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Theorem climeq 12354
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 12291 . 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 2437 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( G `  k ) )
81, 2, 6, 7clim2 12291 . 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 248 1  |-  ( ph  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2698   E.wrex 2699   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   CCcc 8981    < clt 9113    - cmin 9284   ZZcz 10275   ZZ>=cuz 10481   RR+crp 10605   abscabs 12032    ~~> cli 12271
This theorem is referenced by:  climmpt  12358  climres  12362  climshft  12363  climshft2  12369  isumclim3  12536  logtayl  20544  dfef2  20802  iprodclim3  25306  climexp  27699  stirlinglem14  27804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-pre-lttri 9057  ax-pre-lttrn 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-po 4496  df-so 4497  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-neg 9287  df-z 10276  df-uz 10482  df-clim 12275
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