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Theorem iserex 12440
Description: An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.)
Hypotheses
Ref Expression
clim2ser.1  |-  Z  =  ( ZZ>= `  M )
iserex.2  |-  ( ph  ->  N  e.  Z )
iserex.3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
Assertion
Ref Expression
iserex  |-  ( ph  ->  (  seq  M (  +  ,  F )  e.  dom  ~~>  <->  seq  N (  +  ,  F )  e.  dom  ~~>  ) )
Distinct variable groups:    k, F    k, M    k, N    ph, k    k, Z

Proof of Theorem iserex
StepHypRef Expression
1 seqeq1 11316 . . . . 5  |-  ( N  =  M  ->  seq  N (  +  ,  F
)  =  seq  M
(  +  ,  F
) )
21eleq1d 2501 . . . 4  |-  ( N  =  M  ->  (  seq  N (  +  ,  F )  e.  dom  ~~>  <->  seq  M (  +  ,  F
)  e.  dom  ~~>  ) )
32bicomd 193 . . 3  |-  ( N  =  M  ->  (  seq  M (  +  ,  F )  e.  dom  ~~>  <->  seq  N (  +  ,  F
)  e.  dom  ~~>  ) )
43a1i 11 . 2  |-  ( ph  ->  ( N  =  M  ->  (  seq  M
(  +  ,  F
)  e.  dom  ~~>  <->  seq  N (  +  ,  F )  e.  dom  ~~>  ) ) )
5 simpll 731 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  M (  +  ,  F
)  e.  dom  ~~>  )  ->  ph )
6 iserex.2 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  Z )
7 clim2ser.1 . . . . . . . . . . . 12  |-  Z  =  ( ZZ>= `  M )
86, 7syl6eleq 2525 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
9 eluzelz 10486 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
108, 9syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
1110zcnd 10366 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
12 ax-1cn 9038 . . . . . . . . 9  |-  1  e.  CC
13 npcan 9304 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
1411, 12, 13sylancl 644 . . . . . . . 8  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
1514seqeq1d 11319 . . . . . . 7  |-  ( ph  ->  seq  ( ( N  -  1 )  +  1 ) (  +  ,  F )  =  seq  N (  +  ,  F ) )
165, 15syl 16 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  M (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  ( ( N  - 
1 )  +  1 ) (  +  ,  F )  =  seq  N (  +  ,  F
) )
17 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  M (  +  ,  F
)  e.  dom  ~~>  )  -> 
( N  -  1 )  e.  ( ZZ>= `  M ) )
1817, 7syl6eleqr 2526 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  M (  +  ,  F
)  e.  dom  ~~>  )  -> 
( N  -  1 )  e.  Z )
19 iserex.3 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
205, 19sylan 458 . . . . . . 7  |-  ( ( ( ( ph  /\  ( N  -  1
)  e.  ( ZZ>= `  M ) )  /\  seq  M (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  Z
)  ->  ( F `  k )  e.  CC )
21 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  M (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
22 climdm 12338 . . . . . . . 8  |-  (  seq 
M (  +  ,  F )  e.  dom  ~~>  <->  seq  M (  +  ,  F
)  ~~>  (  ~~>  `  seq  M (  +  ,  F
) ) )
2321, 22sylib 189 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  M (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  M (  +  ,  F )  ~~>  (  ~~>  `  seq  M (  +  ,  F
) ) )
247, 18, 20, 23clim2ser 12438 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  M (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  ( ( N  - 
1 )  +  1 ) (  +  ,  F )  ~~>  ( (  ~~>  `
 seq  M (  +  ,  F )
)  -  (  seq 
M (  +  ,  F ) `  ( N  -  1 ) ) ) )
2516, 24eqbrtrrd 4226 . . . . 5  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  M (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  N (  +  ,  F )  ~~>  ( (  ~~>  `
 seq  M (  +  ,  F )
)  -  (  seq 
M (  +  ,  F ) `  ( N  -  1 ) ) ) )
26 climrel 12276 . . . . . 6  |-  Rel  ~~>
2726releldmi 5098 . . . . 5  |-  (  seq 
N (  +  ,  F )  ~~>  ( (  ~~>  `
 seq  M (  +  ,  F )
)  -  (  seq 
M (  +  ,  F ) `  ( N  -  1 ) ) )  ->  seq  N (  +  ,  F
)  e.  dom  ~~>  )
2825, 27syl 16 . . . 4  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  M (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  N (  +  ,  F )  e.  dom  ~~>  )
29 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
3029, 7syl6eleqr 2526 . . . . . . 7  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( N  -  1 )  e.  Z )
3130adantr 452 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  N (  +  ,  F
)  e.  dom  ~~>  )  -> 
( N  -  1 )  e.  Z )
32 simpll 731 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  N (  +  ,  F
)  e.  dom  ~~>  )  ->  ph )
3332, 19sylan 458 . . . . . 6  |-  ( ( ( ( ph  /\  ( N  -  1
)  e.  ( ZZ>= `  M ) )  /\  seq  N (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  Z
)  ->  ( F `  k )  e.  CC )
3432, 15syl 16 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  N (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  ( ( N  - 
1 )  +  1 ) (  +  ,  F )  =  seq  N (  +  ,  F
) )
35 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  N (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  N (  +  ,  F )  e.  dom  ~~>  )
36 climdm 12338 . . . . . . . 8  |-  (  seq 
N (  +  ,  F )  e.  dom  ~~>  <->  seq  N (  +  ,  F
)  ~~>  (  ~~>  `  seq  N (  +  ,  F
) ) )
3735, 36sylib 189 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  N (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  N (  +  ,  F )  ~~>  (  ~~>  `  seq  N (  +  ,  F
) ) )
3834, 37eqbrtrd 4224 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  N (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  ( ( N  - 
1 )  +  1 ) (  +  ,  F )  ~~>  (  ~~>  `  seq  N (  +  ,  F
) ) )
397, 31, 33, 38clim2ser2 12439 . . . . 5  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  N (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  M (  +  ,  F )  ~~>  ( (  ~~>  `
 seq  N (  +  ,  F )
)  +  (  seq 
M (  +  ,  F ) `  ( N  -  1 ) ) ) )
4026releldmi 5098 . . . . 5  |-  (  seq 
M (  +  ,  F )  ~~>  ( (  ~~>  `
 seq  N (  +  ,  F )
)  +  (  seq 
M (  +  ,  F ) `  ( N  -  1 ) ) )  ->  seq  M (  +  ,  F
)  e.  dom  ~~>  )
4139, 40syl 16 . . . 4  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  N (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
4228, 41impbida 806 . . 3  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  (  seq  M (  +  ,  F
)  e.  dom  ~~>  <->  seq  N (  +  ,  F )  e.  dom  ~~>  ) )
4342ex 424 . 2  |-  ( ph  ->  ( ( N  - 
1 )  e.  (
ZZ>= `  M )  -> 
(  seq  M (  +  ,  F )  e.  dom  ~~> 
<->  seq  N (  +  ,  F )  e. 
dom 
~~>  ) ) )
44 uzm1 10506 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  ( N  -  1 )  e.  ( ZZ>= `  M
) ) )
458, 44syl 16 . 2  |-  ( ph  ->  ( N  =  M  \/  ( N  - 
1 )  e.  (
ZZ>= `  M ) ) )
464, 43, 45mpjaod 371 1  |-  ( ph  ->  (  seq  M (  +  ,  F )  e.  dom  ~~>  <->  seq  N (  +  ,  F )  e.  dom  ~~>  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   dom cdm 4870   ` cfv 5446  (class class class)co 6073   CCcc 8978   1c1 8981    + caddc 8983    - cmin 9281   ZZcz 10272   ZZ>=cuz 10478    seq cseq 11313    ~~> cli 12268
This theorem is referenced by:  isumsplit  12610  isumrpcl  12613  climcnds  12621  geolim2  12638  cvgrat  12650  mertenslem1  12651  mertenslem2  12652  mertens  12653  eftlcvg  12697  rpnnen2lem5  12808  prmreclem5  13278  prmreclem6  13279  dvradcnv  20327  abelthlem7  20344  log2tlbnd  20775  lgamgulmlem4  24806
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 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
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-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-fz 11034  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-clim 12272
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