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Theorem fsum3cvg2 11291
Description: 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, 2-Dec-2022.)
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.dc  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  -> DECID  k  e.  A
)
fsumsers.4  |-  ( ph  ->  A  C_  ( M ... N ) )
Assertion
Ref Expression
fsum3cvg2  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  (  seq M (  +  ,  F ) `  N
) )
Distinct variable groups:    A, k    k, F    k, M    k, N    ph, k
Allowed substitution hint:    B( k)

Proof of Theorem fsum3cvg2
Dummy variables  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2299 . . . 4  |-  F/_ m if ( k  e.  A ,  B ,  0 )
2 nfv 1508 . . . . 5  |-  F/ k  m  e.  A
3 nfcsb1v 3064 . . . . 5  |-  F/_ k [_ m  /  k ]_ B
4 nfcv 2299 . . . . 5  |-  F/_ k
0
52, 3, 4nfif 3533 . . . 4  |-  F/_ k if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 )
6 eleq1w 2218 . . . . 5  |-  ( k  =  m  ->  (
k  e.  A  <->  m  e.  A ) )
7 csbeq1a 3040 . . . . 5  |-  ( k  =  m  ->  B  =  [_ m  /  k ]_ B )
86, 7ifbieq1d 3527 . . . 4  |-  ( k  =  m  ->  if ( k  e.  A ,  B ,  0 )  =  if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 ) )
91, 5, 8cbvmpt 4059 . . 3  |-  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )  =  ( m  e.  ZZ  |->  if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 ) )
10 fsumsers.3 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
1110ralrimiva 2530 . . . 4  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
123nfel1 2310 . . . . 5  |-  F/ k
[_ m  /  k ]_ B  e.  CC
137eleq1d 2226 . . . . 5  |-  ( k  =  m  ->  ( B  e.  CC  <->  [_ m  / 
k ]_ B  e.  CC ) )
1412, 13rspc 2810 . . . 4  |-  ( m  e.  A  ->  ( A. k  e.  A  B  e.  CC  ->  [_ m  /  k ]_ B  e.  CC )
)
1511, 14mpan9 279 . . 3  |-  ( (
ph  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  CC )
166dcbid 824 . . . 4  |-  ( k  =  m  ->  (DECID  k  e.  A  <-> DECID  m  e.  A )
)
17 fsumsers.dc . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  -> DECID  k  e.  A
)
1817ralrimiva 2530 . . . . 5  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M )DECID  k  e.  A )
1918adantr 274 . . . 4  |-  ( (
ph  /\  m  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )DECID  k  e.  A )
20 simpr 109 . . . 4  |-  ( (
ph  /\  m  e.  ( ZZ>= `  M )
)  ->  m  e.  ( ZZ>= `  M )
)
2116, 19, 20rspcdva 2821 . . 3  |-  ( (
ph  /\  m  e.  ( ZZ>= `  M )
)  -> DECID  m  e.  A
)
22 fsumsers.2 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
23 fsumsers.4 . . 3  |-  ( ph  ->  A  C_  ( M ... N ) )
249, 15, 21, 22, 23fsum3cvg 11275 . 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
) )
25 eluzel2 9444 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
2622, 25syl 14 . . 3  |-  ( ph  ->  M  e.  ZZ )
27 fveq2 5468 . . . . 5  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
2827eleq1d 2226 . . . 4  |-  ( k  =  x  ->  (
( F `  k
)  e.  CC  <->  ( F `  x )  e.  CC ) )
29 fsumsers.1 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )
3010adantlr 469 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  B  e.  CC )
31 0cnd 7871 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  -.  k  e.  A )  ->  0  e.  CC )
3230, 31, 17ifcldadc 3534 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  if (
k  e.  A ,  B ,  0 )  e.  CC )
3329, 32eqeltrd 2234 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3433ralrimiva 2530 . . . . 5  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  e.  CC )
3534adantr 274 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )
( F `  k
)  e.  CC )
36 simpr 109 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  ( ZZ>= `  M )
)
3728, 35, 36rspcdva 2821 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  CC )
38 eluzelz 9448 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
39 eqid 2157 . . . . . . . 8  |-  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
4039fvmpt2 5551 . . . . . . 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 ) )
4138, 32, 40syl2an2 584 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k )  =  if ( k  e.  A ,  B ,  0 ) )
4229, 41eqtr4d 2193 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k ) )
4342ralrimiva 2530 . . . 4  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k ) )
44 nffvmpt1 5479 . . . . . 6  |-  F/_ k
( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `
 n )
4544nfeq2 2311 . . . . 5  |-  F/ k ( F `  n
)  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n )
46 fveq2 5468 . . . . . 6  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
47 fveq2 5468 . . . . . 6  |-  ( k  =  n  ->  (
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n ) )
4846, 47eqeq12d 2172 . . . . 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 ) ) )
4945, 48rspc 2810 . . . 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 ) ) )
5043, 49mpan9 279 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( F `  n )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n ) )
51 addcl 7857 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
5251adantl 275 . . 3  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
5326, 37, 50, 52seq3feq 10371 . 2  |-  ( ph  ->  seq M (  +  ,  F )  =  seq M (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ) )
5453fveq1d 5470 . 2  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  =  (  seq M (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ) `  N
) )
5524, 53, 543brtr4d 3996 1  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  (  seq M (  +  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103  DECID wdc 820    = wceq 1335    e. wcel 2128   A.wral 2435   [_csb 3031    C_ wss 3102   ifcif 3505   class class class wbr 3965    |-> cmpt 4025   ` cfv 5170  (class class class)co 5824   CCcc 7730   0cc0 7732    + caddc 7735   ZZcz 9167   ZZ>=cuz 9439   ...cfz 9912    seqcseq 10344    ~~> cli 11175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-n0 9091  df-z 9168  df-uz 9440  df-rp 9561  df-fz 9913  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-rsqrt 10898  df-abs 10899  df-clim 11176
This theorem is referenced by:  fsumsersdc  11292  fsum3cvg3  11293  ef0lem  11557
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