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Theorem climeq 14232
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 14169 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦𝑍𝑘 ∈ (ℤ𝑦)((𝐺𝑘) ∈ ℂ ∧ (abs‘((𝐺𝑘) − 𝐴)) < 𝑥))))
6 climeq.3 . . 3 (𝜑𝐺𝑊)
7 eqidd 2622 . . 3 ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐺𝑘))
81, 2, 6, 7clim2 14169 . 2 (𝜑 → (𝐺𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦𝑍𝑘 ∈ (ℤ𝑦)((𝐺𝑘) ∈ ℂ ∧ (abs‘((𝐺𝑘) − 𝐴)) < 𝑥))))
95, 8bitr4d 271 1 (𝜑 → (𝐹𝐴𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908   class class class wbr 4613  cfv 5847  (class class class)co 6604  cc 9878   < clt 10018  cmin 10210  cz 11321  cuz 11631  +crp 11776  abscabs 13908  cli 14149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-neg 10213  df-z 11322  df-uz 11632  df-clim 14153
This theorem is referenced by:  climmpt  14236  climres  14240  climshft  14241  climshft2  14247  isumclim3  14418  iprodclim3  14656  logtayl  24306  dfef2  24597  climexp  39241  climeldmeq  39301  climfveq  39305  climfveqf  39316  climeqf  39324  stirlinglem14  39611  fourierdlem112  39742  vonioolem1  40201
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