MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsumcvg Structured version   Unicode version

Theorem fsumcvg 12498
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
summo.1  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
summo.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
sumrb.3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
fsumcvg.4  |-  ( ph  ->  A  C_  ( M ... N ) )
Assertion
Ref Expression
fsumcvg  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  (  seq 
M (  +  ,  F ) `  N
) )
Distinct variable groups:    A, k    k, F    k, N    ph, k    k, M
Allowed substitution hint:    B( k)

Proof of Theorem fsumcvg
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 sumrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
3 eluzelz 10488 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
42, 3syl 16 . 2  |-  ( ph  ->  N  e.  ZZ )
5 seqex 11317 . . 3  |-  seq  M
(  +  ,  F
)  e.  _V
65a1i 11 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
_V )
7 eqid 2435 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
8 eluzel2 10485 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
92, 8syl 16 . . . 4  |-  ( ph  ->  M  e.  ZZ )
10 eluzelz 10488 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
11 iftrue 3737 . . . . . . . . . 10  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  B )
1211adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  0 )  =  B )
13 summo.2 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
1412, 13eqeltrd 2509 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
1514ex 424 . . . . . . 7  |-  ( ph  ->  ( k  e.  A  ->  if ( k  e.  A ,  B , 
0 )  e.  CC ) )
16 iffalse 3738 . . . . . . . 8  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
17 0cn 9076 . . . . . . . 8  |-  0  e.  CC
1816, 17syl6eqel 2523 . . . . . . 7  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
1915, 18pm2.61d1 153 . . . . . 6  |-  ( ph  ->  if ( k  e.  A ,  B , 
0 )  e.  CC )
20 summo.1 . . . . . . 7  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
2120fvmpt2 5804 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  0 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  0 ) )
2210, 19, 21syl2anr 465 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )
2319adantr 452 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  if (
k  e.  A ,  B ,  0 )  e.  CC )
2422, 23eqeltrd 2509 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
257, 9, 24serf 11343 . . 3  |-  ( ph  ->  seq  M (  +  ,  F ) : ( ZZ>= `  M ) --> CC )
2625, 2ffvelrnd 5863 . 2  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  e.  CC )
27 addid1 9238 . . . . 5  |-  ( m  e.  CC  ->  (
m  +  0 )  =  m )
2827adantl 453 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  CC )  ->  ( m  +  0 )  =  m )
292adantr 452 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  N  e.  ( ZZ>= `  M )
)
30 simpr 448 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ( ZZ>= `  N )
)
3126adantr 452 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  +  ,  F
) `  N )  e.  CC )
32 elfzuz 11047 . . . . . 6  |-  ( m  e.  ( ( N  +  1 ) ... n )  ->  m  e.  ( ZZ>= `  ( N  +  1 ) ) )
33 eluzelz 10488 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  ( N  +  1 ) )  ->  m  e.  ZZ )
3433adantl 453 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ZZ )
35 fsumcvg.4 . . . . . . . . . . . 12  |-  ( ph  ->  A  C_  ( M ... N ) )
3635sseld 3339 . . . . . . . . . . 11  |-  ( ph  ->  ( m  e.  A  ->  m  e.  ( M ... N ) ) )
37 fznuz 11121 . . . . . . . . . . 11  |-  ( m  e.  ( M ... N )  ->  -.  m  e.  ( ZZ>= `  ( N  +  1
) ) )
3836, 37syl6 31 . . . . . . . . . 10  |-  ( ph  ->  ( m  e.  A  ->  -.  m  e.  (
ZZ>= `  ( N  + 
1 ) ) ) )
3938con2d 109 . . . . . . . . 9  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( N  + 
1 ) )  ->  -.  m  e.  A
) )
4039imp 419 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  -.  m  e.  A )
4134, 40eldifd 3323 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ( ZZ  \  A ) )
42 fveq2 5720 . . . . . . . . 9  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
4342eqeq1d 2443 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  =  0  <->  ( F `  m )  =  0 ) )
44 eldifi 3461 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
45 eldifn 3462 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ  \  A )  ->  -.  k  e.  A )
4645, 16syl 16 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
4746, 17syl6eqel 2523 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
4844, 47, 21syl2anc 643 . . . . . . . . 9  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
0 ) )
4948, 46eqtrd 2467 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  0 )
5043, 49vtoclga 3009 . . . . . . 7  |-  ( m  e.  ( ZZ  \  A )  ->  ( F `  m )  =  0 )
5141, 50syl 16 . . . . . 6  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  m )  =  0 )
5232, 51sylan2 461 . . . . 5  |-  ( (
ph  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  0 )
5352adantlr 696 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  0 )
5428, 29, 30, 31, 53seqid2 11361 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  +  ,  F
) `  N )  =  (  seq  M (  +  ,  F ) `
 n ) )
5554eqcomd 2440 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  +  ,  F
) `  n )  =  (  seq  M (  +  ,  F ) `
 N ) )
561, 4, 6, 26, 55climconst 12329 1  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  (  seq 
M (  +  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    C_ wss 3312   ifcif 3731   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035    seq cseq 11315    ~~> cli 12270
This theorem is referenced by:  summolem2a  12501  fsumcvg2  12513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274
  Copyright terms: Public domain W3C validator