Theorem climeqf 43119

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	⊢ Ⅎ𝑘𝜑
climeqf.k	⊢ Ⅎ𝑘𝐹
climeqf.n	⊢ Ⅎ𝑘𝐺
climeqf.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
climeqf.z	⊢ 𝑍 = (ℤ≥‘𝑀)
climeqf.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
climeqf.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)
climeqf.e	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘))

Assertion

Ref	Expression
climeqf	⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))

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.

Step	Hyp	Ref	Expression
1		climeqf.z	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		climeqf.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
3		climeqf.g	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
4		climeqf.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		climeqf.p	. . . . 5 ⊢ Ⅎ𝑘𝜑
6		nfv 1918	. . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
7	5, 6	nfan 1903	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
8		climeqf.k	. . . . . 6 ⊢ Ⅎ𝑘𝐹
9		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑘𝑗
10	8, 9	nffv 6766	. . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗)
11		climeqf.n	. . . . . 6 ⊢ Ⅎ𝑘𝐺
12	11, 9	nffv 6766	. . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗)
13	10, 12	nfeq 2919	. . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗)
14	7, 13	nfim 1900	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗))
15		eleq1w 2821	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
16	15	anbi2d 628	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
17		fveq2 6756	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
18		fveq2 6756	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗))
19	17, 18	eqeq12d 2754	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑗) = (𝐺‘𝑗)))
20	16, 19	imbi12d 344	. . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗))))
21		climeqf.e	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘))
22	14, 20, 21	chvarfv 2236	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗))
23	1, 2, 3, 4, 22	climeq 15204	1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2108  Ⅎwnfc 2886   class class class wbr 5070  ‘cfv 6418  ℤcz 12249  ℤ_≥cuz 12511   ⇝ cli 15121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-pre-lttri 10876  ax-pre-lttrn 10877

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-neg 11138  df-z 12250  df-uz 12512  df-clim 15125

This theorem is referenced by:  climeqmpt  43128