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Theorem climeq 15604
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 = (ℤ𝑀)
climeq.2 (𝜑𝐹𝑉)
climeq.3 (𝜑𝐺𝑊)
climeq.5 (𝜑𝑀 ∈ ℤ)
climeq.6 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))
Assertion
Ref Expression
climeq (𝜑 → (𝐹𝐴𝐺𝐴))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝐺   𝜑,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 = (ℤ𝑀)
2 climeq.5 . . 3 (𝜑𝑀 ∈ ℤ)
3 climeq.2 . . 3 (𝜑𝐹𝑉)
4 climeq.6 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))
51, 2, 3, 4clim2 15541 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦𝑍𝑘 ∈ (ℤ𝑦)((𝐺𝑘) ∈ ℂ ∧ (abs‘((𝐺𝑘) − 𝐴)) < 𝑥))))
6 climeq.3 . . 3 (𝜑𝐺𝑊)
7 eqidd 2737 . . 3 ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐺𝑘))
81, 2, 6, 7clim2 15541 . 2 (𝜑 → (𝐺𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦𝑍𝑘 ∈ (ℤ𝑦)((𝐺𝑘) ∈ ℂ ∧ (abs‘((𝐺𝑘) − 𝐴)) < 𝑥))))
95, 8bitr4d 282 1 (𝜑 → (𝐹𝐴𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  wrex 3069   class class class wbr 5142  cfv 6560  (class class class)co 7432  cc 11154   < clt 11296  cmin 11493  cz 12615  cuz 12879  +crp 13035  abscabs 15274  cli 15521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-pre-lttri 11230  ax-pre-lttrn 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-neg 11496  df-z 12616  df-uz 12880  df-clim 15525
This theorem is referenced by:  climmpt  15608  climres  15612  climshft  15613  climshft2  15619  isumclim3  15796  iprodclim3  16037  logtayl  26703  dfef2  27015  climexp  45625  climeldmeq  45685  climfveq  45689  climfveqf  45700  climeqf  45708  stirlinglem14  46107  fourierdlem112  46238  vonioolem1  46700
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