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Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclimuni 10901 An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( F  ~~>  A  /\  F 
 ~~>  B )  ->  A  =  B )
 
Theoremfclim 10902 The limit relation is function-like, and with range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ~~>  : dom  ~~>  --> CC
 
Theoremclimdm 10903 Two ways to express that a function has a limit. (The expression  (  ~~>  `  F
) is sometimes useful as a shorthand for "the unique limit of the function  F"). (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( F  e.  dom  ~~>  <->  F  ~~>  ( 
 ~~>  `  F ) )
 
Theoremclimeu 10904* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
 |-  ( F  ~~>  A  ->  E! x  F  ~~>  x )
 
Theoremclimreu 10905* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
 |-  ( F  ~~>  A  ->  E! x  e.  CC  F  ~~>  x )
 
Theoremclimmo 10906* An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)
 |- 
 E* x  F  ~~>  x
 
Theoremclimeq 10907* Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  ( G `
  k ) )   =>    |-  ( ph  ->  ( F  ~~>  A 
 <->  G  ~~>  A ) )
 
Theoremclimmpt 10908* Exhibit a function  G with the same convergence properties as the not-quite-function  F. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  G  =  ( k  e.  Z  |->  ( F `  k ) )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
 
Theorem2clim 10909* If two sequences converge to each other, they converge to the same limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( abs `  ( ( F `  k )  -  ( G `  k ) ) )  <  x )   &    |-  ( ph  ->  F  ~~>  A )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremclimshftlemg 10910 A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( F  ~~>  A  ->  ( F  shift  M )  ~~>  A )
 )
 
Theoremclimres 10911 A function restricted to upper integers converges iff the original function converges. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( ( F  |`  ( ZZ>= `  M )
 )  ~~>  A  <->  F  ~~>  A ) )
 
Theoremclimshft 10912 A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( ( F 
 shift  M )  ~~>  A  <->  F  ~~>  A ) )
 
Theoremserclim0 10913 The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
 |-  ( M  e.  ZZ  ->  seq M (  +  ,  ( ( ZZ>= `  M )  X.  { 0 } ) )  ~~>  0 )
 
Theoremclimshft2 10914* A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  F  e.  W )   &    |-  ( ph  ->  G  e.  X )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  ( k  +  K ) )  =  ( F `  k
 ) )   =>    |-  ( ph  ->  ( F 
 ~~>  A  <->  G  ~~>  A ) )
 
Theoremclimabs0 10915* Convergence to zero of the absolute value is equivalent to convergence to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( abs `  ( F `  k ) ) )   =>    |-  ( ph  ->  ( F 
 ~~>  0  <->  G  ~~>  0 ) )
 
Theoremclimcn1 10916* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  z  e.  B )  ->  ( F `  z )  e. 
 CC )   &    |-  ( ph  ->  G  ~~>  A )   &    |-  ( ph  ->  H  e.  W )   &    |-  (
 ( ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  B  ( ( abs `  (
 z  -  A ) )  <  y  ->  ( abs `  ( ( F `  z )  -  ( F `  A ) ) )  <  x ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  B )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( F `  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( F `  A ) )
 
Theoremclimcn2 10917* Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ( ph  /\  ( u  e.  C  /\  v  e.  D ) )  ->  ( u F v )  e. 
 CC )   &    |-  ( ph  ->  G  ~~>  A )   &    |-  ( ph  ->  H  ~~>  B )   &    |-  ( ph  ->  K  e.  W )   &    |-  (
 ( ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  C  A. v  e.  D  ( ( ( abs `  ( u  -  A ) )  <  y  /\  ( abs `  ( v  -  B ) )  < 
 z )  ->  ( abs `  ( ( u F v )  -  ( A F B ) ) )  <  x ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  C )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( H `  k )  e.  D )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( K `  k
 )  =  ( ( G `  k ) F ( H `  k ) ) )   =>    |-  ( ph  ->  K  ~~>  ( A F B ) )
 
Theoremaddcn2 10918* Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (We write out the definition directly because df-cn and df-cncf are not yet available to us. See addcn for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e. 
 CC  ( ( ( abs `  ( u  -  B ) )  < 
 y  /\  ( abs `  ( v  -  C ) )  <  z ) 
 ->  ( abs `  (
 ( u  +  v
 )  -  ( B  +  C ) ) )  <  A ) )
 
Theoremsubcn2 10919* Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e. 
 CC  ( ( ( abs `  ( u  -  B ) )  < 
 y  /\  ( abs `  ( v  -  C ) )  <  z ) 
 ->  ( abs `  (
 ( u  -  v
 )  -  ( B  -  C ) ) )  <  A ) )
 
Theoremmulcn2 10920* Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e. 
 CC  ( ( ( abs `  ( u  -  B ) )  < 
 y  /\  ( abs `  ( v  -  C ) )  <  z ) 
 ->  ( abs `  (
 ( u  x.  v
 )  -  ( B  x.  C ) ) )  <  A ) )
 
Theoremreccn2ap 10921* The reciprocal function is continuous. The class  T is just for convenience in writing the proof and typically would be passed in as an instance of eqid 2100. (Contributed by Mario Carneiro, 9-Feb-2014.) Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
 |-  T  =  (inf ( { 1 ,  (
 ( abs `  A )  x.  B ) } ,  RR ,  <  )  x.  ( ( abs `  A )  /  2 ) )   =>    |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  { w  e.  CC  |  w #  0 }  ( ( abs `  (
 z  -  A ) )  <  y  ->  ( abs `  ( (
 1  /  z )  -  ( 1  /  A ) ) )  <  B ) )
 
Theoremcn1lem 10922* A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  F : CC --> CC   &    |-  (
 ( z  e.  CC  /\  A  e.  CC )  ->  ( abs `  (
 ( F `  z
 )  -  ( F `
  A ) ) )  <_  ( abs `  ( z  -  A ) ) )   =>    |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  ( ( abs `  (
 z  -  A ) )  <  y  ->  ( abs `  ( ( F `  z )  -  ( F `  A ) ) )  <  x ) )
 
Theoremabscn2 10923* The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( abs `  z
 )  -  ( abs `  A ) ) )  <  x ) )
 
Theoremcjcn2 10924* The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( * `  z )  -  ( * `  A ) ) )  <  x ) )
 
Theoremrecn2 10925* The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( Re `  z )  -  ( Re `  A ) ) )  <  x ) )
 
Theoremimcn2 10926* The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( Im `  z )  -  ( Im `  A ) ) )  <  x ) )
 
Theoremclimcn1lem 10927* The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  H : CC --> CC   &    |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( H `  z )  -  ( H `  A ) ) )  <  x ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( H `
  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( H `  A ) )
 
Theoremclimabs 10928* Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( abs `  ( F `  k
 ) ) )   =>    |-  ( ph  ->  G  ~~>  ( abs `  A )
 )
 
Theoremclimcj 10929* Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( * `
  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( * `  A ) )
 
Theoremclimre 10930* Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( Re
 `  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( Re `  A ) )
 
Theoremclimim 10931* Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( Im
 `  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( Im `  A ) )
 
Theoremclimrecl 10932* The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremclimge0 10933* A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  ( F `
  k ) )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremclimadd 10934* Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  +  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  +  B ) )
 
Theoremclimmul 10935* Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  x.  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
 
Theoremclimsub 10936* Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  -  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  -  B ) )
 
Theoremclimaddc1 10937* Limit of a constant  C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `  k )  +  C ) )   =>    |-  ( ph  ->  G  ~~>  ( A  +  C ) )
 
Theoremclimaddc2 10938* Limit of a constant  C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  +  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  +  A ) )
 
Theoremclimmulc2 10939* Limit of a sequence multiplied by a constant  C. Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  x.  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  x.  A ) )
 
Theoremclimsubc1 10940* Limit of a constant  C subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `  k )  -  C ) )   =>    |-  ( ph  ->  G  ~~>  ( A  -  C ) )
 
Theoremclimsubc2 10941* Limit of a constant  C minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  -  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  -  A ) )
 
Theoremclimle 10942* Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremclimsqz 10943* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  <_  A )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremclimsqz2 10944* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  <_  ( F `  k ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  <_  ( G `  k ) )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremclim2ser 10945* 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 10946* 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 10947* 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 10948* Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Jim Kingdon, 8-Apr-2023.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  seq M (  +  ,  F ) 
 ~~>  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  x.  ( F `  k ) ) )   =>    |-  ( ph  ->  seq M (  +  ,  G ) 
 ~~>  ( C  x.  A ) )
 
Theoremclimlec2 10949* 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 10950* 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 10951* 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 10952* 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 10953* The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  seq M (  +  ,  F )  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  ( F `
  k ) )   =>    |-  ( ph  ->  (  seq M (  +  ,  F ) `  N )  <_  A )
 
Theoremiser3shft 10954* Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Jim Kingdon, 17-Oct-2022.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F )  ~~>  A  <->  seq ( M  +  N ) (  .+  ,  ( F  shift  N ) )  ~~>  A ) )
 
Theoremclimcau 10955* A converging sequence of complex numbers is a Cauchy sequence. The converse would require excluded middle or a different definition of Cauchy sequence (for example, fixing a rate of convergence as in climcvg1n 10958). Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  dom  ~~>  ) 
 ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x )
 
Theoremclimrecvg1n 10956* A Cauchy sequence of real numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within  C  /  n of the nth term, where  C is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( abs `  ( ( F `  k )  -  ( F `  n ) ) )  <  ( C  /  n ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremclimcvg1nlem 10957* Lemma for climcvg1n 10958. We construct sequences of the real and imaginary parts of each term of  F, show those converge, and use that to show that  F converges. (Contributed by Jim Kingdon, 24-Aug-2021.)
 |-  ( ph  ->  F : NN --> CC )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( abs `  ( ( F `  k )  -  ( F `  n ) ) )  <  ( C  /  n ) )   &    |-  G  =  ( x  e.  NN  |->  ( Re `  ( F `  x ) ) )   &    |-  H  =  ( x  e.  NN  |->  ( Im `  ( F `
  x ) ) )   &    |-  J  =  ( x  e.  NN  |->  ( _i  x.  ( H `
  x ) ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremclimcvg1n 10958* A Cauchy sequence of complex numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within  C  /  n of the nth term, where  C is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
 |-  ( ph  ->  F : NN --> CC )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( abs `  ( ( F `  k )  -  ( F `  n ) ) )  <  ( C  /  n ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremclimcaucn 10959* A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 10955 but adds the part that  ( F `  k ) is complex. (Contributed by Jim Kingdon, 24-Aug-2021.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  dom  ~~>  ) 
 ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  ( abs `  ( ( F `
  k )  -  ( F `  j ) ) )  <  x ) )
 
Theoremserf0 10960* 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 )
 
3.8.2  Finite and infinite sums
 
Syntaxcsu 10961 Extend class notation to include finite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is a commonly used word in comments.)
 class  sum_ k  e.  A  B
 
Definitiondf-sumdc 10962* 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. In both cases we have an  if expression so that we only need  B to be defined where  k  e.  A. In the infinite case, we also require that the indexing set be a decidable subset of an upperset of integers (that is, membership of integers in it is decidable). These two methods of summation produce the same result on their common region of definition (i.e. finite sets of integers). Examples: 
sum_ k  e.  {
1 ,  2 ,  4 }  k means  1  +  2  +  4  =  7, and  sum_ k  e.  NN  (
1  /  ( 2 ^ k ) )  =  1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 11130). (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 21-May-2023.)
 |- 
 sum_ k  e.  A  B  =  ( iota x ( E. m  e. 
 ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>=
 `  m )DECID  j  e.  A  /\  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
 ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m )
 -1-1-onto-> A  /\  x  =  ( 
 seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ) `  m ) ) ) )
 
Theoremsumeq1 10963 Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  ( A  =  B  -> 
 sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremnfsum1 10964 Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  F/_ k A   =>    |-  F/_ k sum_ k  e.  A  B
 
Theoremnfsum 10965 Bound-variable hypothesis builder for sum: if  x is (effectively) not free in  A and  B, it is not free in  sum_ k  e.  A B. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x sum_ k  e.  A  B
 
Theoremsumdc 10966* Decidability of a subset of upper integers. (Contributed by Jim Kingdon, 1-Jan-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A 
 C_  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A. x  e.  ( ZZ>= `  M )DECID  x  e.  A )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  -> DECID  N  e.  A )
 
Theoremsumeq2 10967* 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 10968 Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  ( j  =  k 
 ->  B  =  C )   &    |-  F/_ k A   &    |-  F/_ j A   &    |-  F/_ k B   &    |-  F/_ j C   =>    |- 
 sum_ j  e.  A  B  =  sum_ k  e.  A  C
 
Theoremcbvsumv 10969* 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 10970* 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 10971* Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
 |-  A  =  B   =>    |-  sum_ k  e.  A  C  =  sum_ k  e.  B  C
 
Theoremsumeq2i 10972* 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 10973* 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 10974* 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 10975* Equality deduction for sum. Note that unlike sumeq2dv 10976, 
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 10976* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2ad 10977* Equality deduction for sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2sdv 10978* 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 10979* 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 10980* 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 10981* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  B ) 
 ->  C  =  D )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  D )
 
Theoremsumfct 10982* A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 18-Sep-2022.)
 |-  ( A. k  e.  A  B  e.  CC  -> 
 sum_ j  e.  A  ( ( k  e.  A  |->  B ) `  j )  =  sum_ k  e.  A  B )
 
Theoremfz1f1o 10983* 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 ) ) )
 
Theoremsumrbdclem 10984* Lemma for sumrbdc 10986. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> DECID  k  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   =>    |-  ( ( ph  /\  A  C_  ( ZZ>= `  N )
 )  ->  (  seq M (  +  ,  F )  |`  ( ZZ>= `  N ) )  =  seq N (  +  ,  F ) )
 
Theoremfsum3cvg 10985* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 12-Nov-2022.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> DECID  k  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A 
 C_  ( M ... N ) )   =>    |-  ( ph  ->  seq M (  +  ,  F ) 
 ~~>  (  seq M (  +  ,  F ) `
  N ) )
 
Theoremsumrbdc 10986* Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  A  C_  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  A  C_  ( ZZ>= `  N )
 )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  -> DECID  k  e.  A )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  N )
 )  -> DECID  k  e.  A )   =>    |-  ( ph  ->  (  seq M (  +  ,  F )  ~~>  C  <->  seq N (  +  ,  F )  ~~>  C )
 )
 
Theoremisummolemnm 10987* Lemma for summodc 10991. (Contributed by Jim Kingdon, 15-Aug-2022.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN )
 )   &    |-  ( ph  ->  f : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ph  ->  K : ( 1 ...
 N ) -1-1-onto-> A )   =>    |-  ( ph  ->  N  =  M )
 
Theoremsummodclem3 10988* Lemma for summodc 10991. (Contributed by Mario Carneiro, 29-Mar-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN )
 )   &    |-  ( ph  ->  f : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ph  ->  K : ( 1 ...
 N ) -1-1-onto-> A )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  M ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   &    |-  H  =  ( n  e.  NN  |->  if ( n  <_  N ,  [_ ( K `  n )  /  k ]_ B ,  0 ) )   =>    |-  ( ph  ->  (  seq 1 (  +  ,  G ) `  M )  =  (  seq 1 (  +  ,  H ) `  N ) )
 
Theoremsummodclem2a 10989* Lemma for summodc 10991. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> DECID  k  e.  A )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   &    |-  H  =  ( n  e.  NN  |->  if ( n  <_  N ,  [_ ( K `
  n )  /  k ]_ B ,  0 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  C_  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  f : ( 1 ...
 N ) -1-1-onto-> A )   &    |-  ( ph  ->  K 
 Isom  <  ,  <  (
 ( 1 ... ( `  A ) ) ,  A ) )   =>    |-  ( ph  ->  seq
 M (  +  ,  F )  ~~>  (  seq 1
 (  +  ,  G ) `  N ) )
 
Theoremsummodclem2 10990* Lemma for summodc 10991. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   =>    |-  ( ( ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  ,  F ) 
 ~~>  x ) )  ->  ( E. m  e.  NN  E. f ( f : ( 1 ... m )
 -1-1-onto-> A  /\  y  =  ( 
 seq 1 (  +  ,  G ) `  m ) )  ->  x  =  y ) )
 
Theoremsummodc 10991* A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  ( `  A ) , 
 [_ ( f `  n )  /  k ]_ B ,  0 ) )   =>    |-  ( ph  ->  E* x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  ,  F ) 
 ~~>  x )  \/  E. m  e.  NN  E. f
 ( f : ( 1 ... m ) -1-1-onto-> A 
 /\  x  =  ( 
 seq 1 (  +  ,  G ) `  m ) ) ) )
 
Theoremzsumdc 10992* Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.) (Revised by Jim Kingdon, 8-Apr-2023.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
 0 ) )   &    |-  ( ph  ->  A. x  e.  Z DECID  x  e.  A )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  (  ~~>  `  seq M (  +  ,  F ) ) )
 
Theoremisum 10993* Series sum with an upper integer index set (i.e. an infinite series). (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 7-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  Z  B  =  (  ~~>  `  seq M (  +  ,  F ) ) )
 
Theoremfsumgcl 10994* Closure for a function used to describe a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
 |-  ( k  =  ( F `  n ) 
 ->  B  =  C )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  n  e.  ( 1 ... M ) )  ->  ( G `
  n )  =  C )   =>    |-  ( ph  ->  A. n  e.  ( 1 ... M ) ( G `  n )  e.  CC )
 
Theoremfsum3 10995* The value of a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
 |-  ( k  =  ( F `  n ) 
 ->  B  =  C )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  n  e.  ( 1 ... M ) )  ->  ( G `
  n )  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  ( 
 seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  M ,  ( G `  n ) ,  0 )
 ) ) `  M ) )
 
Theoremsum0 10996 Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
 |- 
 sum_ k  e.  (/)  A  =  0
 
Theoremisumz 10997* Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
 |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>=
 `  M )DECID  j  e.  A )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  =  0 )
 
Theoremfsumf1o 10998* Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.)
 |-  ( k  =  G  ->  B  =  D )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  ( ph  ->  F : C -1-1-onto-> A )   &    |-  ( ( ph  /\  n  e.  C ) 
 ->  ( F `  n )  =  G )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ n  e.  C  D )
 
Theoremisumss 10999* Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 21-Sep-2022.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( B  \  A ) )  ->  C  =  0 )   &    |-  ( ph  ->  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  B 
 C_  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A. j  e.  ( ZZ>= `  M )DECID  j  e.  B )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremfisumss 11000* Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 23-Sep-2022.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( B  \  A ) )  ->  C  =  0 )   &    |-  ( ph  ->  A. j  e.  B DECID  j  e.  A )   &    |-  ( ph  ->  B  e.  Fin )   =>    |-  ( ph  ->  sum_
 k  e.  A  C  =  sum_ k  e.  B  C )
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