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Theorem fisumcvg2 10786
Description: The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) Use fsum3cvg2 10787 instead. (New usage is discouraged.)
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
fisumcvg2  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  (  seq M (  +  ,  F ,  CC ) `  N ) )
Distinct variable groups:    A, k    k, F    k, M    k, N    ph, k
Allowed substitution hint:    B( k)

Proof of Theorem fisumcvg2
Dummy variables  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2228 . . . 4  |-  F/_ m if ( k  e.  A ,  B ,  0 )
2 nfv 1466 . . . . 5  |-  F/ k  m  e.  A
3 nfcsb1v 2963 . . . . 5  |-  F/_ k [_ m  /  k ]_ B
4 nfcv 2228 . . . . 5  |-  F/_ k
0
52, 3, 4nfif 3419 . . . 4  |-  F/_ k if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 )
6 eleq1w 2148 . . . . 5  |-  ( k  =  m  ->  (
k  e.  A  <->  m  e.  A ) )
7 csbeq1a 2941 . . . . 5  |-  ( k  =  m  ->  B  =  [_ m  /  k ]_ B )
86, 7ifbieq1d 3413 . . . 4  |-  ( k  =  m  ->  if ( k  e.  A ,  B ,  0 )  =  if ( m  e.  A ,  [_ m  /  k ]_ B ,  0 ) )
91, 5, 8cbvmpt 3933 . . 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 2446 . . . 4  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
123nfel1 2239 . . . . 5  |-  F/ k
[_ m  /  k ]_ B  e.  CC
137eleq1d 2156 . . . . 5  |-  ( k  =  m  ->  ( B  e.  CC  <->  [_ m  / 
k ]_ B  e.  CC ) )
1412, 13rspc 2716 . . . 4  |-  ( m  e.  A  ->  ( A. k  e.  A  B  e.  CC  ->  [_ m  /  k ]_ B  e.  CC )
)
1511, 14mpan9 275 . . 3  |-  ( (
ph  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  CC )
166dcbid 786 . . . 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 2446 . . . . 5  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M )DECID  k  e.  A )
1918adantr 270 . . . 4  |-  ( (
ph  /\  m  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )DECID  k  e.  A )
20 simpr 108 . . . 4  |-  ( (
ph  /\  m  e.  ( ZZ>= `  M )
)  ->  m  e.  ( ZZ>= `  M )
)
2116, 19, 20rspcdva 2727 . . 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, 23fisumcvg 10766 . 2  |-  ( ph  ->  seq M (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ,  CC )  ~~>  (  seq M (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ,  CC ) `
 N ) )
25 eluzel2 9024 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
2622, 25syl 14 . . 3  |-  ( ph  ->  M  e.  ZZ )
27 fveq2 5305 . . . . 5  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
2827eleq1d 2156 . . . 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 461 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  B  e.  CC )
31 0cnd 7481 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  -.  k  e.  A )  ->  0  e.  CC )
3230, 31, 17ifcldadc 3420 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  if (
k  e.  A ,  B ,  0 )  e.  CC )
3329, 32eqeltrd 2164 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3433ralrimiva 2446 . . . . 5  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  e.  CC )
3534adantr 270 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )
( F `  k
)  e.  CC )
36 simpr 108 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  ( ZZ>= `  M )
)
3728, 35, 36rspcdva 2727 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  CC )
38 eluzelz 9028 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
39 eqid 2088 . . . . . . . 8  |-  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
4039fvmpt2 5386 . . . . . . 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 561 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k )  =  if ( k  e.  A ,  B ,  0 ) )
4229, 41eqtr4d 2123 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k ) )
4342ralrimiva 2446 . . . 4  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  k ) )
44 nffvmpt1 5316 . . . . . 6  |-  F/_ k
( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `
 n )
4544nfeq2 2240 . . . . 5  |-  F/ k ( F `  n
)  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n )
46 fveq2 5305 . . . . . 6  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
47 fveq2 5305 . . . . . 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 2102 . . . . 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 2716 . . . 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 275 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( F `  n )  =  ( ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) `  n ) )
51 addcl 7467 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
5251adantl 271 . . 3  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
5326, 37, 50, 52iseqfeq 9896 . 2  |-  ( ph  ->  seq M (  +  ,  F ,  CC )  =  seq M (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ,  CC ) )
5453fveq1d 5307 . 2  |-  ( ph  ->  (  seq M (  +  ,  F ,  CC ) `  N )  =  (  seq M
(  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) ,  CC ) `
 N ) )
5524, 53, 543brtr4d 3875 1  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  (  seq M (  +  ,  F ,  CC ) `  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102  DECID wdc 780    = wceq 1289    e. wcel 1438   A.wral 2359   [_csb 2933    C_ wss 2999   ifcif 3393   class class class wbr 3845    |-> cmpt 3899   ` cfv 5015  (class class class)co 5652   CCcc 7348   0cc0 7350    + caddc 7353   ZZcz 8750   ZZ>=cuz 9019   ...cfz 9424    seqcseq4 9851    ~~> cli 10666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-n0 8674  df-z 8751  df-uz 9020  df-rp 9135  df-fz 9425  df-iseq 9853  df-seq3 9854  df-exp 9955  df-cj 10276  df-rsqrt 10431  df-abs 10432  df-clim 10667
This theorem is referenced by:  fisumsers  10788
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