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Theorem climeq 13339
 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
climeq.2
climeq.3
climeq.5
climeq.6
Assertion
Ref Expression
climeq
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem climeq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climeq.1 . . 3
2 climeq.5 . . 3
3 climeq.2 . . 3
4 climeq.6 . . 3
51, 2, 3, 4clim2 13276 . 2
6 climeq.3 . . 3
7 eqidd 2461 . . 3
81, 2, 6, 7clim2 13276 . 2
95, 8bitr4d 256 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1374   wcel 1762  wral 2807  wrex 2808   class class class wbr 4440  cfv 5579  (class class class)co 6275  cc 9479   clt 9617   cmin 9794  cz 10853  cuz 11071  crp 11209  cabs 13017   cli 13256 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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-neg 9797  df-z 10854  df-uz 11072  df-clim 13260 This theorem is referenced by:  climmpt  13343  climres  13347  climshft  13348  climshft2  13354  isumclim3  13523  logtayl  22762  dfef2  23021  iprodclim3  28682  climexp  31102  stirlinglem14  31342  fourierdlem112  31474
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