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Theorem iserex 13428
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 12066 . . . . 5  |-  ( N  =  M  ->  seq N (  +  ,  F )  =  seq M (  +  ,  F ) )
21eleq1d 2529 . . . 4  |-  ( N  =  M  ->  (  seq N (  +  ,  F )  e.  dom  ~~>  <->  seq M (  +  ,  F )  e.  dom  ~~>  ) )
32bicomd 201 . . 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 753 . . . . . . 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 2558 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
9 eluzelz 11080 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
108, 9syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
1110zcnd 10956 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
12 ax-1cn 9539 . . . . . . . . 9  |-  1  e.  CC
13 npcan 9818 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
1411, 12, 13sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
1514seqeq1d 12069 . . . . . . 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 754 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F )  e.  dom  ~~>  )  ->  ( N  - 
1 )  e.  (
ZZ>= `  M ) )
1817, 7syl6eleqr 2559 . . . . . . 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 471 . . . . . . 7  |-  ( ( ( ( ph  /\  ( N  -  1
)  e.  ( ZZ>= `  M ) )  /\  seq M (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  Z
)  ->  ( F `  k )  e.  CC )
21 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F )  e.  dom  ~~>  )  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
22 climdm 13326 . . . . . . . 8  |-  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq M (  +  ,  F )  ~~>  (  ~~>  `  seq M (  +  ,  F ) ) )
2321, 22sylib 196 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F )  e.  dom  ~~>  )  ->  seq M (  +  ,  F )  ~~>  (  ~~>  `  seq M (  +  ,  F ) ) )
247, 18, 20, 23clim2ser 13426 . . . . . 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 4462 . . . . 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 13264 . . . . . 6  |-  Rel  ~~>
2726releldmi 5230 . . . . 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 461 . . . . . . . 8  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
3029, 7syl6eleqr 2559 . . . . . . 7  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( N  -  1 )  e.  Z )
3130adantr 465 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  ( N  - 
1 )  e.  Z
)
32 simpll 753 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  ph )
3332, 19sylan 471 . . . . . 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 461 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq N (  +  ,  F )  e. 
dom 
~~>  )
36 climdm 13326 . . . . . . . 8  |-  (  seq N (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  ~~>  (  ~~>  `  seq N (  +  ,  F ) ) )
3735, 36sylib 196 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq N (  +  ,  F )  ~~>  (  ~~>  `  seq N (  +  ,  F ) ) )
3834, 37eqbrtrd 4460 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq ( ( N  -  1 )  +  1 ) (  +  ,  F )  ~~>  (  ~~>  `  seq N (  +  ,  F ) ) )
397, 31, 33, 38clim2ser2 13427 . . . . 5  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq M (  +  ,  F )  ~~>  ( (  ~~>  `
 seq N (  +  ,  F ) )  +  (  seq M
(  +  ,  F
) `  ( N  -  1 ) ) ) )
4026releldmi 5230 . . . . 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 829 . . 3  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) )
4342ex 434 . 2  |-  ( ph  ->  ( ( N  - 
1 )  e.  (
ZZ>= `  M )  -> 
(  seq M (  +  ,  F )  e. 
dom 
~~> 
<->  seq N (  +  ,  F )  e. 
dom 
~~>  ) ) )
44 uzm1 11101 . . 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 381 1  |-  ( ph  ->  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762   class class class wbr 4440   dom cdm 4992   ` cfv 5579  (class class class)co 6275   CCcc 9479   1c1 9482    + caddc 9484    - cmin 9794   ZZcz 10853   ZZ>=cuz 11071    seqcseq 12063    ~~> cli 13256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260
This theorem is referenced by:  isumsplit  13604  isumrpcl  13607  climcnds  13615  geolim2  13632  cvgrat  13644  mertenslem1  13645  mertenslem2  13646  mertens  13647  eftlcvg  13691  rpnnen2lem5  13802  prmreclem5  14286  prmreclem6  14287  dvradcnv  22543  abelthlem7  22560  log2tlbnd  22997  lgamgulmlem4  28200
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