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Theorem List for Metamath Proof Explorer - 12401-12500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremo1le 12401* Transfer eventual boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 25-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  ( abs `  C )  <_  ( abs `  B ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )
 
Theoremrlimno1 12402* A function whose inverse converges to zero is unbounded. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  sup ( A ,  RR* ,  <  )  =  +oo )   &    |-  ( ph  ->  ( x  e.  A  |->  ( 1  /  B ) )  ~~> r  0 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  =/=  0 )   =>    |-  ( ph  ->  -.  ( x  e.  A  |->  B )  e.  O ( 1 ) )
 
Theoremclim2ser 12403* 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 12404* 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 12405* 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 12406* Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  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 12407* 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 12408* 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 12409* 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 12410* 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 12411* 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 )
 
Theoremisershft 12412 Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  (  seq  M (  .+  ,  F )  ~~>  A  <->  seq  ( M  +  N ) (  .+  ,  ( F  shift  N ) )  ~~>  A ) )
 
Theoremisercolllem1 12413* Lemma for isercoll 12416. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : NN --> Z )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( G `  k
 )  <  ( G `  ( k  +  1 ) ) )   =>    |-  ( ( ph  /\  S  C_  NN )  ->  ( G  |`  S ) 
 Isom  <  ,  <  ( S ,  ( G " S ) ) )
 
Theoremisercolllem2 12414* Lemma for isercoll 12416. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : NN --> Z )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( G `  k
 )  <  ( G `  ( k  +  1 ) ) )   =>    |-  ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  (
 1 ... ( # `  ( G " ( `' G " ( M ... N ) ) ) ) )  =  ( `' G " ( M
 ... N ) ) )
 
Theoremisercolllem3 12415* Lemma for isercoll 12416. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : NN --> Z )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( G `  k
 )  <  ( G `  ( k  +  1 ) ) )   &    |-  (
 ( ph  /\  n  e.  ( Z  \  ran  G ) )  ->  ( F `  n )  =  0 )   &    |-  ( ( ph  /\  n  e.  Z ) 
 ->  ( F `  n )  e.  CC )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( H `  k )  =  ( F `  ( G `  k ) ) )   =>    |-  ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) ) 
 ->  (  seq  M (  +  ,  F ) `
  N )  =  (  seq  1 (  +  ,  H ) `
  ( # `  ( G " ( `' G " ( M ... N ) ) ) ) ) )
 
Theoremisercoll 12416* Rearrange an infinite series by spacing out the terms using an order isomorphism. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : NN --> Z )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( G `  k
 )  <  ( G `  ( k  +  1 ) ) )   &    |-  (
 ( ph  /\  n  e.  ( Z  \  ran  G ) )  ->  ( F `  n )  =  0 )   &    |-  ( ( ph  /\  n  e.  Z ) 
 ->  ( F `  n )  e.  CC )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( H `  k )  =  ( F `  ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq  1 (  +  ,  H )  ~~>  A  <->  seq  M (  +  ,  F )  ~~>  A )
 )
 
Theoremisercoll2 12417* Generalize isercoll 12416 so that both sequences have arbitrary starting point. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  G : Z --> W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( G `  k )  < 
 ( G `  (
 k  +  1 ) ) )   &    |-  ( ( ph  /\  n  e.  ( W 
 \  ran  G )
 )  ->  ( F `  n )  =  0 )   &    |-  ( ( ph  /\  n  e.  W ) 
 ->  ( F `  n )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( F `  ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq  M (  +  ,  H )  ~~>  A  <->  seq  N (  +  ,  F )  ~~>  A )
 )
 
Theoremclimsup 12418* A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   &    |-  ( ph  ->  E. x  e.  RR  A. k  e.  Z  ( F `  k ) 
 <_  x )   =>    |-  ( ph  ->  F  ~~>  sup ( ran  F ,  RR ,  <  ) )
 
Theoremclimcau 12419* A converging sequence of complex numbers is a Cauchy sequence. 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 )
 
Theoremclimbdd 12420* A converging sequence of complex numbers is bounded. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  dom  ~~>  /\ 
 A. k  e.  Z  ( F `  k )  e.  CC )  ->  E. x  e.  RR  A. k  e.  Z  ( abs `  ( F `  k ) )  <_  x )
 
Theoremcaucvgrlem 12421* Lemma for caurcvgr 12422. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  sup ( A ,  RR*
 ,  <  )  =  +oo )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x ) )   &    |-  ( ph  ->  R  e.  RR+ )   =>    |-  ( ph  ->  E. j  e.  A  ( ( limsup `  F )  e.  RR  /\ 
 A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F )
 ) )  <  (
 3  x.  R ) ) ) )
 
Theoremcaurcvgr 12422* A Cauchy sequence of real numbers converges to its limit supremum. The third hypothesis specifies that  F is a Cauchy sequence. (Contributed by Mario Carneiro, 7-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  sup ( A ,  RR*
 ,  <  )  =  +oo )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x ) )   =>    |-  ( ph  ->  F  ~~> r  ( limsup `  F )
 )
 
Theoremcaucvgrlem2 12423* Lemma for caucvgr 12424. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Mario Carneiro, 8-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  sup ( A ,  RR*
 ,  <  )  =  +oo )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x ) )   &    |-  H : CC --> RR   &    |-  ( ( ( F `
  k )  e. 
 CC  /\  ( F `  j )  e.  CC )  ->  ( abs `  (
 ( H `  ( F `  k ) )  -  ( H `  ( F `  j ) ) ) )  <_  ( abs `  ( ( F `  k )  -  ( F `  j ) ) ) )   =>    |-  ( ph  ->  ( n  e.  A  |->  ( H `  ( F `
  n ) ) )  ~~> r  (  ~~> r  `  ( H  o.  F ) ) )
 
Theoremcaucvgr 12424* A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 8-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  sup ( A ,  RR*
 ,  <  )  =  +oo )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x ) )   =>    |-  ( ph  ->  F  e.  dom  ~~> r  )
 
Theoremcaurcvg 12425* A Cauchy sequence of real numbers converges to its limit supremum. The fourth hypothesis specifies that  F is a Cauchy sequence. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 8-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F : Z --> RR )   &    |-  ( ph  ->  A. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x )   =>    |-  ( ph  ->  F  ~~>  ( limsup `  F ) )
 
Theoremcaurcvg2 12426* A Cauchy sequence of real numbers converges, existence version. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 7-Sep-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  RR  /\  ( abs `  ( ( F `
  k )  -  ( F `  j ) ) )  <  x ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremcaucvg 12427* A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Proof shortened by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x )   &    |-  ( ph  ->  F  e.  V )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremcaucvgb 12428* A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Mario Carneiro, 15-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( 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 12429* 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 )
 
Theoremiseraltlem1 12430* Lemma for iseralt 12433. A decreasing sequence with limit zero consists of positive terms. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : Z --> RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  (
 k  +  1 ) )  <_  ( G `  k ) )   &    |-  ( ph  ->  G  ~~>  0 )   =>    |-  (
 ( ph  /\  N  e.  Z )  ->  0  <_  ( G `  N ) )
 
Theoremiseraltlem2 12431* Lemma for iseralt 12433. The terms of an alternating series form a chain of inequalities in alternate terms, so that for example  S ( 1 )  <_  S (
3 )  <_  S
( 5 )  <_  ... and  ...  <_  S
( 4 )  <_  S ( 2 )  <_  S ( 0 ) (assuming  M  =  0 so that these terms are defined). (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : Z --> RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  (
 k  +  1 ) )  <_  ( G `  k ) )   &    |-  ( ph  ->  G  ~~>  0 )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( ( -u 1 ^ k )  x.  ( G `  k
 ) ) )   =>    |-  ( ( ph  /\  N  e.  Z  /\  K  e.  NN0 )  ->  ( ( -u 1 ^ N )  x.  (  seq  M (  +  ,  F ) `  ( N  +  ( 2  x.  K ) ) ) )  <_  ( ( -u 1 ^ N )  x.  (  seq  M (  +  ,  F ) `  N ) ) )
 
Theoremiseraltlem3 12432* Lemma for iseralt 12433. From iseraltlem2 12431, we have  ( -u 1 ^ n )  x.  S ( n  + 
2 k )  <_ 
( -u 1 ^ n
)  x.  S ( n ) and  ( -u 1 ^ n )  x.  S ( n  + 
1 )  <_  ( -u 1 ^ n )  x.  S ( n  +  2 k  +  1 ), and we also have  ( -u 1 ^ n )  x.  S
( n  +  1 )  =  ( -u 1 ^ n )  x.  S ( n )  -  G ( n  +  1 ) for each  n by the definition of the partial sum  S, so combining the inequalities we get  ( -u 1 ^ n )  x.  S ( n )  -  G ( n  +  1 )  =  ( -u 1 ^ n )  x.  S ( n  + 
1 )  <_  ( -u 1 ^ n )  x.  S ( n  + 
2 k  +  1 )  =  ( -u 1 ^ n )  x.  S ( n  + 
2 k )  -  G ( n  + 
2 k  +  1 )  <_  ( -u 1 ^ n )  x.  S ( n  + 
2 k )  <_ 
( -u 1 ^ n
)  x.  S ( n )  <_  ( -u 1 ^ n )  x.  S ( n )  +  G ( n  +  1 ), so  |  ( -u
1 ^ n )  x.  S ( n  +  2 k  +  1 )  -  ( -u 1 ^ n )  x.  S ( n )  |  =  |  S ( n  +  2 k  +  1 )  -  S ( n )  |  <_  G (
n  +  1 ) and  |  ( -u
1 ^ n )  x.  S ( n  +  2 k )  -  ( -u 1 ^ n )  x.  S ( n )  |  =  |  S ( n  +  2 k )  -  S ( n )  |  <_  G ( n  +  1 ). Thus, both even and odd partial sums are Cauchy if  G converges to  0. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : Z --> RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  (
 k  +  1 ) )  <_  ( G `  k ) )   &    |-  ( ph  ->  G  ~~>  0 )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( ( -u 1 ^ k )  x.  ( G `  k
 ) ) )   =>    |-  ( ( ph  /\  N  e.  Z  /\  K  e.  NN0 )  ->  ( ( abs `  (
 (  seq  M (  +  ,  F ) `  ( N  +  (
 2  x.  K ) ) )  -  (  seq  M (  +  ,  F ) `  N ) ) )  <_  ( G `  ( N  +  1 ) ) 
 /\  ( abs `  (
 (  seq  M (  +  ,  F ) `  ( ( N  +  ( 2  x.  K ) )  +  1
 ) )  -  (  seq  M (  +  ,  F ) `  N ) ) )  <_  ( G `  ( N  +  1 ) ) ) )
 
Theoremiseralt 12433* The alternating series test. If  G ( k ) is a decreasing sequence that converges to  0, then  sum_ k  e.  Z
( -u 1 ^ k
)  x.  G ( k ) is a convergent series. (Note that the first term is positive if  M is even, and negative if  M is odd. If the parity of your series does not match up with this, you will need to post-compose the series with multiplication by 
-u 1 using isermulc2 12406.) (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : Z --> RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  (
 k  +  1 ) )  <_  ( G `  k ) )   &    |-  ( ph  ->  G  ~~>  0 )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( ( -u 1 ^ k )  x.  ( G `  k
 ) ) )   =>    |-  ( ph  ->  seq 
 M (  +  ,  F )  e.  dom  ~~>  )
 
5.8.3  Finite and infinite sums
 
Syntaxcsu 12434 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-sum 12435* Define the sum of a series with an index set of integers  A.  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. These two methods of summation produce the same result on their common region of definition (i.e. finite subsets of the upper integers) by summo 12466. 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 12614). (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |- 
 sum_ k  e.  A  B  =  ( iota x ( E. m  e. 
 ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m )
 -1-1-onto-> A  /\  x  =  ( 
 seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) ) ) )
 
Theoremsumex 12436 A sum is a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |- 
 sum_ k  e.  A  B  e.  _V
 
Theoremsumeq1f 12437 Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  F/_ k A   &    |-  F/_ k B   =>    |-  ( A  =  B  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremsumeq1 12438* Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( A  =  B  -> 
 sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremnfsum1 12439* Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  F/_ k A   =>    |-  F/_ k sum_ k  e.  A  B
 
Theoremnfsum 12440* 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-Jul-2013.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x sum_ k  e.  A  B
 
Theoremsumeq2w 12441* Equality theorem for sum, when the class expressions  B and  C are equal everywhere. Proved using only Extensionality. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( A. k  B  =  C  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2ii 12442* Equality theorem for sum, with the class expressions  B and  C guarded by  _I to be always sets. (Contributed by Mario Carneiro, 29-Mar-2014.)
 |-  ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2 12443* 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 12444* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( 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 12445* 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 12446* 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 12447* Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
 |-  A  =  B   =>    |-  sum_ k  e.  A  C  =  sum_ k  e.  B  C
 
Theoremsumeq2i 12448* 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 12449* 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 12450* 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 12451* Equality deduction for sum. Note that unlike sumeq2dv 12452, 
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 12452* 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 )
 
Theoremsumeq2sdv 12453* 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 12454* 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 12455* 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 12456* 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 )
 
Theoremsum2id 12457* The second class argument to a sum can be chosen so that it is always a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |- 
 sum_ k  e.  A  B  =  sum_ k  e.  A  (  _I  `  B )
 
Theoremsumfc 12458* 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 Mario Carneiro, 23-Apr-2014.)
 |- 
 sum_ j  e.  A  ( ( k  e.  A  |->  B ) `  j )  =  sum_ k  e.  A  B
 
Theoremfz1f1o 12459* 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 ) ) )
 
Theoremsumrblem 12460* Lemma for sumrb 12462. (Contributed by Mario Carneiro, 12-Aug-2013.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   =>    |-  ( ( ph  /\  A  C_  ( ZZ>= `  N )
 )  ->  (  seq  M (  +  ,  F )  |`  ( ZZ>= `  N ) )  =  seq  N (  +  ,  F ) )
 
Theoremfsumcvg 12461* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A 
 C_  ( M ... N ) )   =>    |-  ( ph  ->  seq  M (  +  ,  F ) 
 ~~>  (  seq  M (  +  ,  F ) `
  N ) )
 
Theoremsumrb 12462* Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 9-Apr-2014.)
 |-  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  ->  (  seq  M (  +  ,  F )  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C )
 )
 
Theoremsummolem3 12463* Lemma for summo 12466. (Contributed by Mario Carneiro, 29-Mar-2014.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  G  =  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B )   &    |-  H  =  ( n  e.  NN  |->  [_ ( K `  n )  /  k ]_ B )   &    |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN ) )   &    |-  ( ph  ->  f : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ph  ->  K : ( 1 ...
 N ) -1-1-onto-> A )   =>    |-  ( ph  ->  (  seq  1 (  +  ,  G ) `  M )  =  (  seq  1 (  +  ,  H ) `  N ) )
 
Theoremsummolem2a 12464* Lemma for summo 12466. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 20-Apr-2014.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  G  =  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B )   &    |-  H  =  ( n  e.  NN  |->  [_ ( K `  n )  /  k ]_ B )   &    |-  ( 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 ) )
 
Theoremsummolem2 12465* Lemma for summo 12466. (Contributed by Mario Carneiro, 3-Apr-2014.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  G  =  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B )   =>    |-  ( ( ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f ( f : ( 1 ... m )
 -1-1-onto-> A  /\  y  =  ( 
 seq  1 (  +  ,  G ) `  m ) )  ->  x  =  y ) )
 
Theoremsummo 12466* A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  G  =  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B )   =>    |-  ( ph  ->  E* x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  F ) 
 ~~>  x )  \/  E. m  e.  NN  E. f
 ( f : ( 1 ... m ) -1-1-onto-> A 
 /\  x  =  ( 
 seq  1 (  +  ,  G ) `  m ) ) ) )
 
Theoremzsum 12467* Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 9-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
 0 ) )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  (  ~~>  ` 
 seq  M (  +  ,  F ) ) )
 
Theoremisum 12468* 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 ) ) )
 
Theoremfsum 12469* The value of a sum over a nonempty finite set. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( 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 (  +  ,  G ) `  M ) )
 
Theoremsum0 12470 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
 
Theoremsumz 12471* Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
 |-  ( ( A  C_  ( ZZ>= `  M )  \/  A  e.  Fin )  -> 
 sum_ k  e.  A  0  =  0 )
 
Theoremfsumf1o 12472* 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 )
 
Theoremsumss 12473* Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( B  \  A ) )  ->  C  =  0 )   &    |-  ( ph  ->  B 
 C_  ( ZZ>= `  M ) )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremfsumss 12474* Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( B  \  A ) )  ->  C  =  0 )   &    |-  ( ph  ->  B  e.  Fin )   =>    |-  ( ph  ->  sum_
 k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremsumss2 12475* Change the index set of a sum by adding zeroes. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
 |-  ( ( ( A 
 C_  B  /\  A. k  e.  A  C  e.  CC )  /\  ( B  C_  ( ZZ>= `  M )  \/  B  e.  Fin ) )  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  if (
 k  e.  A ,  C ,  0 )
 )
 
Theoremfsumcvg2 12476* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
 |-  ( ( 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  ->  A  C_  ( M ... N ) )   =>    |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  (  seq  M (  +  ,  F ) `  N ) )
 
Theoremfsumsers 12477* Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 21-Apr-2014.)
 |-  ( ( 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  ->  A  C_  ( M ... N ) )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  ( 
 seq  M (  +  ,  F ) `  N ) )
 
Theoremfsumcvg3 12478* A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( 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  ~~>  )
 
Theoremfsumser 12479* A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition of follows as fsum1 12490 and fsump1i 12508, which should make our notation clear and from which, along with closure fsumcl 12482, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
 |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  =  A )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  ( M ... N ) A  =  (  seq  M (  +  ,  F ) `  N ) )
 
Theoremfsumcl2lem 12480* - 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 12481* - 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 12482* 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 12483* 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 12484* 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 12485* 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 12486* 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+ )
 
Theoremfsumadd 12487* 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 12488* 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 ) )
 
Theoremsumsn 12489* 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 12490* 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 )
 
Theoremsumsns 12491* 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 12492* 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 12493* 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 12494* 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 ) )
 
Theoremfsump1 12495* 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 12496* 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 )
 
Theoremisumclim2 12497* A converging series converges to its infinite sum. (Contributed by NM, 2-Jan-2006.) (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 )  e.  dom  ~~>  )   =>    |-  ( ph  ->  seq  M (  +  ,  F ) 
 ~~>  sum_ k  e.  Z  A )
 
Theoremisumclim3 12498* The sequence of partial finite sums of a converging infinite series converge to the infinite sum of the series. Note that  j must not occur in  A. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  dom  ~~>  )   &    |-  ( ( ph  /\  k  e.  Z )  ->  A  e.  CC )   &    |-  ( ( ph  /\  j  e.  Z ) 
 ->  ( F `  j
 )  =  sum_ k  e.  ( M ... j
 ) A )   =>    |-  ( ph  ->  F  ~~>  sum_
 k  e.  Z  A )
 
Theoremsumnul 12499* The sum of a non-convergent infinite series evaluates to the empty set. (Contributed by Paul Chapman, 4-Nov-2007.) (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 )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  =  (/) )
 
Theoremisumcl 12500* The sum of a converging infinite series is a complex number. (Contributed by NM, 13-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 )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  e.  CC )
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