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Theorem fsum3cvg2 11170
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 2281 . . . 4  |-  F/_ m if ( k  e.  A ,  B ,  0 )
2 nfv 1508 . . . . 5  |-  F/ k  m  e.  A
3 nfcsb1v 3035 . . . . 5  |-  F/_ k [_ m  /  k ]_ B
4 nfcv 2281 . . . . 5  |-  F/_ k
0
52, 3, 4nfif 3500 . . . 4  |-  F/_ k if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 )
6 eleq1w 2200 . . . . 5  |-  ( k  =  m  ->  (
k  e.  A  <->  m  e.  A ) )
7 csbeq1a 3012 . . . . 5  |-  ( k  =  m  ->  B  =  [_ m  /  k ]_ B )
86, 7ifbieq1d 3494 . . . 4  |-  ( k  =  m  ->  if ( k  e.  A ,  B ,  0 )  =  if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 ) )
91, 5, 8cbvmpt 4023 . . 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 2505 . . . 4  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
123nfel1 2292 . . . . 5  |-  F/ k
[_ m  /  k ]_ B  e.  CC
137eleq1d 2208 . . . . 5  |-  ( k  =  m  ->  ( B  e.  CC  <->  [_ m  / 
k ]_ B  e.  CC ) )
1412, 13rspc 2783 . . . 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 823 . . . 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 2505 . . . . 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 2794 . . 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 11154 . 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 9338 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
2622, 25syl 14 . . 3  |-  ( ph  ->  M  e.  ZZ )
27 fveq2 5421 . . . . 5  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
2827eleq1d 2208 . . . 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 468 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  B  e.  CC )
31 0cnd 7766 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  -.  k  e.  A )  ->  0  e.  CC )
3230, 31, 17ifcldadc 3501 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  if (
k  e.  A ,  B ,  0 )  e.  CC )
3329, 32eqeltrd 2216 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3433ralrimiva 2505 . . . . 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 2794 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  CC )
38 eluzelz 9342 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
39 eqid 2139 . . . . . . . 8  |-  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
4039fvmpt2 5504 . . . . . . 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 583 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k )  =  if ( k  e.  A ,  B ,  0 ) )
4229, 41eqtr4d 2175 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k ) )
4342ralrimiva 2505 . . . 4  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k ) )
44 nffvmpt1 5432 . . . . . 6  |-  F/_ k
( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `
 n )
4544nfeq2 2293 . . . . 5  |-  F/ k ( F `  n
)  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n )
46 fveq2 5421 . . . . . 6  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
47 fveq2 5421 . . . . . 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 2154 . . . . 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 2783 . . . 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 7752 . . . 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 10252 . 2  |-  ( ph  ->  seq M (  +  ,  F )  =  seq M (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ) )
5453fveq1d 5423 . 2  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  =  (  seq M (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ) `  N
) )
5524, 53, 543brtr4d 3960 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 819    = wceq 1331    e. wcel 1480   A.wral 2416   [_csb 3003    C_ wss 3071   ifcif 3474   class class class wbr 3929    |-> cmpt 3989   ` cfv 5123  (class class class)co 5774   CCcc 7625   0cc0 7627    + caddc 7630   ZZcz 9061   ZZ>=cuz 9333   ...cfz 9797    seqcseq 10225    ~~> cli 11054
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-n0 8985  df-z 9062  df-uz 9334  df-rp 9449  df-fz 9798  df-seqfrec 10226  df-exp 10300  df-cj 10621  df-rsqrt 10777  df-abs 10778  df-clim 11055
This theorem is referenced by:  fsumsersdc  11171  fsum3cvg3  11172  ef0lem  11373
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