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Theorem fsumcvg 12426
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 2380 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 sumrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
3 eluzelz 10421 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
42, 3syl 16 . 2  |-  ( ph  ->  N  e.  ZZ )
5 seqex 11245 . . 3  |-  seq  M
(  +  ,  F
)  e.  _V
65a1i 11 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
_V )
7 eqid 2380 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
8 eluzel2 10418 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
92, 8syl 16 . . . 4  |-  ( ph  ->  M  e.  ZZ )
10 eluzelz 10421 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
11 iftrue 3681 . . . . . . . . . 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 2454 . . . . . . . 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 3682 . . . . . . . 8  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
17 0cn 9010 . . . . . . . 8  |-  0  e.  CC
1816, 17syl6eqel 2468 . . . . . . 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 5744 . . . . . 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 2454 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
257, 9, 24serf 11271 . . 3  |-  ( ph  ->  seq  M (  +  ,  F ) : ( ZZ>= `  M ) --> CC )
2625, 2ffvelrnd 5803 . 2  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  e.  CC )
27 addid1 9171 . . . . 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 10980 . . . . . 6  |-  ( m  e.  ( ( N  +  1 ) ... n )  ->  m  e.  ( ZZ>= `  ( N  +  1 ) ) )
33 eluzelz 10421 . . . . . . . . 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 3283 . . . . . . . . . . 11  |-  ( ph  ->  ( m  e.  A  ->  m  e.  ( M ... N ) ) )
37 fznuz 11052 . . . . . . . . . . 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 3267 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ( ZZ  \  A ) )
42 fveq2 5661 . . . . . . . . 9  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
4342eqeq1d 2388 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  =  0  <->  ( F `  m )  =  0 ) )
44 eldifi 3405 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
45 eldifn 3406 . . . . . . . . . . . 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 2468 . . . . . . . . . 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 2412 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  0 )
5043, 49vtoclga 2953 . . . . . . 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 11289 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  +  ,  F
) `  N )  =  (  seq  M (  +  ,  F ) `
 n ) )
5554eqcomd 2385 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  +  ,  F
) `  n )  =  (  seq  M (  +  ,  F ) `
 N ) )
561, 4, 6, 26, 55climconst 12257 1  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  (  seq 
M (  +  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892    \ cdif 3253    C_ wss 3256   ifcif 3675   class class class wbr 4146    e. cmpt 4200   ` cfv 5387  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917    + caddc 8919   ZZcz 10207   ZZ>=cuz 10413   ...cfz 10968    seq cseq 11243    ~~> cli 12198
This theorem is referenced by:  summolem2a  12429  fsumcvg2  12441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fz 10969  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202
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