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Theorem climeqf 40238
 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 1883 . . . . 5 𝑘 𝑗𝑍
75, 6nfan 1868 . . . 4 𝑘(𝜑𝑗𝑍)
8 climeqf.k . . . . . 6 𝑘𝐹
9 nfcv 2793 . . . . . 6 𝑘𝑗
108, 9nffv 6236 . . . . 5 𝑘(𝐹𝑗)
11 climeqf.n . . . . . 6 𝑘𝐺
1211, 9nffv 6236 . . . . 5 𝑘(𝐺𝑗)
1310, 12nfeq 2805 . . . 4 𝑘(𝐹𝑗) = (𝐺𝑗)
147, 13nfim 1865 . . 3 𝑘((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))
15 eleq1 2718 . . . . 5 (𝑘 = 𝑗 → (𝑘𝑍𝑗𝑍))
1615anbi2d 740 . . . 4 (𝑘 = 𝑗 → ((𝜑𝑘𝑍) ↔ (𝜑𝑗𝑍)))
17 fveq2 6229 . . . . 5 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
18 fveq2 6229 . . . . 5 (𝑘 = 𝑗 → (𝐺𝑘) = (𝐺𝑗))
1917, 18eqeq12d 2666 . . . 4 (𝑘 = 𝑗 → ((𝐹𝑘) = (𝐺𝑘) ↔ (𝐹𝑗) = (𝐺𝑗)))
2016, 19imbi12d 333 . . 3 (𝑘 = 𝑗 → (((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘)) ↔ ((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))))
21 climeqf.e . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))
2214, 20, 21chvar 2298 . 2 ((𝜑𝑗𝑍) → (𝐹𝑗) = (𝐺𝑗))
231, 2, 3, 4, 22climeq 14342 1 (𝜑 → (𝐹𝐴𝐺𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  Ⅎwnf 1748   ∈ wcel 2030  Ⅎwnfc 2780   class class class wbr 4685  ‘cfv 5926  ℤcz 11415  ℤ≥cuz 11725   ⇝ cli 14259 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-neg 10307  df-z 11416  df-uz 11726  df-clim 14263 This theorem is referenced by:  climeqmpt  40247
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