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Theorem iserex 13137
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 11812 . . . . 5  |-  ( N  =  M  ->  seq N (  +  ,  F )  =  seq M (  +  ,  F ) )
21eleq1d 2509 . . . 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 2533 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
9 eluzelz 10873 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
108, 9syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
1110zcnd 10751 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
12 ax-1cn 9343 . . . . . . . . 9  |-  1  e.  CC
13 npcan 9622 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
1411, 12, 13sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
1514seqeq1d 11815 . . . . . . 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 2534 . . . . . . 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 13035 . . . . . . . 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 13135 . . . . . 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 4317 . . . . 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 12973 . . . . . 6  |-  Rel  ~~>
2726releldmi 5079 . . . . 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 2534 . . . . . . 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 13035 . . . . . . . 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 4315 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq ( ( N  -  1 )  +  1 ) (  +  ,  F )  ~~>  (  ~~>  `  seq N (  +  ,  F ) ) )
397, 31, 33, 38clim2ser2 13136 . . . . 5  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq M (  +  ,  F )  ~~>  ( (  ~~>  `
 seq N (  +  ,  F ) )  +  (  seq M
(  +  ,  F
) `  ( N  -  1 ) ) ) )
4026releldmi 5079 . . . . 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 828 . . 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 10894 . . 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 1369    e. wcel 1756   class class class wbr 4295   dom cdm 4843   ` cfv 5421  (class class class)co 6094   CCcc 9283   1c1 9286    + caddc 9288    - cmin 9598   ZZcz 10649   ZZ>=cuz 10864    seqcseq 11809    ~~> cli 12965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-fz 11441  df-seq 11810  df-exp 11869  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969
This theorem is referenced by:  isumsplit  13306  isumrpcl  13309  climcnds  13317  geolim2  13334  cvgrat  13346  mertenslem1  13347  mertenslem2  13348  mertens  13349  eftlcvg  13393  rpnnen2lem5  13504  prmreclem5  13984  prmreclem6  13985  dvradcnv  21889  abelthlem7  21906  log2tlbnd  22343  lgamgulmlem4  27021
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