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Theorem List for Intuitionistic Logic Explorer - 11401-11500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisumclim2 11401* 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 11402* 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 11403* 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 11404* 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 11405* 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 11406* 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 11407* 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 11408* 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 11409* 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 11410* 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 11411* 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 11412* Optimized version of fsump1 11399 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 11413* Lemma for fsum2d 11414- 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 11414* 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 11415* 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 11416* 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 11417* 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 11418* 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 11419* Lemma for fisum0diag 11420. (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 11420* 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 11421* 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 11422* 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 11423* 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 11424* 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 11425* 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 11426* 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 11427* 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 11428* 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 11429* 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 11430* 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 11431* 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 11432* 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 11433* 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 11434* 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 11435* Lemma for modfsummod 11437. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
 |-  ( A. k  e.  ( A  u.  {
 z } ) B  e.  ZZ  ->  [_ z  /  k ]_ B  e.  ZZ )
 
Theoremmodfsummodlemstep 11436* Induction step for modfsummod 11437. (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 11437* 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 11438* 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 11439* 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 11440* 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 11441* 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 11442* 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 11443* 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 11444* 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 11445* 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 11446* 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 11447* 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 11448* 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 11449* 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 11450* Lemma for fsumre 11451, fsumim 11452, and fsumcj 11453. (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 11451* 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 11452* 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 11453* 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 11454* 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 11455* 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 11456* 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 11457* 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 11458* 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 11459* 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 11460* 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 11461* 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 11462* Lemma for binom 11463 (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 11463* 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 11462. 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 11464* 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 11465* 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 11466* 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 11467 Lemma for bcxmas 11468. (Contributed by Paul Chapman, 18-May-2007.)
 |-  ( A  =  B  ->  ( ( N  +  A )  _C  A )  =  ( ( N  +  B )  _C  B ) )
 
Theorembcxmas 11468* 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 11469* 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 11470* 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 11471* 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 11472* 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 11473* 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 11474* 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 11475* 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 11476* 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 11477* 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 11478* 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 11479 Lemma for trirecip 11480. 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
 
Theoremtrirecip 11480 The sum of the reciprocals of the triangle numbers converge to two. This is Metamath 100 proof #42. (Contributed by Scott Fenton, 23-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |- 
 sum_ k  e.  NN  ( 2  /  (
 k  x.  ( k  +  1 ) ) )  =  2
 
4.8.7  Geometric series
 
Theoremexpcnvap0 11481* A sequence of powers of a complex number  A with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 23-Oct-2022.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
 
Theoremexpcnvre 11482* A sequence of powers of a nonnegative real number less than one converges to zero. (Contributed by Jim Kingdon, 28-Oct-2022.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  1 )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
 
Theoremexpcnv 11483* A sequence of powers of a complex number  A with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 28-Oct-2022.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   =>    |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
 
Theoremexplecnv 11484* A sequence of terms converges to zero when it is less than powers of a number  A whose absolute value is smaller than 1. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  <_  ( A ^ k ) )   =>    |-  ( ph  ->  F  ~~>  0 )
 
Theoremgeosergap 11485* The value of the finite geometric series  A ^ M  +  A ^ ( M  + 
1 )  +...  +  A ^
( N  -  1 ). (Contributed by Mario Carneiro, 2-May-2016.) (Revised by Jim Kingdon, 24-Oct-2022.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  1 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   =>    |-  ( ph  ->  sum_ k  e.  ( M..^ N ) ( A ^ k
 )  =  ( ( ( A ^ M )  -  ( A ^ N ) )  /  ( 1  -  A ) ) )
 
Theoremgeoserap 11486* The value of the finite geometric series  1  +  A ^
1  +  A ^
2  +...  +  A ^
( N  -  1 ). This is Metamath 100 proof #66. (Contributed by NM, 12-May-2006.) (Revised by Jim Kingdon, 24-Oct-2022.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  1 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
 )  =  ( ( 1  -  ( A ^ N ) ) 
 /  ( 1  -  A ) ) )
 
Theorempwm1geoserap1 11487* The n-th power of a number decreased by 1 expressed by the finite geometric series  1  +  A ^ 1  +  A ^ 2  +...  +  A ^ ( N  - 
1 ). (Contributed by AV, 14-Aug-2021.) (Revised by Jim Kingdon, 24-Oct-2022.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A #  1 )   =>    |-  ( ph  ->  (
 ( A ^ N )  -  1 )  =  ( ( A  -  1 )  x.  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
 ) ) )
 
Theoremabsltap 11488 Less-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  A )  <  B )   =>    |-  ( ph  ->  A #  B )
 
Theoremabsgtap 11489 Greater-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  B  <  ( abs `  A ) )   =>    |-  ( ph  ->  A #  B )
 
Theoremgeolim 11490* The partial sums in the infinite series  1  +  A ^
1  +  A ^
2... converge to  ( 1  /  (
1  -  A ) ). (Contributed by NM, 15-May-2006.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( F `  k
 )  =  ( A ^ k ) )   =>    |-  ( ph  ->  seq 0
 (  +  ,  F ) 
 ~~>  ( 1  /  (
 1  -  A ) ) )
 
Theoremgeolim2 11491* The partial sums in the geometric series  A ^ M  +  A ^ ( M  + 
1 )... converge to  ( ( A ^ M )  / 
( 1  -  A
) ). (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  ->  ( F `  k )  =  ( A ^
 k ) )   =>    |-  ( ph  ->  seq
 M (  +  ,  F )  ~~>  ( ( A ^ M )  /  ( 1  -  A ) ) )
 
Theoremgeoreclim 11492* The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  1  <  ( abs `  A ) )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( F `  k
 )  =  ( ( 1  /  A ) ^ k ) )   =>    |-  ( ph  ->  seq 0
 (  +  ,  F ) 
 ~~>  ( A  /  ( A  -  1 ) ) )
 
Theoremgeo2sum 11493* The value of the finite geometric series  2 ^ -u 1  +  2 ^ -u 2  +...  +  2 ^
-u N, multiplied by a constant. (Contributed by Mario Carneiro, 17-Mar-2014.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( N  e.  NN  /\  A  e.  CC )  ->  sum_ k  e.  (
 1 ... N ) ( A  /  ( 2 ^ k ) )  =  ( A  -  ( A  /  (
 2 ^ N ) ) ) )
 
Theoremgeo2sum2 11494* The value of the finite geometric series  1  +  2  + 
4  +  8  +...  +  2 ^ ( N  -  1 ). (Contributed by Mario Carneiro, 7-Sep-2016.)
 |-  ( N  e.  NN0  ->  sum_ k  e.  ( 0..^ N ) ( 2 ^ k )  =  ( ( 2 ^ N )  -  1
 ) )
 
Theoremgeo2lim 11495* The value of the infinite geometric series  2 ^ -u 1  +  2 ^ -u 2  +... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  F  =  ( k  e.  NN  |->  ( A 
 /  ( 2 ^
 k ) ) )   =>    |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  A )
 
Theoremgeoisum 11496* The infinite sum of  1  +  A ^ 1  +  A ^ 2... is  ( 1  /  ( 1  -  A ) ). (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  sum_ k  e.  NN0  ( A ^ k )  =  ( 1  /  (
 1  -  A ) ) )
 
Theoremgeoisumr 11497* The infinite sum of reciprocals  1  +  ( 1  /  A ) ^ 1  +  ( 1  /  A ) ^ 2... is  A  / 
( A  -  1 ). (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  1  <  ( abs `  A ) ) 
 ->  sum_ k  e.  NN0  ( ( 1  /  A ) ^ k
 )  =  ( A 
 /  ( A  -  1 ) ) )
 
Theoremgeoisum1 11498* The infinite sum of  A ^ 1  +  A ^ 2... is  ( A  /  ( 1  -  A ) ). (Contributed by NM, 1-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  sum_ k  e.  NN  ( A ^ k )  =  ( A  /  (
 1  -  A ) ) )
 
Theoremgeoisum1c 11499* The infinite sum of  A  x.  ( R ^ 1 )  +  A  x.  ( R ^ 2 )... is  ( A  x.  R )  /  (
1  -  R ). (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( A  x.  ( R ^
 k ) )  =  ( ( A  x.  R )  /  (
 1  -  R ) ) )
 
Theorem0.999... 11500 The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e.  9  /  1 0 ^ 1  +  9  /  1 0 ^ 2  +  9  / 
1 0 ^ 3  +  ..., is exactly equal to 1. (Contributed by NM, 2-Nov-2007.) (Revised by AV, 8-Sep-2021.)
 |- 
 sum_ k  e.  NN  ( 9  /  (; 1 0 ^ k ) )  =  1
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