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Theorem climeq 13041
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 12978 . 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 2442 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( G `  k ) )
81, 2, 6, 7clim2 12978 . 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 256 1  |-  ( ph  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276    < clt 9414    - cmin 9591   ZZcz 10642   ZZ>=cuz 10857   RR+crp 10987   abscabs 12719    ~~> cli 12958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-pre-lttri 9352  ax-pre-lttrn 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-neg 9594  df-z 10643  df-uz 10858  df-clim 12962
This theorem is referenced by:  climmpt  13045  climres  13049  climshft  13050  climshft2  13056  isumclim3  13222  logtayl  22064  dfef2  22323  iprodclim3  27429  climexp  29703  stirlinglem14  29807
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