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Theorem iiserex 10315
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
iiserex  |-  ( ph  ->  (  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  <->  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  ) )
Distinct variable groups:    k, F    k, M    k, N    ph, k    k, Z

Proof of Theorem iiserex
StepHypRef Expression
1 iseqeq1 9524 . . . . 5  |-  ( N  =  M  ->  seq N (  +  ,  F ,  CC )  =  seq M (  +  ,  F ,  CC ) )
21eleq1d 2148 . . . 4  |-  ( N  =  M  ->  (  seq N (  +  ,  F ,  CC )  e.  dom  ~~> 
<->  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  ) )
32bicomd 139 . . 3  |-  ( N  =  M  ->  (  seq M (  +  ,  F ,  CC )  e.  dom  ~~> 
<->  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  ) )
43a1i 9 . 2  |-  ( ph  ->  ( N  =  M  ->  (  seq M
(  +  ,  F ,  CC )  e.  dom  ~~>  <->  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  ) ) )
5 simpll 496 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  ph )
6 iserex.2 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  Z )
7 clim2ser.1 . . . . . . . . . . . 12  |-  Z  =  ( ZZ>= `  M )
86, 7syl6eleq 2172 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
9 eluzelz 8709 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
108, 9syl 14 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
1110zcnd 8551 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
12 ax-1cn 7131 . . . . . . . . 9  |-  1  e.  CC
13 npcan 7384 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
1411, 12, 13sylancl 404 . . . . . . . 8  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
15 iseqeq1 9524 . . . . . . . 8  |-  ( ( ( N  -  1 )  +  1 )  =  N  ->  seq ( ( N  - 
1 )  +  1 ) (  +  ,  F ,  CC )  =  seq N (  +  ,  F ,  CC ) )
1614, 15syl 14 . . . . . . 7  |-  ( ph  ->  seq ( ( N  -  1 )  +  1 ) (  +  ,  F ,  CC )  =  seq N (  +  ,  F ,  CC ) )
175, 16syl 14 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq ( ( N  - 
1 )  +  1 ) (  +  ,  F ,  CC )  =  seq N (  +  ,  F ,  CC ) )
18 simplr 497 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  ( N  -  1 )  e.  ( ZZ>= `  M
) )
1918, 7syl6eleqr 2173 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  ( N  -  1 )  e.  Z )
20 iserex.3 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
215, 20sylan 277 . . . . . . 7  |-  ( ( ( ( ph  /\  ( N  -  1
)  e.  ( ZZ>= `  M ) )  /\  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
22 simpr 108 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )
23 climdm 10272 . . . . . . . 8  |-  (  seq M (  +  ,  F ,  CC )  e.  dom  ~~> 
<->  seq M (  +  ,  F ,  CC ) 
~~>  (  ~~>  `  seq M
(  +  ,  F ,  CC ) ) )
2422, 23sylib 120 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq M (  +  ,  F ,  CC )  ~~>  ( 
~~>  `  seq M (  +  ,  F ,  CC ) ) )
257, 19, 21, 24clim2iser 10313 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq ( ( N  - 
1 )  +  1 ) (  +  ,  F ,  CC )  ~~>  ( (  ~~>  `  seq M (  +  ,  F ,  CC )
)  -  (  seq M (  +  ,  F ,  CC ) `  ( N  -  1 ) ) ) )
2617, 25eqbrtrrd 3815 . . . . 5  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq N (  +  ,  F ,  CC )  ~~>  ( (  ~~>  `  seq M (  +  ,  F ,  CC )
)  -  (  seq M (  +  ,  F ,  CC ) `  ( N  -  1 ) ) ) )
27 climrel 10257 . . . . . 6  |-  Rel  ~~>
2827releldmi 4601 . . . . 5  |-  (  seq N (  +  ,  F ,  CC )  ~~>  ( (  ~~>  `  seq M (  +  ,  F ,  CC )
)  -  (  seq M (  +  ,  F ,  CC ) `  ( N  -  1 ) ) )  ->  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )
2926, 28syl 14 . . . 4  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )
30 simpr 108 . . . . . . . 8  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
3130, 7syl6eleqr 2173 . . . . . . 7  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( N  -  1 )  e.  Z )
3231adantr 270 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  ( N  -  1 )  e.  Z )
33 simpll 496 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  ph )
3433, 20sylan 277 . . . . . 6  |-  ( ( ( ( ph  /\  ( N  -  1
)  e.  ( ZZ>= `  M ) )  /\  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
3533, 16syl 14 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq ( ( N  - 
1 )  +  1 ) (  +  ,  F ,  CC )  =  seq N (  +  ,  F ,  CC ) )
36 simpr 108 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )
37 climdm 10272 . . . . . . . 8  |-  (  seq N (  +  ,  F ,  CC )  e.  dom  ~~> 
<->  seq N (  +  ,  F ,  CC ) 
~~>  (  ~~>  `  seq N
(  +  ,  F ,  CC ) ) )
3836, 37sylib 120 . . . . . . 7  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq N (  +  ,  F ,  CC )  ~~>  ( 
~~>  `  seq N (  +  ,  F ,  CC ) ) )
3935, 38eqbrtrd 3813 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq ( ( N  - 
1 )  +  1 ) (  +  ,  F ,  CC )  ~~>  ( 
~~>  `  seq N (  +  ,  F ,  CC ) ) )
407, 32, 34, 39clim2iser2 10314 . . . . 5  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq M (  +  ,  F ,  CC )  ~~>  ( (  ~~>  `  seq N (  +  ,  F ,  CC )
)  +  (  seq M (  +  ,  F ,  CC ) `  ( N  -  1 ) ) ) )
4127releldmi 4601 . . . . 5  |-  (  seq M (  +  ,  F ,  CC )  ~~>  ( (  ~~>  `  seq N (  +  ,  F ,  CC )
)  +  (  seq M (  +  ,  F ,  CC ) `  ( N  -  1 ) ) )  ->  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )
4240, 41syl 14 . . . 4  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  )  ->  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )
4329, 42impbida 561 . . 3  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  (  seq M (  +  ,  F ,  CC )  e.  dom  ~~> 
<->  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  ) )
4443ex 113 . 2  |-  ( ph  ->  ( ( N  - 
1 )  e.  (
ZZ>= `  M )  -> 
(  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  <->  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  ) ) )
45 uzm1 8730 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  ( N  -  1 )  e.  ( ZZ>= `  M
) ) )
468, 45syl 14 . 2  |-  ( ph  ->  ( N  =  M  \/  ( N  - 
1 )  e.  (
ZZ>= `  M ) ) )
474, 44, 46mpjaod 671 1  |-  ( ph  ->  (  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  <->  seq N (  +  ,  F ,  CC )  e.  dom  ~~>  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   class class class wbr 3793   dom cdm 4371   ` cfv 4932  (class class class)co 5543   CCcc 7041   1c1 7044    + caddc 7046    - cmin 7346   ZZcz 8432   ZZ>=cuz 8700    seqcseq 9521    ~~> cli 10255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157  ax-caucvg 7158
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-3 8166  df-4 8167  df-n0 8356  df-z 8433  df-uz 8701  df-rp 8816  df-fz 9106  df-iseq 9522  df-iexp 9573  df-cj 9867  df-re 9868  df-im 9869  df-rsqrt 10022  df-abs 10023  df-clim 10256
This theorem is referenced by: (None)
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