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Theorem climmpt 13033
Description: Exhibit a function  G with the same convergence properties as the not-quite-function  F. (Contributed by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
2clim.1  |-  Z  =  ( ZZ>= `  M )
climmpt.2  |-  G  =  ( k  e.  Z  |->  ( F `  k
) )
Assertion
Ref Expression
climmpt  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Distinct variable groups:    A, k    k, F    k, Z
Allowed substitution hints:    G( k)    M( k)    V( k)

Proof of Theorem climmpt
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 2clim.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 simpr 458 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  F  e.  V )
3 climmpt.2 . . . 4  |-  G  =  ( k  e.  Z  |->  ( F `  k
) )
4 fvex 5689 . . . . . 6  |-  ( ZZ>= `  M )  e.  _V
51, 4eqeltri 2503 . . . . 5  |-  Z  e. 
_V
65mptex 5935 . . . 4  |-  ( k  e.  Z  |->  ( F `
 k ) )  e.  _V
73, 6eqeltri 2503 . . 3  |-  G  e. 
_V
87a1i 11 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  G  e.  _V )
9 simpl 454 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  M  e.  ZZ )
10 fveq2 5679 . . . . 5  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
11 fvex 5689 . . . . 5  |-  ( F `
 m )  e. 
_V
1210, 3, 11fvmpt 5762 . . . 4  |-  ( m  e.  Z  ->  ( G `  m )  =  ( F `  m ) )
1312eqcomd 2438 . . 3  |-  ( m  e.  Z  ->  ( F `  m )  =  ( G `  m ) )
1413adantl 463 . 2  |-  ( ( ( M  e.  ZZ  /\  F  e.  V )  /\  m  e.  Z
)  ->  ( F `  m )  =  ( G `  m ) )
151, 2, 8, 9, 14climeq 13029 1  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755   _Vcvv 2962   class class class wbr 4280    e. cmpt 4338   ` cfv 5406   ZZcz 10634   ZZ>=cuz 10849    ~~> cli 12946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-pre-lttri 9344  ax-pre-lttrn 9345
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-neg 9586  df-z 10635  df-uz 10850  df-clim 12950
This theorem is referenced by:  climmpt2  13035  climrecl  13045  climge0  13046  caurcvg2  13139  caucvg  13140  climfsum  13266  dstfrvclim1  26708
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