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Theorem fisumcvg 10572
Description: The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
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
fisumcvg  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  (  seq M (  +  ,  F ,  CC ) `  N ) )
Distinct variable groups:    A, k    B, k    k, N    ph, k    k, M    k, F

Proof of Theorem fisumcvg
Dummy variables  n  z  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2083 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 isumrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
3 eluzelz 8921 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
42, 3syl 14 . 2  |-  ( ph  ->  N  e.  ZZ )
5 iseqex 9740 . . 3  |-  seq M
(  +  ,  F ,  CC )  e.  _V
65a1i 9 . 2  |-  ( ph  ->  seq M (  +  ,  F ,  CC )  e.  _V )
7 eqid 2083 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
8 eluzel2 8917 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
92, 8syl 14 . . . 4  |-  ( ph  ->  M  e.  ZZ )
10 eluzelz 8921 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
1110adantl 271 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ZZ )
12 iftrue 3378 . . . . . . . . . . 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 2159 . . . . . . . . 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 3381 . . . . . . . . 9  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
19 0cn 7381 . . . . . . . . 9  |-  0  e.  CC
2018, 19syl6eqel 2173 . . . . . . . 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 778 . . . . . . . 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 671 . . . . . 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 5329 . . . . . 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 2159 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
307, 9, 29iserf 9766 . . 3  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) : ( ZZ>= `  M
) --> CC )
3130, 2ffvelrnd 5378 . 2  |-  ( ph  ->  (  seq M (  +  ,  F ,  CC ) `  N )  e.  CC )
32 addid1 7521 . . . . 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 ,  CC ) `  N )  e.  CC )
37 elfzuz 9329 . . . . . 6  |-  ( m  e.  ( ( N  +  1 ) ... n )  ->  m  e.  ( ZZ>= `  ( N  +  1 ) ) )
38 eluzelz 8921 . . . . . . . . 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 3009 . . . . . . . . . . 11  |-  ( ph  ->  ( m  e.  A  ->  m  e.  ( M ... N ) ) )
42 fznuz 9407 . . . . . . . . . . 11  |-  ( m  e.  ( M ... N )  ->  -.  m  e.  ( ZZ>= `  ( N  +  1
) ) )
4341, 42syl6 33 . . . . . . . . . 10  |-  ( ph  ->  ( m  e.  A  ->  -.  m  e.  (
ZZ>= `  ( N  + 
1 ) ) ) )
4443con2d 587 . . . . . . . . 9  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( N  + 
1 ) )  ->  -.  m  e.  A
) )
4544imp 122 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  -.  m  e.  A )
4639, 45eldifd 2994 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ( ZZ  \  A ) )
47 fveq2 5251 . . . . . . . . 9  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
4847eqeq1d 2091 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  =  0  <->  ( F `  m )  =  0 ) )
49 eldifi 3106 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
50 eldifn 3107 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ  \  A )  ->  -.  k  e.  A )
5150, 18syl 14 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
5251, 19syl6eqel 2173 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
5349, 52, 27syl2anc 403 . . . . . . . . 9  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
0 ) )
5453, 51eqtrd 2115 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  0 )
5548, 54vtoclga 2675 . . . . . . 7  |-  ( m  e.  ( ZZ  \  A )  ->  ( F `  m )  =  0 )
5646, 55syl 14 . . . . . 6  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  m )  =  0 )
5737, 56sylan2 280 . . . . 5  |-  ( (
ph  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  0 )
5857adantlr 461 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  0 )
59 cnex 7367 . . . . 5  |-  CC  e.  _V
6059a1i 9 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  CC  e.  _V )
6147eleq1d 2151 . . . . 5  |-  ( k  =  m  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
6229ralrimiva 2440 . . . . . 6  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  e.  CC )
6362ad2antrr 472 . . . . 5  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )
( F `  k
)  e.  CC )
64 simpr 108 . . . . 5  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  m  e.  ( ZZ>= `  M )
)
6561, 63, 64rspcdva 2717 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ZZ>= `  M )
)  ->  ( F `  m )  e.  CC )
66 addcl 7368 . . . . 5  |-  ( ( m  e.  CC  /\  z  e.  CC )  ->  ( m  +  z )  e.  CC )
6766adantl 271 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  ( m  e.  CC  /\  z  e.  CC ) )  -> 
( m  +  z )  e.  CC )
6833, 34, 35, 36, 58, 60, 65, 67iseqid2 9781 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  +  ,  F ,  CC ) `  N )  =  (  seq M (  +  ,  F ,  CC ) `  n )
)
6968eqcomd 2088 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  +  ,  F ,  CC ) `  n )  =  (  seq M (  +  ,  F ,  CC ) `  N )
)
701, 4, 6, 31, 69climconst 10501 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    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434   A.wral 2353   _Vcvv 2612    \ cdif 2981    C_ wss 2984   ifcif 3373   class class class wbr 3811    |-> cmpt 3865   ` cfv 4967  (class class class)co 5589   CCcc 7249   0cc0 7251   1c1 7252    + caddc 7254   ZZcz 8644   ZZ>=cuz 8912   ...cfz 9317    seqcseq 9738    ~~> cli 10489
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7337  ax-resscn 7338  ax-1cn 7339  ax-1re 7340  ax-icn 7341  ax-addcl 7342  ax-addrcl 7343  ax-mulcl 7344  ax-mulrcl 7345  ax-addcom 7346  ax-mulcom 7347  ax-addass 7348  ax-mulass 7349  ax-distr 7350  ax-i2m1 7351  ax-0lt1 7352  ax-1rid 7353  ax-0id 7354  ax-rnegex 7355  ax-precex 7356  ax-cnre 7357  ax-pre-ltirr 7358  ax-pre-ltwlin 7359  ax-pre-lttrn 7360  ax-pre-apti 7361  ax-pre-ltadd 7362  ax-pre-mulgt0 7363  ax-pre-mulext 7364
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-riota 5545  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-frec 6086  df-pnf 7425  df-mnf 7426  df-xr 7427  df-ltxr 7428  df-le 7429  df-sub 7556  df-neg 7557  df-reap 7950  df-ap 7957  df-div 8036  df-inn 8315  df-2 8373  df-n0 8564  df-z 8645  df-uz 8913  df-rp 9028  df-fz 9318  df-iseq 9739  df-iexp 9790  df-cj 10101  df-rsqrt 10256  df-abs 10257  df-clim 10490
This theorem is referenced by: (None)
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