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Theorem fsumcvg2 12200
Description: The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
Hypotheses
Ref Expression
fsumsers.1  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )
fsumsers.2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
fsumsers.3  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
fsumsers.4  |-  ( ph  ->  A  C_  ( M ... N ) )
Assertion
Ref Expression
fsumcvg2  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  (  seq 
M (  +  ,  F ) `  N
) )
Distinct variable groups:    A, k    k, F    k, N    ph, k    k, M
Allowed substitution hint:    B( k)

Proof of Theorem fsumcvg2
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2419 . . . 4  |-  F/_ m if ( k  e.  A ,  B ,  0 )
2 nfv 1605 . . . . 5  |-  F/ k  m  e.  A
3 nfcsb1v 3113 . . . . 5  |-  F/_ k [_ m  /  k ]_ B
4 nfcv 2419 . . . . 5  |-  F/_ k
0
52, 3, 4nfif 3589 . . . 4  |-  F/_ k if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 )
6 eleq1 2343 . . . . 5  |-  ( k  =  m  ->  (
k  e.  A  <->  m  e.  A ) )
7 csbeq1a 3089 . . . . 5  |-  ( k  =  m  ->  B  =  [_ m  /  k ]_ B )
8 eqidd 2284 . . . . 5  |-  ( k  =  m  ->  0  =  0 )
96, 7, 8ifbieq12d 3587 . . . 4  |-  ( k  =  m  ->  if ( k  e.  A ,  B ,  0 )  =  if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 ) )
101, 5, 9cbvmpt 4110 . . 3  |-  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )  =  ( m  e.  ZZ  |->  if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 ) )
11 fsumsers.3 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
1211ralrimiva 2626 . . . 4  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
133nfel1 2429 . . . . 5  |-  F/ k
[_ m  /  k ]_ B  e.  CC
147eleq1d 2349 . . . . 5  |-  ( k  =  m  ->  ( B  e.  CC  <->  [_ m  / 
k ]_ B  e.  CC ) )
1513, 14rspc 2878 . . . 4  |-  ( m  e.  A  ->  ( A. k  e.  A  B  e.  CC  ->  [_ m  /  k ]_ B  e.  CC )
)
1612, 15mpan9 455 . . 3  |-  ( (
ph  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  CC )
17 fsumsers.2 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
18 fsumsers.4 . . 3  |-  ( ph  ->  A  C_  ( M ... N ) )
1910, 16, 17, 18fsumcvg 12185 . 2  |-  ( ph  ->  seq  M (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  (  seq  M
(  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ) `  N
) )
20 eluzel2 10235 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
2117, 20syl 15 . . 3  |-  ( ph  ->  M  e.  ZZ )
22 fsumsers.1 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )
23 eluzelz 10238 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
24 iftrue 3571 . . . . . . . . . . 11  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  B )
2524adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  0 )  =  B )
2625, 11eqeltrd 2357 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
2726ex 423 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  ->  if ( k  e.  A ,  B , 
0 )  e.  CC ) )
28 iffalse 3572 . . . . . . . . 9  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
29 0cn 8831 . . . . . . . . 9  |-  0  e.  CC
3028, 29syl6eqel 2371 . . . . . . . 8  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
3127, 30pm2.61d1 151 . . . . . . 7  |-  ( ph  ->  if ( k  e.  A ,  B , 
0 )  e.  CC )
32 eqid 2283 . . . . . . . 8  |-  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
3332fvmpt2 5608 . . . . . . 7  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  0 )  e.  CC )  -> 
( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `
 k )  =  if ( k  e.  A ,  B , 
0 ) )
3423, 31, 33syl2anr 464 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k )  =  if ( k  e.  A ,  B ,  0 ) )
3522, 34eqtr4d 2318 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k ) )
3635ralrimiva 2626 . . . 4  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k ) )
37 nfmpt1 4109 . . . . . . 7  |-  F/_ k
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
38 nfcv 2419 . . . . . . 7  |-  F/_ k
n
3937, 38nffv 5532 . . . . . 6  |-  F/_ k
( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `
 n )
4039nfeq2 2430 . . . . 5  |-  F/ k ( F `  n
)  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n )
41 fveq2 5525 . . . . . 6  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
42 fveq2 5525 . . . . . 6  |-  ( k  =  n  ->  (
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n ) )
4341, 42eqeq12d 2297 . . . . 5  |-  ( k  =  n  ->  (
( F `  k
)  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k )  <-> 
( F `  n
)  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n ) ) )
4440, 43rspc 2878 . . . 4  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( A. k  e.  ( ZZ>= `  M ) ( F `
 k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `
 k )  -> 
( F `  n
)  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n ) ) )
4536, 44mpan9 455 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( F `  n )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n ) )
4621, 45seqfeq 11071 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  =  seq  M (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ) )
4746fveq1d 5527 . 2  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  =  (  seq  M (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ) `  N
) )
4819, 46, 473brtr4d 4053 1  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  (  seq 
M (  +  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   [_csb 3081    C_ wss 3152   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737    + caddc 8740   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046    ~~> cli 11958
This theorem is referenced by:  fsumsers  12201  fsumcvg3  12202  ef0lem  12360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962
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