Theorem climeqf 44958

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 1909	. . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
7	5, 6	nfan 1894	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
8		climeqf.k	. . . . . 6 ⊢ Ⅎ𝑘𝐹
9		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑘𝑗
10	8, 9	nffv 6894	. . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗)
11		climeqf.n	. . . . . 6 ⊢ Ⅎ𝑘𝐺
12	11, 9	nffv 6894	. . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗)
13	10, 12	nfeq 2910	. . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗)
14	7, 13	nfim 1891	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗))
15		eleq1w 2810	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
16	15	anbi2d 628	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
17		fveq2 6884	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
18		fveq2 6884	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗))
19	17, 18	eqeq12d 2742	. . . 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 2225	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗))
23	1, 2, 3, 4, 22	climeq 15514	1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2877   class class class wbr 5141  ‘cfv 6536  ℤcz 12559  ℤ_≥cuz 12823   ⇝ cli 15431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-pre-lttri 11183  ax-pre-lttrn 11184

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-neg 11448  df-z 12560  df-uz 12824  df-clim 15435

This theorem is referenced by:  climeqmpt  44967