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Theorem climmpt 12367
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 449 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  F  e.  V )
3 climmpt.2 . . . 4  |-  G  =  ( k  e.  Z  |->  ( F `  k
) )
4 fvex 5744 . . . . . 6  |-  ( ZZ>= `  M )  e.  _V
51, 4eqeltri 2508 . . . . 5  |-  Z  e. 
_V
65mptex 5968 . . . 4  |-  ( k  e.  Z  |->  ( F `
 k ) )  e.  _V
73, 6eqeltri 2508 . . 3  |-  G  e. 
_V
87a1i 11 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  G  e.  _V )
9 simpl 445 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  M  e.  ZZ )
10 fveq2 5730 . . . . 5  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
11 fvex 5744 . . . . 5  |-  ( F `
 m )  e. 
_V
1210, 3, 11fvmpt 5808 . . . 4  |-  ( m  e.  Z  ->  ( G `  m )  =  ( F `  m ) )
1312eqcomd 2443 . . 3  |-  ( m  e.  Z  ->  ( F `  m )  =  ( G `  m ) )
1413adantl 454 . 2  |-  ( ( ( M  e.  ZZ  /\  F  e.  V )  /\  m  e.  Z
)  ->  ( F `  m )  =  ( G `  m ) )
151, 2, 8, 9, 14climeq 12363 1  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   class class class wbr 4214    e. cmpt 4268   ` cfv 5456   ZZcz 10284   ZZ>=cuz 10490    ~~> cli 12280
This theorem is referenced by:  climmpt2  12369  climrecl  12379  climge0  12380  caurcvg2  12473  caucvg  12474  climfsum  12601  dstfrvclim1  24737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-pre-lttri 9066  ax-pre-lttrn 9067
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-neg 9296  df-z 10285  df-uz 10491  df-clim 12284
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