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Theorem climeq 12248
Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
climeq.1  |-  Z  =  ( ZZ>= `  M )
climeq.2  |-  ( ph  ->  F  e.  V )
climeq.3  |-  ( ph  ->  G  e.  W )
climeq.5  |-  ( ph  ->  M  e.  ZZ )
climeq.6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( G `  k ) )
Assertion
Ref Expression
climeq  |-  ( ph  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Distinct variable groups:    A, k    k, F    k, G    ph, k    k, Z
Allowed substitution hints:    M( k)    V( k)    W( k)

Proof of Theorem climeq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climeq.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 climeq.5 . . 3  |-  ( ph  ->  M  e.  ZZ )
3 climeq.2 . . 3  |-  ( ph  ->  F  e.  V )
4 climeq.6 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( G `  k ) )
51, 2, 3, 4clim2 12185 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. y  e.  Z  A. k  e.  ( ZZ>= `  y )
( ( G `  k )  e.  CC  /\  ( abs `  (
( G `  k
)  -  A ) )  <  x ) ) ) )
6 climeq.3 . . 3  |-  ( ph  ->  G  e.  W )
7 eqidd 2367 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( G `  k ) )
81, 2, 6, 7clim2 12185 . 2  |-  ( ph  ->  ( G  ~~>  A  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. y  e.  Z  A. k  e.  ( ZZ>= `  y )
( ( G `  k )  e.  CC  /\  ( abs `  (
( G `  k
)  -  A ) )  <  x ) ) ) )
95, 8bitr4d 247 1  |-  ( ph  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   CCcc 8882    < clt 9014    - cmin 9184   ZZcz 10175   ZZ>=cuz 10381   RR+crp 10505   abscabs 11926    ~~> cli 12165
This theorem is referenced by:  climmpt  12252  climres  12256  climshft  12257  climshft2  12263  isumclim3  12430  logtayl  20229  dfef2  20487  iprodclim3  24790  climexp  27237  stirlinglem14  27342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-pre-lttri 8958  ax-pre-lttrn 8959
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-neg 9187  df-z 10176  df-uz 10382  df-clim 12169
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