MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  climres Unicode version

Theorem climres 12045
Description: A function restricted to upper integers converges iff the original function converges. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
Assertion
Ref Expression
climres  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( ( F  |`  ( ZZ>= `  M )
)  ~~>  A  <->  F  ~~>  A ) )

Proof of Theorem climres
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 eqid 2284 . 2  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 resexg 4993 . . 3  |-  ( F  e.  V  ->  ( F  |`  ( ZZ>= `  M
) )  e.  _V )
32adantl 452 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( F  |`  ( ZZ>=
`  M ) )  e.  _V )
4 simpr 447 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  F  e.  V )
5 simpl 443 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  M  e.  ZZ )
6 fvres 5503 . . 3  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ( F  |`  ( ZZ>= `  M
) ) `  k
)  =  ( F `
 k ) )
76adantl 452 . 2  |-  ( ( ( M  e.  ZZ  /\  F  e.  V )  /\  k  e.  (
ZZ>= `  M ) )  ->  ( ( F  |`  ( ZZ>= `  M )
) `  k )  =  ( F `  k ) )
81, 3, 4, 5, 7climeq 12037 1  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( ( F  |`  ( ZZ>= `  M )
)  ~~>  A  <->  F  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   _Vcvv 2789   class class class wbr 4024    |` cres 4690   ` cfv 5221   ZZcz 10020   ZZ>=cuz 10226    ~~> cli 11954
This theorem is referenced by:  sumrb  12182  iscmet3lem3  18712  leibpilem2  20233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-pre-lttri 8807  ax-pre-lttrn 8808
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-neg 9036  df-z 10021  df-uz 10227  df-clim 11958
  Copyright terms: Public domain W3C validator