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Theorem climeqf 45644
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 1912 . . . . 5 𝑘 𝑗𝑍
75, 6nfan 1897 . . . 4 𝑘(𝜑𝑗𝑍)
8 climeqf.k . . . . . 6 𝑘𝐹
9 nfcv 2903 . . . . . 6 𝑘𝑗
108, 9nffv 6917 . . . . 5 𝑘(𝐹𝑗)
11 climeqf.n . . . . . 6 𝑘𝐺
1211, 9nffv 6917 . . . . 5 𝑘(𝐺𝑗)
1310, 12nfeq 2917 . . . 4 𝑘(𝐹𝑗) = (𝐺𝑗)
147, 13nfim 1894 . . 3 𝑘((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))
15 eleq1w 2822 . . . . 5 (𝑘 = 𝑗 → (𝑘𝑍𝑗𝑍))
1615anbi2d 630 . . . 4 (𝑘 = 𝑗 → ((𝜑𝑘𝑍) ↔ (𝜑𝑗𝑍)))
17 fveq2 6907 . . . . 5 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
18 fveq2 6907 . . . . 5 (𝑘 = 𝑗 → (𝐺𝑘) = (𝐺𝑗))
1917, 18eqeq12d 2751 . . . 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 2238 . 2 ((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))
231, 2, 3, 4, 22climeq 15600 1 (𝜑 → (𝐹𝐴𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wnf 1780  wcel 2106  wnfc 2888   class class class wbr 5148  cfv 6563  cz 12611  cuz 12876  cli 15517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-neg 11493  df-z 12612  df-uz 12877  df-clim 15521
This theorem is referenced by:  climeqmpt  45653
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