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Theorem fsumcvg 12511
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 2438 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 sumrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
3 eluzelz 10501 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
42, 3syl 16 . 2  |-  ( ph  ->  N  e.  ZZ )
5 seqex 11330 . . 3  |-  seq  M
(  +  ,  F
)  e.  _V
65a1i 11 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
_V )
7 eqid 2438 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
8 eluzel2 10498 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
92, 8syl 16 . . . 4  |-  ( ph  ->  M  e.  ZZ )
10 eluzelz 10501 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
11 iftrue 3747 . . . . . . . . . 10  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  B )
1211adantl 454 . . . . . . . . 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 2512 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
1514ex 425 . . . . . . 7  |-  ( ph  ->  ( k  e.  A  ->  if ( k  e.  A ,  B , 
0 )  e.  CC ) )
16 iffalse 3748 . . . . . . . 8  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
17 0cn 9089 . . . . . . . 8  |-  0  e.  CC
1816, 17syl6eqel 2526 . . . . . . 7  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
1915, 18pm2.61d1 154 . . . . . 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 5815 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  0 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  0 ) )
2210, 19, 21syl2anr 466 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )
2319adantr 453 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  if (
k  e.  A ,  B ,  0 )  e.  CC )
2422, 23eqeltrd 2512 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
257, 9, 24serf 11356 . . 3  |-  ( ph  ->  seq  M (  +  ,  F ) : ( ZZ>= `  M ) --> CC )
2625, 2ffvelrnd 5874 . 2  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  e.  CC )
27 addid1 9251 . . . . 5  |-  ( m  e.  CC  ->  (
m  +  0 )  =  m )
2827adantl 454 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  CC )  ->  ( m  +  0 )  =  m )
292adantr 453 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  N  e.  ( ZZ>= `  M )
)
30 simpr 449 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ( ZZ>= `  N )
)
3126adantr 453 . . . 4  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  +  ,  F
) `  N )  e.  CC )
32 elfzuz 11060 . . . . . 6  |-  ( m  e.  ( ( N  +  1 ) ... n )  ->  m  e.  ( ZZ>= `  ( N  +  1 ) ) )
33 eluzelz 10501 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  ( N  +  1 ) )  ->  m  e.  ZZ )
3433adantl 454 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ZZ )
35 fsumcvg.4 . . . . . . . . . . . 12  |-  ( ph  ->  A  C_  ( M ... N ) )
3635sseld 3349 . . . . . . . . . . 11  |-  ( ph  ->  ( m  e.  A  ->  m  e.  ( M ... N ) ) )
37 fznuz 11134 . . . . . . . . . . 11  |-  ( m  e.  ( M ... N )  ->  -.  m  e.  ( ZZ>= `  ( N  +  1
) ) )
3836, 37syl6 32 . . . . . . . . . 10  |-  ( ph  ->  ( m  e.  A  ->  -.  m  e.  (
ZZ>= `  ( N  + 
1 ) ) ) )
3938con2d 110 . . . . . . . . 9  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( N  + 
1 ) )  ->  -.  m  e.  A
) )
4039imp 420 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  -.  m  e.  A )
4134, 40eldifd 3333 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  m  e.  ( ZZ  \  A ) )
42 fveq2 5731 . . . . . . . . 9  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
4342eqeq1d 2446 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  =  0  <->  ( F `  m )  =  0 ) )
44 eldifi 3471 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
45 eldifn 3472 . . . . . . . . . . . 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 2526 . . . . . . . . . 10  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
4844, 47, 21syl2anc 644 . . . . . . . . 9  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
0 ) )
4948, 46eqtrd 2470 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  0 )
5043, 49vtoclga 3019 . . . . . . 7  |-  ( m  e.  ( ZZ  \  A )  ->  ( F `  m )  =  0 )
5141, 50syl 16 . . . . . 6  |-  ( (
ph  /\  m  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  m )  =  0 )
5232, 51sylan2 462 . . . . 5  |-  ( (
ph  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  0 )
5352adantlr 697 . . . 4  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  N )
)  /\  m  e.  ( ( N  + 
1 ) ... n
) )  ->  ( F `  m )  =  0 )
5428, 29, 30, 31, 53seqid2 11374 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  +  ,  F
) `  N )  =  (  seq  M (  +  ,  F ) `
 n ) )
5554eqcomd 2443 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  +  ,  F
) `  n )  =  (  seq  M (  +  ,  F ) `
 N ) )
561, 4, 6, 26, 55climconst 12342 1  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  (  seq 
M (  +  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    C_ wss 3322   ifcif 3741   class class class wbr 4215    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   CCcc 8993   0cc0 8995   1c1 8996    + caddc 8998   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048    seq cseq 11328    ~~> cli 12283
This theorem is referenced by:  summolem2a  12514  fsumcvg2  12526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-seq 11329  df-exp 11388  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287
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