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Theorem fisumcvg 10730
Description: The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) Use fsum3cvg 10731 instead. (New usage is discouraged.)
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    k, N    ph, k    k, M   
k, F
Allowed substitution hint:    B( k)

Proof of Theorem fisumcvg
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 8997 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
42, 3syl 14 . 2  |-  ( ph  ->  N  e.  ZZ )
5 iseqex 9821 . . 3  |-  seq M
(  +  ,  F ,  CC )  e.  _V
65a1i 9 . 2  |-  ( ph  ->  seq M (  +  ,  F ,  CC )  e.  _V )
7 eqid 2088 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
8 eluzel2 8993 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
92, 8syl 14 . . . 4  |-  ( ph  ->  M  e.  ZZ )
10 eluzelz 8997 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
1110adantl 271 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ZZ )
12 iftrue 3394 . . . . . . . . . . 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 3397 . . . . . . . . 9  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
19 0cn 7459 . . . . . . . . 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 5370 . . . . . 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, 29iserf 9868 . . 3  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) : ( ZZ>= `  M
) --> CC )
3130, 2ffvelrnd 5419 . 2  |-  ( ph  ->  (  seq M (  +  ,  F ,  CC ) `  N )  e.  CC )
32 addid1 7599 . . . . 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 9405 . . . . . 6  |-  ( m  e.  ( ( N  +  1 ) ... n )  ->  m  e.  ( ZZ>= `  ( N  +  1 ) ) )
38 eluzelz 8997 . . . . . . . . 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 3022 . . . . . . . . . . 11  |-  ( ph  ->  ( m  e.  A  ->  m  e.  ( M ... N ) ) )
42 fznuz 9483 . . . . . . . . . . 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 3007 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ( ZZ  \  A ) )
47 fveq2 5289 . . . . . . . . 9  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
4847eqeq1d 2096 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  =  0  <->  ( F `  m )  =  0 ) )
49 eldifi 3120 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
50 eldifn 3121 . . . . . . . . . . . 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 2178 . . . . . . . . . 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 2120 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  0 )
5548, 54vtoclga 2685 . . . . . . 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 )
5947eleq1d 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 7446 . . . . 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, 58, 63, 65iseqid2 9906 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  +  ,  F ,  CC ) `  N )  =  (  seq M (  +  ,  F ,  CC ) `  n )
)
6766eqcomd 2093 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq M (  +  ,  F ,  CC ) `  n )  =  (  seq M (  +  ,  F ,  CC ) `  N )
)
681, 4, 6, 31, 67climconst 10642 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 664  DECID wdc 780    = wceq 1289    e. wcel 1438   A.wral 2359   _Vcvv 2619    \ cdif 2994    C_ wss 2997   ifcif 3389   class class class wbr 3837    |-> cmpt 3891   ` cfv 5002  (class class class)co 5634   CCcc 7327   0cc0 7329   1c1 7330    + caddc 7332   ZZcz 8720   ZZ>=cuz 8988   ...cfz 9393    seqcseq4 9816    ~~> cli 10630
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 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
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 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721  df-uz 8989  df-rp 9104  df-fz 9394  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-rsqrt 10396  df-abs 10397  df-clim 10631
This theorem is referenced by:  isummolem2a  10735  fisumcvg2  10750
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