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Theorem List for Intuitionistic Logic Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfsumrecl 11201* 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 11202* 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 11203* 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 11204* 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 11205* 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 11206* 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 11207* 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 11208* Split a sum into two parts. A version of fsumsplit 11207 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 11209* A sum of a singleton is the term. A version of sumsn 11211 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 11210* 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 11211* 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 11212* 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 11213* 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 11214* 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 11215* 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 11216* 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 11217* 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 11218* 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 11219* 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 11220* 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 11221* 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 11222* 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 11223* The sequence of partial finite sums of a converging infinite series converges 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 11224* 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 11225* 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 )
 
Theoremisummulc2 11226* An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-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  ->  B  e.  CC )   =>    |-  ( ph  ->  ( B  x.  sum_ k  e.  Z  A )  =  sum_ k  e.  Z  ( B  x.  A ) )
 
Theoremisummulc1 11227* An infinite sum multiplied by a constant. (Contributed by NM, 13-Nov-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  ->  B  e.  CC )   =>    |-  ( ph  ->  ( sum_ k  e.  Z  A  x.  B )  =  sum_ k  e.  Z  ( A  x.  B ) )
 
Theoremisumdivapc 11228* An infinite sum divided by a constant. (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  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( sum_ k  e.  Z  A  /  B )  =  sum_ k  e.  Z  ( A 
 /  B ) )
 
Theoremisumrecl 11229* The sum of a converging infinite real series is a real number. (Contributed by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  RR )   &    |-  ( ph  ->  seq M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  e.  RR )
 
Theoremisumge0 11230* An infinite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 28-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  RR )   &    |-  ( ph  ->  seq M (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ( ph  /\  k  e.  Z )  ->  0  <_  A )   =>    |-  ( ph  ->  0  <_ 
 sum_ k  e.  Z  A )
 
Theoremisumadd 11231* Addition of infinite sums. (Contributed by Mario Carneiro, 18-Aug-2013.) (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  /\  k  e.  Z )  ->  ( G `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  CC )   &    |-  ( ph  ->  seq M (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq M (  +  ,  G )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  ( A  +  B )  =  ( sum_ k  e.  Z  A  +  sum_ k  e.  Z  B ) )
 
Theoremsumsplitdc 11232* Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  ( A  u.  B )  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z )  -> DECID  k  e.  A )   &    |-  ( ( ph  /\  k  e.  Z )  -> DECID  k  e.  B )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  if ( k  e.  A ,  C , 
 0 ) )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  if ( k  e.  B ,  C , 
 0 ) )   &    |-  (
 ( ph  /\  k  e.  ( A  u.  B ) )  ->  C  e.  CC )   &    |-  ( ph  ->  seq
 M (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq M (  +  ,  G )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  ( A  u.  B ) C  =  ( sum_ k  e.  A  C  +  sum_ k  e.  B  C ) )
 
Theoremfsump1i 11233* Optimized version of fsump1 11220 for making sums of a concrete number of terms. (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  N  =  ( K  +  1 )   &    |-  ( k  =  N  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  Z )  ->  A  e.  CC )   &    |-  ( ph  ->  ( K  e.  Z  /\  sum_
 k  e.  ( M
 ... K ) A  =  S ) )   &    |-  ( ph  ->  ( S  +  B )  =  T )   =>    |-  ( ph  ->  ( N  e.  Z  /\  sum_
 k  e.  ( M
 ... N ) A  =  T ) )
 
Theoremfsum2dlemstep 11234* Lemma for fsum2d 11235- induction step. (Contributed by Mario Carneiro, 23-Apr-2014.) (Revised by Jim Kingdon, 8-Oct-2022.)
 |-  ( z  =  <. j ,  k >.  ->  D  =  C )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  j  e.  A )  ->  B  e.  Fin )   &    |-  ( ( ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  C  e.  CC )   &    |-  ( ph  ->  -.  y  e.  x )   &    |-  ( ph  ->  ( x  u.  { y } )  C_  A )   &    |-  ( ph  ->  x  e.  Fin )   &    |-  ( ps 
 <-> 
 sum_ j  e.  x  sum_
 k  e.  B  C  =  sum_ z  e.  U_  j  e.  x  ( { j }  X.  B ) D )   =>    |-  ( ( ph  /\  ps )  ->  sum_ j  e.  ( x  u.  { y }
 ) sum_ k  e.  B  C  =  sum_ z  e.  U_  j  e.  ( x  u.  { y }
 ) ( { j }  X.  B ) D )
 
Theoremfsum2d 11235* Write a double sum as a sum over a two-dimensional region. Note that  B ( j ) is a function of  j. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( z  =  <. j ,  k >.  ->  D  =  C )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  j  e.  A )  ->  B  e.  Fin )   &    |-  ( ( ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_ k  e.  B  C  =  sum_ z  e.  U_  j  e.  A  ( { j }  X.  B ) D )
 
Theoremfsumxp 11236* Combine two sums into a single sum over the cartesian product. (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  ( z  =  <. j ,  k >.  ->  D  =  C )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ( ph  /\  (
 j  e.  A  /\  k  e.  B )
 )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_ k  e.  B  C  =  sum_ z  e.  ( A  X.  B ) D )
 
Theoremfsumcnv 11237* Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  ( x  =  <. j ,  k >.  ->  B  =  D )   &    |-  ( y  = 
 <. k ,  j >.  ->  C  =  D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  Rel  A )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ x  e.  A  B  =  sum_ y  e.  `'  A C )
 
Theoremfisumcom2 11238* Interchange order of summation. Note that  B ( j ) and  D
( k ) are not necessarily constant expressions. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.) (Proof shortened by JJ, 2-Aug-2021.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  (
 ( ph  /\  j  e.  A )  ->  B  e.  Fin )   &    |-  ( ( ph  /\  k  e.  C ) 
 ->  D  e.  Fin )   &    |-  ( ph  ->  ( ( j  e.  A  /\  k  e.  B )  <->  ( k  e.  C  /\  j  e.  D ) ) )   &    |-  ( ( ph  /\  (
 j  e.  A  /\  k  e.  B )
 )  ->  E  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_ k  e.  B  E  =  sum_ k  e.  C  sum_ j  e.  D  E )
 
Theoremfsumcom 11239* Interchange order of summation. (Contributed by NM, 15-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  (
 ( ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_
 k  e.  B  C  =  sum_ k  e.  B  sum_
 j  e.  A  C )
 
Theoremfsum0diaglem 11240* Lemma for fisum0diag 11241. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
 |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  (
 0 ... ( N  -  j ) ) ) 
 ->  ( k  e.  (
 0 ... N )  /\  j  e.  ( 0 ... ( N  -  k
 ) ) ) )
 
Theoremfisum0diag 11241* Two ways to express "the sum of  A ( j ,  k ) over the triangular region  M  <_  j,  M  <_  k,  j  +  k  <_  N." (Contributed by NM, 31-Dec-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
 |-  ( ( ph  /\  (
 j  e.  ( 0
 ... N )  /\  k  e.  ( 0 ... ( N  -  j
 ) ) ) ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  sum_ j  e.  ( 0 ... N ) sum_ k  e.  (
 0 ... ( N  -  j ) ) A  =  sum_ k  e.  (
 0 ... N ) sum_ j  e.  ( 0 ... ( N  -  k
 ) ) A )
 
Theoremmptfzshft 11242* 1-1 onto function in maps-to notation which shifts a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  (
 j  e.  ( ( M  +  K )
 ... ( N  +  K ) )  |->  ( j  -  K ) ) : ( ( M  +  K )
 ... ( N  +  K ) ) -1-1-onto-> ( M
 ... N ) )
 
Theoremfsumrev 11243* Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( j  =  ( K  -  k
 )  ->  A  =  B )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( K  -  N ) ... ( K  -  M ) ) B )
 
Theoremfsumshft 11244* Index shift of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) (Proof shortened by AV, 8-Sep-2019.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( j  =  ( k  -  K )  ->  A  =  B )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( M  +  K ) ... ( N  +  K ) ) B )
 
Theoremfsumshftm 11245* Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( j  =  ( k  +  K )  ->  A  =  B )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( M  -  K ) ... ( N  -  K ) ) B )
 
Theoremfisumrev2 11246* Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 13-Apr-2016.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  (
 ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( j  =  ( ( M  +  N )  -  k
 )  ->  A  =  B )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( M ... N ) B )
 
Theoremfisum0diag2 11247* Two ways to express "the sum of  A ( j ,  k ) over the triangular region  0  <_  j, 
0  <_  k,  j  +  k  <_  N." (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( x  =  k 
 ->  B  =  A )   &    |-  ( x  =  (
 k  -  j ) 
 ->  B  =  C )   &    |-  ( ( ph  /\  (
 j  e.  ( 0
 ... N )  /\  k  e.  ( 0 ... ( N  -  j
 ) ) ) ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  sum_ j  e.  ( 0 ... N ) sum_ k  e.  (
 0 ... ( N  -  j ) ) A  =  sum_ k  e.  (
 0 ... N ) sum_ j  e.  ( 0 ... k ) C )
 
Theoremfsummulc2 11248* A finite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  ( C  x.  sum_ k  e.  A  B )  =  sum_ k  e.  A  ( C  x.  B ) )
 
Theoremfsummulc1 11249* A finite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  ( sum_ k  e.  A  B  x.  C )  =  sum_ k  e.  A  ( B  x.  C ) )
 
Theoremfsumdivapc 11250* A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  ( sum_ k  e.  A  B  /  C )  =  sum_ k  e.  A  ( B 
 /  C ) )
 
Theoremfsumneg 11251* Negation of a finite sum. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  -u B  =  -u sum_ k  e.  A  B )
 
Theoremfsumsub 11252* Split a finite sum over a subtraction. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-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 ) )
 
Theoremfsum2mul 11253* Separate the nested sum of the product  C ( j )  x.  D ( k ). (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  (
 ( ph  /\  j  e.  A )  ->  C  e.  CC )   &    |-  ( ( ph  /\  k  e.  B ) 
 ->  D  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_
 k  e.  B  ( C  x.  D )  =  ( sum_ j  e.  A  C  x.  sum_ k  e.  B  D ) )
 
Theoremfsumconst 11254* The sum of constant terms ( k is not free in  B). (Contributed by NM, 24-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ( A  e.  Fin  /\  B  e.  CC )  -> 
 sum_ k  e.  A  B  =  ( ( `  A )  x.  B ) )
 
Theoremfsumdifsnconst 11255* The sum of constant terms ( k is not free in  C) over an index set excluding a singleton. (Contributed by AV, 7-Jan-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  CC )  -> 
 sum_ k  e.  ( A  \  { B }
 ) C  =  ( ( ( `  A )  -  1 )  x.  C ) )
 
Theoremmodfsummodlem1 11256* Lemma for modfsummod 11258. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
 |-  ( A. k  e.  ( A  u.  {
 z } ) B  e.  ZZ  ->  [_ z  /  k ]_ B  e.  ZZ )
 
Theoremmodfsummodlemstep 11257* Induction step for modfsummod 11258. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by Jim Kingdon, 12-Oct-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A. k  e.  ( A  u.  { z }
 ) B  e.  ZZ )   &    |-  ( ph  ->  -.  z  e.  A )   &    |-  ( ph  ->  (
 sum_ k  e.  A  B  mod  N )  =  ( sum_ k  e.  A  ( B  mod  N ) 
 mod  N ) )   =>    |-  ( ph  ->  (
 sum_ k  e.  ( A  u.  { z }
 ) B  mod  N )  =  ( sum_ k  e.  ( A  u.  { z } ) ( B  mod  N ) 
 mod  N ) )
 
Theoremmodfsummod 11258* A finite sum modulo a positive integer equals the finite sum of their summands modulo the positive integer, modulo the positive integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A. k  e.  A  B  e.  ZZ )   =>    |-  ( ph  ->  ( sum_ k  e.  A  B  mod  N )  =  ( sum_ k  e.  A  ( B 
 mod  N )  mod  N ) )
 
Theoremfsumge0 11259* If all of the terms of a finite sum are nonnegative, so is the sum. (Contributed by NM, 26-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  0  <_  B )   =>    |-  ( ph  ->  0  <_ 
 sum_ k  e.  A  B )
 
Theoremfsumlessfi 11260* A shorter sum of nonnegative terms is no greater than a longer one. (Contributed by NM, 26-Dec-2005.) (Revised by Jim Kingdon, 12-Oct-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  0  <_  B )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  C  e.  Fin )   =>    |-  ( ph  ->  sum_ k  e.  C  B  <_  sum_ k  e.  A  B )
 
Theoremfsumge1 11261* A sum of nonnegative numbers is greater than or equal to any one of its terms. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 4-Jun-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  0  <_  B )   &    |-  ( k  =  M  ->  B  =  C )   &    |-  ( ph  ->  M  e.  A )   =>    |-  ( ph  ->  C 
 <_  sum_ k  e.  A  B )
 
Theoremfsum00 11262* A sum of nonnegative numbers is zero iff all terms are zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  0  <_  B )   =>    |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  <->  A. k  e.  A  B  =  0 )
 )
 
Theoremfsumle 11263* If all of the terms of finite sums compare, so do the sums. (Contributed by NM, 11-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  <_  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  <_  sum_ k  e.  A  C )
 
Theoremfsumlt 11264* If every term in one finite sum is less than the corresponding term in another, then the first sum is less than the second. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  <  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  <  sum_ k  e.  A  C )
 
Theoremfsumabs 11265* Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  sum_ k  e.  A  B )  <_  sum_ k  e.  A  ( abs `  B )
 )
 
Theoremtelfsumo 11266* Sum of a telescoping series, using half-open intervals. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( k  =  j 
 ->  A  =  B )   &    |-  ( k  =  (
 j  +  1 ) 
 ->  A  =  C )   &    |-  ( k  =  M  ->  A  =  D )   &    |-  ( k  =  N  ->  A  =  E )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  ( M..^ N ) ( B  -  C )  =  ( D  -  E ) )
 
Theoremtelfsumo2 11267* Sum of a telescoping series. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( k  =  j 
 ->  A  =  B )   &    |-  ( k  =  (
 j  +  1 ) 
 ->  A  =  C )   &    |-  ( k  =  M  ->  A  =  D )   &    |-  ( k  =  N  ->  A  =  E )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  ( M..^ N ) ( C  -  B )  =  ( E  -  D ) )
 
Theoremtelfsum 11268* Sum of a telescoping series. (Contributed by Scott Fenton, 24-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)
 |-  ( k  =  j 
 ->  A  =  B )   &    |-  ( k  =  (
 j  +  1 ) 
 ->  A  =  C )   &    |-  ( k  =  M  ->  A  =  D )   &    |-  ( k  =  ( N  +  1 )  ->  A  =  E )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( N  +  1 ) ) ) 
 ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) ( B  -  C )  =  ( D  -  E ) )
 
Theoremtelfsum2 11269* Sum of a telescoping series. (Contributed by Mario Carneiro, 15-Jun-2014.) (Revised by Mario Carneiro, 2-May-2016.)
 |-  ( k  =  j 
 ->  A  =  B )   &    |-  ( k  =  (
 j  +  1 ) 
 ->  A  =  C )   &    |-  ( k  =  M  ->  A  =  D )   &    |-  ( k  =  ( N  +  1 )  ->  A  =  E )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( N  +  1 ) ) ) 
 ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) ( C  -  B )  =  ( E  -  D ) )
 
Theoremfsumparts 11270* Summation by parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( k  =  j 
 ->  ( A  =  B  /\  V  =  W ) )   &    |-  ( k  =  ( j  +  1 )  ->  ( A  =  C  /\  V  =  X ) )   &    |-  (
 k  =  M  ->  ( A  =  D  /\  V  =  Y )
 )   &    |-  ( k  =  N  ->  ( A  =  E  /\  V  =  Z ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( M ... N ) )  ->  V  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  ( M..^ N ) ( B  x.  ( X  -  W ) )  =  ( ( ( E  x.  Z )  -  ( D  x.  Y ) )  -  sum_
 j  e.  ( M..^ N ) ( ( C  -  B )  x.  X ) ) )
 
Theoremfsumrelem 11271* Lemma for fsumre 11272, fsumim 11273, and fsumcj 11274. (Contributed by Mario Carneiro, 25-Jul-2014.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  F : CC --> CC   &    |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( F `  ( x  +  y ) )  =  ( ( F `  x )  +  ( F `  y ) ) )   =>    |-  ( ph  ->  ( F `  sum_ k  e.  A  B )  =  sum_ k  e.  A  ( F `
  B ) )
 
Theoremfsumre 11272* The real part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  sum_
 k  e.  A  B )  =  sum_ k  e.  A  ( Re `  B ) )
 
Theoremfsumim 11273* The imaginary part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  sum_
 k  e.  A  B )  =  sum_ k  e.  A  ( Im `  B ) )
 
Theoremfsumcj 11274* The complex conjugate of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  ( * `  sum_
 k  e.  A  B )  =  sum_ k  e.  A  ( * `  B ) )
 
Theoremiserabs 11275* Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Jim Kingdon, 14-Dec-2022.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  seq
 M (  +  ,  F )  ~~>  A )   &    |-  ( ph  ->  seq M (  +  ,  G )  ~~>  B )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( abs `  ( F `  k
 ) ) )   =>    |-  ( ph  ->  ( abs `  A )  <_  B )
 
Theoremcvgcmpub 11276* An upper bound for the limit of a real infinite series. This theorem can also be used to compare two infinite series. (Contributed by Mario Carneiro, 24-Mar-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  ( ph  ->  seq M (  +  ,  F )  ~~>  A )   &    |-  ( ph  ->  seq M (  +  ,  G )  ~~>  B )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  <_  ( F `  k ) )   =>    |-  ( ph  ->  B  <_  A )
 
Theoremfsumiun 11277* Sum over a disjoint indexed union. (Contributed by Mario Carneiro, 1-Jul-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  Fin )   &    |-  ( ph  -> Disj  x  e.  A  B )   &    |-  ( ( ph  /\  ( x  e.  A  /\  k  e.  B )
 )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  U_  x  e.  A  B C  =  sum_ x  e.  A  sum_ k  e.  B  C )
 
Theoremhashiun 11278* The cardinality of a disjoint indexed union. (Contributed by Mario Carneiro, 24-Jan-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  Fin )   &    |-  ( ph  -> Disj  x  e.  A  B )   =>    |-  ( ph  ->  ( ` 
 U_ x  e.  A  B )  =  sum_ x  e.  A  ( `  B ) )
 
Theoremhash2iun 11279* The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  Fin )   &    |-  (
 ( ph  /\  x  e.  A  /\  y  e.  B )  ->  C  e.  Fin )   &    |-  ( ph  -> Disj  x  e.  A  U_ y  e.  B  C )   &    |-  (
 ( ph  /\  x  e.  A )  -> Disj  y  e.  B  C )   =>    |-  ( ph  ->  ( `  U_ x  e.  A  U_ y  e.  B  C )  =  sum_ x  e.  A  sum_ y  e.  B  ( `  C ) )
 
Theoremhash2iun1dif1 11280* The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  B  =  ( A  \  { x } )   &    |-  ( ( ph  /\  x  e.  A  /\  y  e.  B )  ->  C  e.  Fin )   &    |-  ( ph  -> Disj  x  e.  A  U_ y  e.  B  C )   &    |-  ( ( ph  /\  x  e.  A )  -> Disj  y  e.  B  C )   &    |-  (
 ( ph  /\  x  e.  A  /\  y  e.  B )  ->  ( `  C )  =  1 )   =>    |-  ( ph  ->  ( ` 
 U_ x  e.  A  U_ y  e.  B  C )  =  ( ( `  A )  x.  (
 ( `  A )  -  1 ) ) )
 
Theoremhashrabrex 11281* The number of elements in a class abstraction with a restricted existential quantification. (Contributed by Alexander van der Vekens, 29-Jul-2018.)
 |-  ( ph  ->  Y  e.  Fin )   &    |-  ( ( ph  /\  y  e.  Y ) 
 ->  { x  e.  X  |  ps }  e.  Fin )   &    |-  ( ph  -> Disj  y  e.  Y  { x  e.  X  |  ps }
 )   =>    |-  ( ph  ->  ( ` 
 { x  e.  X  |  E. y  e.  Y  ps } )  =  sum_ y  e.  Y  ( `  { x  e.  X  |  ps }
 ) )
 
Theoremhashuni 11282* The cardinality of a disjoint union. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Fin )   &    |-  ( ph  -> Disj  x  e.  A  x )   =>    |-  ( ph  ->  ( `  U. A )  = 
 sum_ x  e.  A  ( `  x ) )
 
4.8.3  The binomial theorem
 
Theorembinomlem 11283* Lemma for binom 11284 (binomial theorem). Inductive step. (Contributed by NM, 6-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ps  ->  (
 ( A  +  B ) ^ N )  = 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( A ^ ( N  -  k ) )  x.  ( B ^
 k ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( A  +  B ) ^
 ( N  +  1 ) )  =  sum_ k  e.  ( 0 ... ( N  +  1 ) ) ( ( ( N  +  1 )  _C  k )  x.  ( ( A ^ ( ( N  +  1 )  -  k ) )  x.  ( B ^ k
 ) ) ) )
 
Theorembinom 11284* The binomial theorem:  ( A  +  B
) ^ N is the sum from  k  =  0 to  N of  ( N  _C  k )  x.  ( ( A ^
k )  x.  ( B ^ ( N  -  k ) ). Theorem 15-2.8 of [Gleason] p. 296. This part of the proof sets up the induction and does the base case, with the bulk of the work (the induction step) in binomlem 11283. This is Metamath 100 proof #44. (Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  +  B ) ^ N )  =  sum_ k  e.  ( 0 ... N ) ( ( N  _C  k )  x.  ( ( A ^
 ( N  -  k
 ) )  x.  ( B ^ k ) ) ) )
 
Theorembinom1p 11285* Special case of the binomial theorem for  ( 1  +  A
) ^ N. (Contributed by Paul Chapman, 10-May-2007.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( ( 1  +  A ) ^ N )  =  sum_ k  e.  ( 0 ...
 N ) ( ( N  _C  k )  x.  ( A ^
 k ) ) )
 
Theorembinom11 11286* Special case of the binomial theorem for  2 ^ N. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( N  e.  NN0  ->  ( 2 ^ N )  =  sum_ k  e.  ( 0 ... N ) ( N  _C  k ) )
 
Theorembinom1dif 11287* A summation for the difference between  ( ( A  + 
1 ) ^ N
) and  ( A ^ N ). (Contributed by Scott Fenton, 9-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( ( ( A  +  1 ) ^ N )  -  ( A ^ N ) )  =  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( A ^ k ) ) )
 
Theorembcxmaslem1 11288 Lemma for bcxmas 11289. (Contributed by Paul Chapman, 18-May-2007.)
 |-  ( A  =  B  ->  ( ( N  +  A )  _C  A )  =  ( ( N  +  B )  _C  B ) )
 
Theorembcxmas 11289* Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  ( ( ( N  +  1 )  +  M )  _C  M )  =  sum_ j  e.  (
 0 ... M ) ( ( N  +  j
 )  _C  j )
 )
 
4.8.4  Infinite sums (cont.)
 
Theoremisumshft 11290* Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  ( M  +  K ) )   &    |-  (
 j  =  ( K  +  k )  ->  A  =  B )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  j  e.  W ) 
 ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  W  A  =  sum_ k  e.  Z  B )
 
Theoremisumsplit 11291* Split off the first  N terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 21-Oct-2022.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ( 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  =  (
 sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A ) )
 
Theoremisum1p 11292* The infinite sum of a converging infinite series equals the first term plus the infinite sum of the rest of it. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-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  =  ( ( F `  M )  +  sum_ k  e.  ( ZZ>= `  ( M  +  1 ) ) A ) )
 
Theoremisumnn0nn 11293* Sum from 0 to infinity in terms of sum from 1 to infinity. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( k  =  0 
 ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  A  e.  CC )   &    |-  ( ph  ->  seq 0 (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e. 
 NN0  A  =  ( B  +  sum_ k  e. 
 NN  A ) )
 
Theoremisumrpcl 11294* The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  RR+ )   &    |-  ( ph  ->  seq M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  W  A  e.  RR+ )
 
Theoremisumle 11295* Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  A  <_  B )   &    |-  ( ph  ->  seq
 M (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq M (  +  ,  G )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  <_  sum_ k  e.  Z  B )
 
Theoremisumlessdc 11296* A finite sum of nonnegative numbers is less than or equal to its limit. (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
 )  =  B )   &    |-  ( ph  ->  A. k  e.  Z DECID  k  e.  A )   &    |-  ( ( ph  /\  k  e.  Z )  ->  B  e.  RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  B )   &    |-  ( ph  ->  seq M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  A  B  <_  sum_ k  e.  Z  B )
 
4.8.5  Miscellaneous converging and diverging sequences
 
Theoremdivcnv 11297* The sequence of reciprocals of positive integers, multiplied by the factor  A, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Jim Kingdon, 22-Oct-2022.)
 |-  ( A  e.  CC  ->  ( n  e.  NN  |->  ( A  /  n ) )  ~~>  0 )
 
4.8.6  Arithmetic series
 
Theoremarisum 11298* Arithmetic series sum of the first 
N positive integers. This is Metamath 100 proof #68. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 22-May-2014.)
 |-  ( N  e.  NN0  ->  sum_ k  e.  ( 1
 ... N ) k  =  ( ( ( N ^ 2 )  +  N )  / 
 2 ) )
 
Theoremarisum2 11299* Arithmetic series sum of the first 
N nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 2-Aug-2021.)
 |-  ( N  e.  NN0  ->  sum_ k  e.  ( 0
 ... ( N  -  1 ) ) k  =  ( ( ( N ^ 2 )  -  N )  / 
 2 ) )
 
Theoremtrireciplem 11300 Lemma for trirecip 11301. Show that the sum converges. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  F  =  ( n  e.  NN  |->  ( 1 
 /  ( n  x.  ( n  +  1
 ) ) ) )   =>    |-  seq 1 (  +  ,  F )  ~~>  1
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