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Theorem climres 12051
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 2285 . 2  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 resexg 4996 . . 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 5544 . . 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 12043 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 1625    e. wcel 1686   _Vcvv 2790   class class class wbr 4025    |` cres 4693   ` cfv 5257   ZZcz 10026   ZZ>=cuz 10232    ~~> cli 11960
This theorem is referenced by:  sumrb  12188  iscmet3lem3  18718  leibpilem2  20239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-pre-lttri 8813  ax-pre-lttrn 8814
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-neg 9042  df-z 10027  df-uz 10233  df-clim 11964
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