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Theorem climeq 12043
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 11980 . 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 2286 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( G `  k ) )
81, 2, 6, 7clim2 11980 . 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 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   CCcc 8737    < clt 8869    - cmin 9039   ZZcz 10026   ZZ>=cuz 10232   RR+crp 10356   abscabs 11721    ~~> cli 11960
This theorem is referenced by:  climmpt  12047  climres  12051  climshft  12052  climshft2  12058  isumclim3  12224  logtayl  20009  dfef2  20267  climexp  27742  stirlinglem14  27847
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-pre-lttri 8813  ax-pre-lttrn 8814
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-neg 9042  df-z 10027  df-uz 10233  df-clim 11964
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