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Theorem iserex 12379
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 11255 . . . . 5  |-  ( N  =  M  ->  seq  N (  +  ,  F
)  =  seq  M
(  +  ,  F
) )
21eleq1d 2455 . . . 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 2479 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
9 eluzelz 10430 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
108, 9syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
1110zcnd 10310 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
12 ax-1cn 8983 . . . . . . . . 9  |-  1  e.  CC
13 npcan 9248 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
1411, 12, 13sylancl 644 . . . . . . . 8  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
1514seqeq1d 11258 . . . . . . 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 2480 . . . . . . 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 12277 . . . . . . . 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 12377 . . . . . 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 4177 . . . . 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 12215 . . . . . 6  |-  Rel  ~~>
2726releldmi 5048 . . . . 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 2480 . . . . . . 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 12277 . . . . . . . 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 4175 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  N (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  ( ( N  - 
1 )  +  1 ) (  +  ,  F )  ~~>  (  ~~>  `  seq  N (  +  ,  F
) ) )
397, 31, 33, 38clim2ser2 12378 . . . . 5  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq  N (  +  ,  F
)  e.  dom  ~~>  )  ->  seq  M (  +  ,  F )  ~~>  ( (  ~~>  `
 seq  N (  +  ,  F )
)  +  (  seq 
M (  +  ,  F ) `  ( N  -  1 ) ) ) )
4026releldmi 5048 . . . . 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 10450 . . 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 1649    e. wcel 1717   class class class wbr 4155   dom cdm 4820   ` cfv 5396  (class class class)co 6022   CCcc 8923   1c1 8926    + caddc 8928    - cmin 9225   ZZcz 10216   ZZ>=cuz 10422    seq cseq 11252    ~~> cli 12207
This theorem is referenced by:  isumsplit  12549  isumrpcl  12552  climcnds  12560  geolim2  12577  cvgrat  12589  mertenslem1  12590  mertenslem2  12591  mertens  12592  eftlcvg  12636  rpnnen2lem5  12747  prmreclem5  13217  prmreclem6  13218  dvradcnv  20206  abelthlem7  20223  log2tlbnd  20654  lgamgulmlem4  24597
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-fz 10978  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211
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