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Theorem climeqf 45703
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 1914 . . . . 5 𝑘 𝑗𝑍
75, 6nfan 1899 . . . 4 𝑘(𝜑𝑗𝑍)
8 climeqf.k . . . . . 6 𝑘𝐹
9 nfcv 2905 . . . . . 6 𝑘𝑗
108, 9nffv 6916 . . . . 5 𝑘(𝐹𝑗)
11 climeqf.n . . . . . 6 𝑘𝐺
1211, 9nffv 6916 . . . . 5 𝑘(𝐺𝑗)
1310, 12nfeq 2919 . . . 4 𝑘(𝐹𝑗) = (𝐺𝑗)
147, 13nfim 1896 . . 3 𝑘((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))
15 eleq1w 2824 . . . . 5 (𝑘 = 𝑗 → (𝑘𝑍𝑗𝑍))
1615anbi2d 630 . . . 4 (𝑘 = 𝑗 → ((𝜑𝑘𝑍) ↔ (𝜑𝑗𝑍)))
17 fveq2 6906 . . . . 5 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
18 fveq2 6906 . . . . 5 (𝑘 = 𝑗 → (𝐺𝑘) = (𝐺𝑗))
1917, 18eqeq12d 2753 . . . 4 (𝑘 = 𝑗 → ((𝐹𝑘) = (𝐺𝑘) ↔ (𝐹𝑗) = (𝐺𝑗)))
2016, 19imbi12d 344 . . 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 15603 1 (𝜑 → (𝐹𝐴𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wnf 1783  wcel 2108  wnfc 2890   class class class wbr 5143  cfv 6561  cz 12613  cuz 12878  cli 15520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-neg 11495  df-z 12614  df-uz 12879  df-clim 15524
This theorem is referenced by:  climeqmpt  45712
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