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Theorem climeqf 41845
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 1906 . . . . 5 𝑘 𝑗𝑍
75, 6nfan 1891 . . . 4 𝑘(𝜑𝑗𝑍)
8 climeqf.k . . . . . 6 𝑘𝐹
9 nfcv 2974 . . . . . 6 𝑘𝑗
108, 9nffv 6673 . . . . 5 𝑘(𝐹𝑗)
11 climeqf.n . . . . . 6 𝑘𝐺
1211, 9nffv 6673 . . . . 5 𝑘(𝐺𝑗)
1310, 12nfeq 2988 . . . 4 𝑘(𝐹𝑗) = (𝐺𝑗)
147, 13nfim 1888 . . 3 𝑘((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))
15 eleq1w 2892 . . . . 5 (𝑘 = 𝑗 → (𝑘𝑍𝑗𝑍))
1615anbi2d 628 . . . 4 (𝑘 = 𝑗 → ((𝜑𝑘𝑍) ↔ (𝜑𝑗𝑍)))
17 fveq2 6663 . . . . 5 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
18 fveq2 6663 . . . . 5 (𝑘 = 𝑗 → (𝐺𝑘) = (𝐺𝑗))
1917, 18eqeq12d 2834 . . . 4 (𝑘 = 𝑗 → ((𝐹𝑘) = (𝐺𝑘) ↔ (𝐹𝑗) = (𝐺𝑗)))
2016, 19imbi12d 346 . . 3 (𝑘 = 𝑗 → (((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘)) ↔ ((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))))
21 climeqf.e . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))
2214, 20, 21chvarfv 2232 . 2 ((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))
231, 2, 3, 4, 22climeq 14912 1 (𝜑 → (𝐹𝐴𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wnf 1775  wcel 2105  wnfc 2958   class class class wbr 5057  cfv 6348  cz 11969  cuz 12231  cli 14829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-pre-lttri 10599  ax-pre-lttrn 10600
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-neg 10861  df-z 11970  df-uz 12232  df-clim 14833
This theorem is referenced by:  climeqmpt  41854
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