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Theorem fsumser 12250
Description: A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition of follows as fsum1 12261 and fsump1i 12279, which should make our notation clear and from which, along with closure fsumcl 12253, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
Hypotheses
Ref Expression
fsumser.1  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  =  A )
fsumser.2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
fsumser.3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
Assertion
Ref Expression
fsumser  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  (  seq 
M (  +  ,  F ) `  N
) )
Distinct variable groups:    k, F    k, M    k, N    ph, k
Allowed substitution hint:    A( k)

Proof of Theorem fsumser
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 eleq1 2376 . . . . . 6  |-  ( m  =  k  ->  (
m  e.  ( M ... N )  <->  k  e.  ( M ... N ) ) )
2 fveq2 5563 . . . . . 6  |-  ( m  =  k  ->  ( F `  m )  =  ( F `  k ) )
3 eqidd 2317 . . . . . 6  |-  ( m  =  k  ->  0  =  0 )
41, 2, 3ifbieq12d 3621 . . . . 5  |-  ( m  =  k  ->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 )  =  if ( k  e.  ( M ... N ) ,  ( F `  k ) ,  0 ) )
5 eqid 2316 . . . . 5  |-  ( m  e.  ( ZZ>= `  M
)  |->  if ( m  e.  ( M ... N ) ,  ( F `  m ) ,  0 ) )  =  ( m  e.  ( ZZ>= `  M )  |->  if ( m  e.  ( M ... N
) ,  ( F `
 m ) ,  0 ) )
6 fvex 5577 . . . . . 6  |-  ( F `
 k )  e. 
_V
7 c0ex 8877 . . . . . 6  |-  0  e.  _V
86, 7ifex 3657 . . . . 5  |-  if ( k  e.  ( M ... N ) ,  ( F `  k
) ,  0 )  e.  _V
94, 5, 8fvmpt 5640 . . . 4  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
m  e.  ( ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 ) ) `  k )  =  if ( k  e.  ( M ... N ) ,  ( F `  k ) ,  0 ) )
10 fsumser.1 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  =  A )
1110ifeq1da 3624 . . . 4  |-  ( ph  ->  if ( k  e.  ( M ... N
) ,  ( F `
 k ) ,  0 )  =  if ( k  e.  ( M ... N ) ,  A ,  0 ) )
129, 11sylan9eqr 2370 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
m  e.  ( ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 ) ) `  k )  =  if ( k  e.  ( M ... N ) ,  A ,  0 ) )
13 fsumser.2 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
14 fsumser.3 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
15 ssid 3231 . . . 4  |-  ( M ... N )  C_  ( M ... N )
1615a1i 10 . . 3  |-  ( ph  ->  ( M ... N
)  C_  ( M ... N ) )
1712, 13, 14, 16fsumsers 12248 . 2  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  (  seq 
M (  +  , 
( m  e.  (
ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m ) ,  0 ) ) ) `  N ) )
18 elfzuz 10841 . . . . . 6  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
1918, 9syl 15 . . . . 5  |-  ( k  e.  ( M ... N )  ->  (
( m  e.  (
ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m ) ,  0 ) ) `  k
)  =  if ( k  e.  ( M ... N ) ,  ( F `  k
) ,  0 ) )
20 iftrue 3605 . . . . 5  |-  ( k  e.  ( M ... N )  ->  if ( k  e.  ( M ... N ) ,  ( F `  k ) ,  0 )  =  ( F `
 k ) )
2119, 20eqtrd 2348 . . . 4  |-  ( k  e.  ( M ... N )  ->  (
( m  e.  (
ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m ) ,  0 ) ) `  k
)  =  ( F `
 k ) )
2221adantl 452 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
m  e.  ( ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 ) ) `  k )  =  ( F `  k ) )
2313, 22seqfveq 11117 . 2  |-  ( ph  ->  (  seq  M (  +  ,  ( m  e.  ( ZZ>= `  M
)  |->  if ( m  e.  ( M ... N ) ,  ( F `  m ) ,  0 ) ) ) `  N )  =  (  seq  M
(  +  ,  F
) `  N )
)
2417, 23eqtrd 2348 1  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  (  seq 
M (  +  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701    C_ wss 3186   ifcif 3599    e. cmpt 4114   ` cfv 5292  (class class class)co 5900   CCcc 8780   0cc0 8782    + caddc 8785   ZZ>=cuz 10277   ...cfz 10829    seq cseq 11093   sum_csu 12205
This theorem is referenced by:  isumclim3  12269  seqabs  12319  cvgcmpce  12323  isumsplit  12346  climcndslem1  12355  climcndslem2  12356  climcnds  12357  trireciplem  12367  geolim  12373  geo2lim  12378  mertenslem2  12388  mertens  12389  efcvgfsum  12414  effsumlt  12438  prmreclem6  13015  prmrec  13016  ovollb2lem  18900  ovoliunlem1  18914  ovoliun2  18918  ovolscalem1  18925  ovolicc2lem4  18932  uniioovol  18987  uniioombllem3  18993  uniioombllem6  18996  mtest  19834  psercn2  19852  pserdvlem2  19857  abelthlem6  19865  logfac  20007  emcllem5  20346  basellem8  20378  prmorcht  20469  pclogsum  20507  dchrisumlem2  20692  dchrmusum2  20696  dchrvmasumiflem1  20703  dchrisum0re  20715  dchrisum0lem1b  20717  dchrisum0lem2a  20719  dchrisum0lem2  20720  esumpcvgval  23644  esumcvg  23652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-oi 7270  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-fz 10830  df-fzo 10918  df-seq 11094  df-exp 11152  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-clim 12009  df-sum 12206
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