_  N ,  [_ ( K `
  n )  /  k ]_ B ,  0 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  C_  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  f : ( 1 ...
 N ) -1-1-onto-> A )   &    |-  ( ph  ->  K 
 Isom  <  ,  <  (
 ( 1 ... ( `  A ) ) ,  A ) )   =>    |-  ( ph  ->  seq
 M (  +  ,  F )  ~~>  (  seq 1
 (  +  ,  G ) `  N ) )
 
Theoremsummodclem2 11361* Lemma for summodc 11362. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   =>    |-  ( ( ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  ,  F ) 
 ~~>  x ) )  ->  ( E. m  e.  NN  E. f ( f : ( 1 ... m )
 -1-1-onto-> A  /\  y  =  ( 
 seq 1 (  +  ,  G ) `  m ) )  ->  x  =  y ) )
 
Theoremsummodc 11362* A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  ( `  A ) , 
 [_ ( f `  n )  /  k ]_ B ,  0 ) )   =>    |-  ( ph  ->  E* x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  ,  F ) 
 ~~>  x )  \/  E. m  e.  NN  E. f
 ( f : ( 1 ... m ) -1-1-onto-> A 
 /\  x  =  ( 
 seq 1 (  +  ,  G ) `  m ) ) ) )
 
Theoremzsumdc 11363* Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.) (Revised by Jim Kingdon, 8-Apr-2023.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
 0 ) )   &    |-  ( ph  ->  A. x  e.  Z DECID  x  e.  A )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  (  ~~>  `  seq M (  +  ,  F ) ) )
 
Theoremisum 11364* Series sum with an upper integer index set (i.e. an infinite series). (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 7-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  Z  B  =  (  ~~>  `  seq M (  +  ,  F ) ) )
 
Theoremfsumgcl 11365* Closure for a function used to describe a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
 |-  ( k  =  ( F `  n ) 
 ->  B  =  C )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  n  e.  ( 1 ... M ) )  ->  ( G `
  n )  =  C )   =>    |-  ( ph  ->  A. n  e.  ( 1 ... M ) ( G `  n )  e.  CC )
 
Theoremfsum3 11366* The value of a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
 |-  ( k  =  ( F `  n ) 
 ->  B  =  C )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  n  e.  ( 1 ... M ) )  ->  ( G `
  n )  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  ( 
 seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  M ,  ( G `  n ) ,  0 )
 ) ) `  M ) )
 
Theoremsum0 11367 Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
 |- 
 sum_ k  e.  (/)  A  =  0
 
Theoremisumz 11368* Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
 |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>=
 `  M )DECID  j  e.  A )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  =  0 )
 
Theoremfsumf1o 11369* Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.)
 |-  ( k  =  G  ->  B  =  D )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  ( ph  ->  F : C -1-1-onto-> A )   &    |-  ( ( ph  /\  n  e.  C ) 
 ->  ( F `  n )  =  G )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ n  e.  C  D )
 
Theoremisumss 11370* Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 21-Sep-2022.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( B  \  A ) )  ->  C  =  0 )   &    |-  ( ph  ->  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  B 
 C_  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A. j  e.  ( ZZ>= `  M )DECID  j  e.  B )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremfisumss 11371* Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 23-Sep-2022.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( B  \  A ) )  ->  C  =  0 )   &    |-  ( ph  ->  A. j  e.  B DECID  j  e.  A )   &    |-  ( ph  ->  B  e.  Fin )   =>    |-  ( ph  ->  sum_
 k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremisumss2 11372* Change the index set of a sum by adding zeroes. The nonzero elements are in the contained set  A and the added zeroes compose the rest of the containing set  B which needs to be summable. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Jim Kingdon, 24-Sep-2022.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  A. j  e.  B DECID  j  e.  A )   &    |-  ( ph  ->  A. k  e.  A  C  e.  CC )   &    |-  ( ph  ->  ( ( M  e.  ZZ  /\  B  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  B )  \/  B  e.  Fin ) )   =>    |-  ( ph  ->  sum_
 k  e.  A  C  =  sum_ k  e.  B  if ( k  e.  A ,  C ,  0 ) )
 
Theoremfsum3cvg2 11373* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
 |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  M )
 )  -> DECID  k  e.  A )   &    |-  ( ph  ->  A  C_  ( M ... N ) )   =>    |-  ( ph  ->  seq M (  +  ,  F ) 
 ~~>  (  seq M (  +  ,  F ) `
  N ) )
 
Theoremfsumsersdc 11374* Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Jim Kingdon, 5-May-2023.)
 |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  M )
 )  -> DECID  k  e.  A )   &    |-  ( ph  ->  A  C_  ( M ... N ) )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  ( 
 seq M (  +  ,  F ) `  N ) )
 
Theoremfsum3cvg3 11375* A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> DECID  k  e.  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  seq M (  +  ,  F )  e.  dom  ~~>  )
 
Theoremfsum3ser 11376* A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 11391 and fsump1 11399, which should make our notation clear and from which, along with closure fsumcl 11379, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 1-Oct-2022.)
 |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  =  A )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  ( M ... N ) A  =  (  seq M (  +  ,  F ) `  N ) )
 
Theoremfsumcl2lem 11377* - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by Mario Carneiro, 3-Jun-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  +  y )  e.  S )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  S )   &    |-  ( ph  ->  A  =/=  (/) )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  S )
 
Theoremfsumcllem 11378* - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 3-Jun-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  +  y )  e.  S )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  S )   &    |-  ( ph  ->  0  e.  S )   =>    |-  ( ph  ->  sum_
 k  e.  A  B  e.  S )
 
Theoremfsumcl 11379* Closure of a finite sum of complex numbers  A ( k ). (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  CC )
 
Theoremfsumrecl 11380* Closure of a finite sum of reals. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  RR )
 
Theoremfsumzcl 11381* Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  ZZ )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  ZZ )
 
Theoremfsumnn0cl 11382* Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  NN0 )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  NN0 )
 
Theoremfsumrpcl 11383* Closure of a finite sum of positive reals. (Contributed by Mario Carneiro, 3-Jun-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR+ )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  RR+ )
 
Theoremfsumzcl2 11384* A finite sum with integer summands is an integer. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
 |-  ( ( A  e.  Fin  /\  A. k  e.  A  B  e.  ZZ )  -> 
 sum_ k  e.  A  B  e.  ZZ )
 
Theoremfsumadd 11385* The sum of two finite sums. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  ( B  +  C )  =  ( sum_ k  e.  A  B  +  sum_ k  e.  A  C ) )
 
Theoremfsumsplit 11386* Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.)
 |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  U  =  ( A  u.  B ) )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ( ph  /\  k  e.  U )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  U  C  =  (
 sum_ k  e.  A  C  +  sum_ k  e.  B  C ) )
 
Theoremfsumsplitf 11387* Split a sum into two parts. A version of fsumsplit 11386 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  U  =  ( A  u.  B ) )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ( ph  /\  k  e.  U )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  U  C  =  (
 sum_ k  e.  A  C  +  sum_ k  e.  B  C ) )
 
Theoremsumsnf 11388* A sum of a singleton is the term. A version of sumsn 11390 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/_ k B   &    |-  ( k  =  M  ->  A  =  B )   =>    |-  ( ( M  e.  V  /\  B  e.  CC )  ->  sum_ k  e.  { M } A  =  B )
 
Theoremfsumsplitsn 11389* Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  F/_ k D   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  -.  B  e.  A )   &    |-  ( ( ph  /\  k  e.  A )  ->  C  e.  CC )   &    |-  ( k  =  B  ->  C  =  D )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  sum_
 k  e.  ( A  u.  { B }
 ) C  =  (
 sum_ k  e.  A  C  +  D )
 )
 
Theoremsumsn 11390* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
 |-  ( k  =  M  ->  A  =  B )   =>    |-  ( ( M  e.  V  /\  B  e.  CC )  ->  sum_ k  e.  { M } A  =  B )
 
Theoremfsum1 11391* The finite sum of  A ( k ) from  k  =  M to  M (i.e. a sum with only one term) is  B i.e.  A ( M ). (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
 |-  ( k  =  M  ->  A  =  B )   =>    |-  ( ( M  e.  ZZ  /\  B  e.  CC )  ->  sum_ k  e.  ( M ... M ) A  =  B )
 
Theoremsumpr 11392* A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  ( k  =  A  ->  C  =  D )   &    |-  ( k  =  B  ->  C  =  E )   &    |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC ) )   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  W )
 )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  sum_ k  e.  { A ,  B } C  =  ( D  +  E )
 )
 
Theoremsumtp 11393* A sum over a triple is the sum of the elements. (Contributed by AV, 24-Jul-2020.)
 |-  ( k  =  A  ->  D  =  E )   &    |-  ( k  =  B  ->  D  =  F )   &    |-  ( k  =  C  ->  D  =  G )   &    |-  ( ph  ->  ( E  e.  CC  /\  F  e.  CC  /\  G  e.  CC ) )   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  W  /\  C  e.  X )
 )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  A  =/=  C )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  sum_ k  e.  { A ,  B ,  C } D  =  ( ( E  +  F )  +  G ) )
 
Theoremsumsns 11394* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
 |-  ( ( M  e.  V  /\  [_ M  /  k ]_ A  e.  CC )  ->  sum_ k  e.  { M } A  =  [_ M  /  k ]_ A )
 
Theoremfsumm1 11395* Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( k  =  N  ->  A  =  B )   =>    |-  ( ph  ->  sum_ k  e.  ( M ... N ) A  =  ( sum_ k  e.  ( M
 ... ( N  -  1 ) ) A  +  B ) )
 
Theoremfzosump1 11396* Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( k  =  N  ->  A  =  B )   =>    |-  ( ph  ->  sum_ k  e.  ( M..^ ( N  +  1 ) ) A  =  ( sum_ k  e.  ( M..^ N ) A  +  B ) )
 
Theoremfsum1p 11397* Separate out the first term in a finite sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( k  =  M  ->  A  =  B )   =>    |-  ( ph  ->  sum_ k  e.  ( M ... N ) A  =  ( B  +  sum_ k  e.  ( ( M  +  1 ) ... N ) A ) )
 
Theoremfsumsplitsnun 11398* Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 17-Dec-2021.)
 |-  ( ( A  e.  Fin  /\  ( Z  e.  V  /\  Z  e/  A ) 
 /\  A. k  e.  ( A  u.  { Z }
 ) B  e.  ZZ )  ->  sum_ k  e.  ( A  u.  { Z }
 ) B  =  (
 sum_ k  e.  A  B  +  [_ Z  /  k ]_ B ) )
 
Theoremfsump1 11399* The addition of the next term in a finite sum of  A ( k ) is the current term plus  B i.e.  A ( N  +  1 ). (Contributed by NM, 4-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... ( N  +  1 )
 ) )  ->  A  e.  CC )   &    |-  ( k  =  ( N  +  1 )  ->  A  =  B )   =>    |-  ( ph  ->  sum_ k  e.  ( M ... ( N  +  1 )
 ) A  =  (
 sum_ k  e.  ( M ... N ) A  +  B ) )
 
Theoremisumclim 11400* An infinite sum equals the value its series converges to. (Contributed by NM, 25-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  seq M (  +  ,  F )  ~~>  B )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  =  B )
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