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Theorem climres 12352
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 2430 . 2  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 resexg 5171 . . 3  |-  ( F  e.  V  ->  ( F  |`  ( ZZ>= `  M
) )  e.  _V )
32adantl 453 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( F  |`  ( ZZ>=
`  M ) )  e.  _V )
4 simpr 448 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  F  e.  V )
5 simpl 444 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  M  e.  ZZ )
6 fvres 5731 . . 3  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ( F  |`  ( ZZ>= `  M
) ) `  k
)  =  ( F `
 k ) )
76adantl 453 . 2  |-  ( ( ( M  e.  ZZ  /\  F  e.  V )  /\  k  e.  (
ZZ>= `  M ) )  ->  ( ( F  |`  ( ZZ>= `  M )
) `  k )  =  ( F `  k ) )
81, 3, 4, 5, 7climeq 12344 1  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( ( F  |`  ( ZZ>= `  M )
)  ~~>  A  <->  F  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2943   class class class wbr 4199    |` cres 4866   ` cfv 5440   ZZcz 10266   ZZ>=cuz 10472    ~~> cli 12261
This theorem is referenced by:  sumrb  12490  iscmet3lem3  19226  leibpilem2  20764  lgamcvg2  24822  divcnvshft  25194  divcnvlin  25195  prodrblem2  25241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-pre-lttri 9048  ax-pre-lttrn 9049
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-po 4490  df-so 4491  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-neg 9278  df-z 10267  df-uz 10473  df-clim 12265
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