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Theorem fsum3cvg 10767
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, 12-Nov-2022.)
Hypotheses
Ref Expression
isummo.1  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
isummo.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
isummo.dc  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  -> DECID  k  e.  A
)
isumrb.3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
fisumcvg.4  |-  ( ph  ->  A  C_  ( M ... N ) )
Assertion
Ref Expression
fsum3cvg  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  (  seq M (  +  ,  F ) `  N
) )
Distinct variable groups:    A, k    k, N    ph, k    k, M   
k, F
Allowed substitution hint:    B( k)

Proof of Theorem fsum3cvg
Dummy variables  n  z  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2088 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 isumrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
3 eluzelz 9028 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
42, 3syl 14 . 2  |-  ( ph  ->  N  e.  ZZ )
5 seqex 9857 . . 3  |-  seq M
(  +  ,  F
)  e.  _V
65a1i 9 . 2  |-  ( ph  ->  seq M (  +  ,  F )  e. 
_V )
7 eqid 2088 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
8 eluzel2 9024 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
92, 8syl 14 . . . 4  |-  ( ph  ->  M  e.  ZZ )
10 eluzelz 9028 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
1110adantl 271 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ZZ )
12 iftrue 3398 . . . . . . . . . . 11  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  B )
1312adantl 271 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  0 )  =  B )
14 isummo.2 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
1513, 14eqeltrd 2164 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
1615ex 113 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  ->  if ( k  e.  A ,  B , 
0 )  e.  CC ) )
1716adantr 270 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  e.  CC ) )
18 iffalse 3401 . . . . . . . . 9  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
19 0cn 7480 . . . . . . . . 9  |-  0  e.  CC
2018, 19syl6eqel 2178 . . . . . . . 8  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
2120a1i 9 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  e.  CC ) )
22 isummo.dc . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  -> DECID  k  e.  A
)
23 exmiddc 782 . . . . . . . 8  |-  (DECID  k  e.  A  ->  ( k  e.  A  \/  -.  k  e.  A )
)
2422, 23syl 14 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( k  e.  A  \/  -.  k  e.  A )
)
2517, 21, 24mpjaod 673 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  if (
k  e.  A ,  B ,  0 )  e.  CC )
26 isummo.1 . . . . . . 7  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
2726fvmpt2 5386 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  0 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  0 ) )
2811, 25, 27syl2anc 403 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )
2928, 25eqeltrd 2164 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
307, 9, 29serf 9900 . . 3  |-  ( ph  ->  seq M (  +  ,  F ) : ( ZZ>= `  M ) --> CC )
3130, 2ffvelrnd 5435 . 2  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  e.  CC )
32 addid1 7620 . . . . 5  |-  ( m  e.  CC  ->  (
m  +  0 )  =  m )
3332adantl 271 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  CC )  ->  ( m  +  0 )  =  m )
342adantr 270 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  N  e.  ( ZZ>= `  M )
)
35 simpr 108 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ( ZZ>= `  N )
)
3631adantr 270 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  +  ,  F ) `  N
)  e.  CC )
37 elfzuz 9436 . . . . . 6  |-  ( m  e.  ( ( N  +  1 ) ... n )  ->  m  e.  ( ZZ>= `  ( N  +  1 ) ) )
38 eluzelz 9028 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  ( N  +  1 ) )  ->  m  e.  ZZ )
3938adantl 271 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ZZ )
40 fisumcvg.4 . . . . . . . . . . . 12  |-  ( ph  ->  A  C_  ( M ... N ) )
4140sseld 3024 . . . . . . . . . . 11  |-  ( ph  ->  ( m  e.  A  ->  m  e.  ( M ... N ) ) )
42 fznuz 9516 . . . . . . . . . . 11  |-  ( m  e.  ( M ... N )  ->  -.  m  e.  ( ZZ>= `  ( N  +  1
) ) )
4341, 42syl6 33 . . . . . . . . . 10  |-  ( ph  ->  ( m  e.  A  ->  -.  m  e.  (
ZZ>= `  ( N  + 
1 ) ) ) )
4443con2d 589 . . . . . . . . 9  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( N  + 
1 ) )  ->  -.  m  e.  A
) )
4544imp 122 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  -.  m  e.  A )
4639, 45eldifd 3009 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ( ZZ  \  A ) )
47 fveqeq2 5314 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  =  0  <->  ( F `  m )  =  0 ) )
48 eldifi 3122 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
49 eldifn 3123 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ  \  A )  ->  -.  k  e.  A )
5049, 18syl 14 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
5150, 19syl6eqel 2178 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
5248, 51, 27syl2anc 403 . . . . . . . . 9  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
0 ) )
5352, 50eqtrd 2120 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  0 )
5447, 53vtoclga 2685 . . . . . . 7  |-  ( m  e.  ( ZZ  \  A )  ->  ( F `  m )  =  0 )
5546, 54syl 14 . . . . . 6  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  m )  =  0 )
5637, 55sylan2 280 . . . . 5  |-  ( (
ph  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  0 )
5756adantlr 461 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  0 )
58 fveq2 5305 . . . . . 6  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
5958eleq1d 2156 . . . . 5  |-  ( k  =  m  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
6029ralrimiva 2446 . . . . . 6  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  e.  CC )
6160ad2antrr 472 . . . . 5  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )
( F `  k
)  e.  CC )
62 simpr 108 . . . . 5  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  m  e.  ( ZZ>= `  M )
)
6359, 61, 62rspcdva 2727 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  ( F `  m )  e.  CC )
64 addcl 7467 . . . . 5  |-  ( ( m  e.  CC  /\  z  e.  CC )  ->  ( m  +  z )  e.  CC )
6564adantl 271 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  ( m  e.  CC  /\  z  e.  CC ) )  -> 
( m  +  z )  e.  CC )
6633, 34, 35, 36, 57, 63, 65seq3id2 9940 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  +  ,  F ) `  N
)  =  (  seq M (  +  ,  F ) `  n
) )
6766eqcomd 2093 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  +  ,  F ) `  n
)  =  (  seq M (  +  ,  F ) `  N
) )
681, 4, 6, 31, 67climconst 10678 1  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  (  seq M (  +  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 664  DECID wdc 780    = wceq 1289    e. wcel 1438   A.wral 2359   _Vcvv 2619    \ cdif 2996    C_ wss 2999   ifcif 3393   class class class wbr 3845    |-> cmpt 3899   ` cfv 5015  (class class class)co 5652   CCcc 7348   0cc0 7350   1c1 7351    + caddc 7353   ZZcz 8750   ZZ>=cuz 9019   ...cfz 9424    seqcseq 9852    ~~> 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:  fsum3cvg2  10787
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