MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  climmpt Unicode version

Theorem climmpt 12061
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 447 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  F  e.  V )
3 climmpt.2 . . . 4  |-  G  =  ( k  e.  Z  |->  ( F `  k
) )
4 fvex 5555 . . . . . 6  |-  ( ZZ>= `  M )  e.  _V
51, 4eqeltri 2366 . . . . 5  |-  Z  e. 
_V
65mptex 5762 . . . 4  |-  ( k  e.  Z  |->  ( F `
 k ) )  e.  _V
73, 6eqeltri 2366 . . 3  |-  G  e. 
_V
87a1i 10 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  G  e.  _V )
9 simpl 443 . 2  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  M  e.  ZZ )
10 fveq2 5541 . . . . 5  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
11 fvex 5555 . . . . 5  |-  ( F `
 m )  e. 
_V
1210, 3, 11fvmpt 5618 . . . 4  |-  ( m  e.  Z  ->  ( G `  m )  =  ( F `  m ) )
1312eqcomd 2301 . . 3  |-  ( m  e.  Z  ->  ( F `  m )  =  ( G `  m ) )
1413adantl 452 . 2  |-  ( ( ( M  e.  ZZ  /\  F  e.  V )  /\  m  e.  Z
)  ->  ( F `  m )  =  ( G `  m ) )
151, 2, 8, 9, 14climeq 12057 1  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   ` cfv 5271   ZZcz 10040   ZZ>=cuz 10246    ~~> cli 11974
This theorem is referenced by:  climmpt2  12063  climrecl  12073  climge0  12074  caurcvg2  12166  caucvg  12167  climfsum  12294  dstfrvclim1  23693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-neg 9056  df-z 10041  df-uz 10247  df-clim 11978
  Copyright terms: Public domain W3C validator