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Theorem iserex 11342
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 10445 . . . . 5  |-  ( N  =  M  ->  seq N (  +  ,  F )  =  seq M (  +  ,  F ) )
21eleq1d 2246 . . . 4  |-  ( N  =  M  ->  (  seq N (  +  ,  F )  e.  dom  ~~>  <->  seq M (  +  ,  F )  e.  dom  ~~>  ) )
32bicomd 141 . . 3  |-  ( N  =  M  ->  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) )
43a1i 9 . 2  |-  ( ph  ->  ( N  =  M  ->  (  seq M
(  +  ,  F
)  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) ) )
5 simpll 527 . . . . . . 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, 7eleqtrdi 2270 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
9 eluzelz 9535 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
108, 9syl 14 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
1110zcnd 9374 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
12 ax-1cn 7903 . . . . . . . . 9  |-  1  e.  CC
13 npcan 8164 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
1411, 12, 13sylancl 413 . . . . . . . 8  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
1514seqeq1d 10448 . . . . . . 7  |-  ( ph  ->  seq ( ( N  -  1 )  +  1 ) (  +  ,  F )  =  seq N (  +  ,  F ) )
165, 15syl 14 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F )  e.  dom  ~~>  )  ->  seq ( ( N  -  1 )  +  1 ) (  +  ,  F )  =  seq N (  +  ,  F ) )
17 simplr 528 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F )  e.  dom  ~~>  )  ->  ( N  - 
1 )  e.  (
ZZ>= `  M ) )
1817, 7eleqtrrdi 2271 . . . . . . 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 283 . . . . . . 7  |-  ( ( ( ( ph  /\  ( N  -  1
)  e.  ( ZZ>= `  M ) )  /\  seq M (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  Z
)  ->  ( F `  k )  e.  CC )
21 simpr 110 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F )  e.  dom  ~~>  )  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
22 climdm 11298 . . . . . . . 8  |-  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq M (  +  ,  F )  ~~>  (  ~~>  `  seq M (  +  ,  F ) ) )
2321, 22sylib 122 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F )  e.  dom  ~~>  )  ->  seq M (  +  ,  F )  ~~>  (  ~~>  `  seq M (  +  ,  F ) ) )
247, 18, 20, 23clim2ser 11340 . . . . . 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 4027 . . . . 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 11283 . . . . . 6  |-  Rel  ~~>
2726releldmi 4866 . . . . 5  |-  (  seq N (  +  ,  F )  ~~>  ( (  ~~>  `
 seq M (  +  ,  F ) )  -  (  seq M
(  +  ,  F
) `  ( N  -  1 ) ) )  ->  seq N (  +  ,  F )  e.  dom  ~~>  )
2825, 27syl 14 . . . 4  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F )  e.  dom  ~~>  )  ->  seq N (  +  ,  F )  e. 
dom 
~~>  )
29 simpr 110 . . . . . . . 8  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
3029, 7eleqtrrdi 2271 . . . . . . 7  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( N  -  1 )  e.  Z )
3130adantr 276 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  ( N  - 
1 )  e.  Z
)
32 simpll 527 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  ph )
3332, 19sylan 283 . . . . . 6  |-  ( ( ( ( ph  /\  ( N  -  1
)  e.  ( ZZ>= `  M ) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  Z
)  ->  ( F `  k )  e.  CC )
3432, 15syl 14 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq ( ( N  -  1 )  +  1 ) (  +  ,  F )  =  seq N (  +  ,  F ) )
35 simpr 110 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq N (  +  ,  F )  e. 
dom 
~~>  )
36 climdm 11298 . . . . . . . 8  |-  (  seq N (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  ~~>  (  ~~>  `  seq N (  +  ,  F ) ) )
3735, 36sylib 122 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq N (  +  ,  F )  ~~>  (  ~~>  `  seq N (  +  ,  F ) ) )
3834, 37eqbrtrd 4025 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq ( ( N  -  1 )  +  1 ) (  +  ,  F )  ~~>  (  ~~>  `  seq N (  +  ,  F ) ) )
397, 31, 33, 38clim2ser2 11341 . . . . 5  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq M (  +  ,  F )  ~~>  ( (  ~~>  `
 seq N (  +  ,  F ) )  +  (  seq M
(  +  ,  F
) `  ( N  -  1 ) ) ) )
4026releldmi 4866 . . . . 5  |-  (  seq M (  +  ,  F )  ~~>  ( (  ~~>  `
 seq N (  +  ,  F ) )  +  (  seq M
(  +  ,  F
) `  ( N  -  1 ) ) )  ->  seq M (  +  ,  F )  e.  dom  ~~>  )
4139, 40syl 14 . . . 4  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F )  e.  dom  ~~>  )  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
4228, 41impbida 596 . . 3  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) )
4342ex 115 . 2  |-  ( ph  ->  ( ( N  - 
1 )  e.  (
ZZ>= `  M )  -> 
(  seq M (  +  ,  F )  e. 
dom 
~~> 
<->  seq N (  +  ,  F )  e. 
dom 
~~>  ) ) )
44 uzm1 9556 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  ( N  -  1 )  e.  ( ZZ>= `  M
) ) )
458, 44syl 14 . 2  |-  ( ph  ->  ( N  =  M  \/  ( N  - 
1 )  e.  (
ZZ>= `  M ) ) )
464, 43, 45mpjaod 718 1  |-  ( ph  ->  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148   class class class wbr 4003   dom cdm 4626   ` cfv 5216  (class class class)co 5874   CCcc 7808   1c1 7811    + caddc 7813    - cmin 8126   ZZcz 9251   ZZ>=cuz 9526    seqcseq 10442    ~~> cli 11281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-rp 9652  df-fz 10007  df-seqfrec 10443  df-exp 10517  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282
This theorem is referenced by:  isumsplit  11494  isumrpcl  11497  geolim2  11515  cvgratz  11535  cvgratgt0  11536  mertenslemub  11537  mertenslemi1  11538  mertenslem2  11539  mertensabs  11540  eftlcvg  11690  trilpolemisumle  14668  trilpolemeq1  14670  trilpolemlt1  14671  nconstwlpolemgt0  14693
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