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Theorem climmpt 13171
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 461 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  F  e.  V )
3 climmpt.2 . . . 4  |-  G  =  ( k  e.  Z  |->  ( F `  k
) )
4 fvex 5812 . . . . . 6  |-  ( ZZ>= `  M )  e.  _V
51, 4eqeltri 2538 . . . . 5  |-  Z  e. 
_V
65mptex 6060 . . . 4  |-  ( k  e.  Z  |->  ( F `
 k ) )  e.  _V
73, 6eqeltri 2538 . . 3  |-  G  e. 
_V
87a1i 11 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  G  e.  _V )
9 simpl 457 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  M  e.  ZZ )
10 fveq2 5802 . . . . 5  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
11 fvex 5812 . . . . 5  |-  ( F `
 m )  e. 
_V
1210, 3, 11fvmpt 5886 . . . 4  |-  ( m  e.  Z  ->  ( G `  m )  =  ( F `  m ) )
1312eqcomd 2462 . . 3  |-  ( m  e.  Z  ->  ( F `  m )  =  ( G `  m ) )
1413adantl 466 . 2  |-  ( ( ( M  e.  ZZ  /\  F  e.  V )  /\  m  e.  Z
)  ->  ( F `  m )  =  ( G `  m ) )
151, 2, 8, 9, 14climeq 13167 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 1370    e. wcel 1758   _Vcvv 3078   class class class wbr 4403    |-> cmpt 4461   ` cfv 5529   ZZcz 10761   ZZ>=cuz 10976    ~~> cli 13084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-pre-lttri 9471  ax-pre-lttrn 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-neg 9713  df-z 10762  df-uz 10977  df-clim 13088
This theorem is referenced by:  climmpt2  13173  climrecl  13183  climge0  13184  caurcvg2  13277  caucvg  13278  climfsum  13405  dstfrvclim1  27027
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