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Theorem climeqf 42498
 Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
climeqf.p 𝑘𝜑
climeqf.k 𝑘𝐹
climeqf.n 𝑘𝐺
climeqf.m (𝜑𝑀 ∈ ℤ)
climeqf.z 𝑍 = (ℤ𝑀)
climeqf.f (𝜑𝐹𝑉)
climeqf.g (𝜑𝐺𝑊)
climeqf.e ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))
Assertion
Ref Expression
climeqf (𝜑 → (𝐹𝐴𝐺𝐴))
Distinct variable group:   𝑘,𝑍
Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐹(𝑘)   𝐺(𝑘)   𝑀(𝑘)   𝑉(𝑘)   𝑊(𝑘)

Proof of Theorem climeqf
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 climeqf.z . 2 𝑍 = (ℤ𝑀)
2 climeqf.f . 2 (𝜑𝐹𝑉)
3 climeqf.g . 2 (𝜑𝐺𝑊)
4 climeqf.m . 2 (𝜑𝑀 ∈ ℤ)
5 climeqf.p . . . . 5 𝑘𝜑
6 nfv 1915 . . . . 5 𝑘 𝑗𝑍
75, 6nfan 1900 . . . 4 𝑘(𝜑𝑗𝑍)
8 climeqf.k . . . . . 6 𝑘𝐹
9 nfcv 2955 . . . . . 6 𝑘𝑗
108, 9nffv 6665 . . . . 5 𝑘(𝐹𝑗)
11 climeqf.n . . . . . 6 𝑘𝐺
1211, 9nffv 6665 . . . . 5 𝑘(𝐺𝑗)
1310, 12nfeq 2968 . . . 4 𝑘(𝐹𝑗) = (𝐺𝑗)
147, 13nfim 1897 . . 3 𝑘((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))
15 eleq1w 2872 . . . . 5 (𝑘 = 𝑗 → (𝑘𝑍𝑗𝑍))
1615anbi2d 631 . . . 4 (𝑘 = 𝑗 → ((𝜑𝑘𝑍) ↔ (𝜑𝑗𝑍)))
17 fveq2 6655 . . . . 5 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
18 fveq2 6655 . . . . 5 (𝑘 = 𝑗 → (𝐺𝑘) = (𝐺𝑗))
1917, 18eqeq12d 2814 . . . 4 (𝑘 = 𝑗 → ((𝐹𝑘) = (𝐺𝑘) ↔ (𝐹𝑗) = (𝐺𝑗)))
2016, 19imbi12d 348 . . 3 (𝑘 = 𝑗 → (((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘)) ↔ ((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))))
21 climeqf.e . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))
2214, 20, 21chvarfv 2240 . 2 ((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))
231, 2, 3, 4, 22climeq 14936 1 (𝜑 → (𝐹𝐴𝐺𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  Ⅎwnfc 2936   class class class wbr 5034  ‘cfv 6332  ℤcz 11989  ℤ≥cuz 12251   ⇝ cli 14853 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-pre-lttri 10618  ax-pre-lttrn 10619 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-neg 10880  df-z 11990  df-uz 12252  df-clim 14857 This theorem is referenced by:  climeqmpt  42507
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