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Theorem fsum3cvg2 11429
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 2332 . . . 4  |-  F/_ m if ( k  e.  A ,  B ,  0 )
2 nfv 1539 . . . . 5  |-  F/ k  m  e.  A
3 nfcsb1v 3105 . . . . 5  |-  F/_ k [_ m  /  k ]_ B
4 nfcv 2332 . . . . 5  |-  F/_ k
0
52, 3, 4nfif 3577 . . . 4  |-  F/_ k if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 )
6 eleq1w 2250 . . . . 5  |-  ( k  =  m  ->  (
k  e.  A  <->  m  e.  A ) )
7 csbeq1a 3081 . . . . 5  |-  ( k  =  m  ->  B  =  [_ m  /  k ]_ B )
86, 7ifbieq1d 3571 . . . 4  |-  ( k  =  m  ->  if ( k  e.  A ,  B ,  0 )  =  if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 ) )
91, 5, 8cbvmpt 4113 . . 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 2563 . . . 4  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
123nfel1 2343 . . . . 5  |-  F/ k
[_ m  /  k ]_ B  e.  CC
137eleq1d 2258 . . . . 5  |-  ( k  =  m  ->  ( B  e.  CC  <->  [_ m  / 
k ]_ B  e.  CC ) )
1412, 13rspc 2850 . . . 4  |-  ( m  e.  A  ->  ( A. k  e.  A  B  e.  CC  ->  [_ m  /  k ]_ B  e.  CC )
)
1511, 14mpan9 281 . . 3  |-  ( (
ph  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  CC )
166dcbid 839 . . . 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 2563 . . . . 5  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M )DECID  k  e.  A )
1918adantr 276 . . . 4  |-  ( (
ph  /\  m  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )DECID  k  e.  A )
20 simpr 110 . . . 4  |-  ( (
ph  /\  m  e.  ( ZZ>= `  M )
)  ->  m  e.  ( ZZ>= `  M )
)
2116, 19, 20rspcdva 2861 . . 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 11413 . 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 9558 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
2622, 25syl 14 . . 3  |-  ( ph  ->  M  e.  ZZ )
27 fveq2 5531 . . . . 5  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
2827eleq1d 2258 . . . 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 477 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  B  e.  CC )
31 0cnd 7975 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  -.  k  e.  A )  ->  0  e.  CC )
3230, 31, 17ifcldadc 3578 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  if (
k  e.  A ,  B ,  0 )  e.  CC )
3329, 32eqeltrd 2266 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3433ralrimiva 2563 . . . . 5  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  e.  CC )
3534adantr 276 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )
( F `  k
)  e.  CC )
36 simpr 110 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  ( ZZ>= `  M )
)
3728, 35, 36rspcdva 2861 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  CC )
38 eluzelz 9562 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
39 eqid 2189 . . . . . . . 8  |-  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
4039fvmpt2 5616 . . . . . . 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 594 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k )  =  if ( k  e.  A ,  B ,  0 ) )
4229, 41eqtr4d 2225 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k ) )
4342ralrimiva 2563 . . . 4  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k ) )
44 nffvmpt1 5542 . . . . . 6  |-  F/_ k
( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `
 n )
4544nfeq2 2344 . . . . 5  |-  F/ k ( F `  n
)  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n )
46 fveq2 5531 . . . . . 6  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
47 fveq2 5531 . . . . . 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 2204 . . . . 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 2850 . . . 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 281 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( F `  n )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n ) )
51 addcl 7961 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
5251adantl 277 . . 3  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
5326, 37, 50, 52seq3feq 10498 . 2  |-  ( ph  ->  seq M (  +  ,  F )  =  seq M (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ) )
5453fveq1d 5533 . 2  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  =  (  seq M (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ) `  N
) )
5524, 53, 543brtr4d 4050 1  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  (  seq M (  +  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104  DECID wdc 835    = wceq 1364    e. wcel 2160   A.wral 2468   [_csb 3072    C_ wss 3144   ifcif 3549   class class class wbr 4018    |-> cmpt 4079   ` cfv 5232  (class class class)co 5892   CCcc 7834   0cc0 7836    + caddc 7839   ZZcz 9278   ZZ>=cuz 9553   ...cfz 10033    seqcseq 10471    ~~> cli 11313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-mulrcl 7935  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-precex 7946  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-apti 7951  ax-pre-ltadd 7952  ax-pre-mulgt0 7953  ax-pre-mulext 7954
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-frec 6411  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-reap 8557  df-ap 8564  df-div 8655  df-inn 8945  df-2 9003  df-n0 9202  df-z 9279  df-uz 9554  df-rp 9679  df-fz 10034  df-seqfrec 10472  df-exp 10546  df-cj 10878  df-rsqrt 11034  df-abs 11035  df-clim 11314
This theorem is referenced by:  fsumsersdc  11430  fsum3cvg3  11431  ef0lem  11695
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