Theorem climeqf 46901

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 1949	. . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
7	5, 6	nfan 1934	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
8		climeqf.k	. . . . . 6 ⊢ Ⅎ𝑘𝐹
9		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑘𝑗
10	8, 9	nffv 6915	. . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗)
11		climeqf.n	. . . . . 6 ⊢ Ⅎ𝑘𝐺
12	11, 9	nffv 6915	. . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗)
13	10, 12	nfeq 2939	. . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗)
14	7, 13	nfim 1931	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗))
15		eleq1w 2847	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
16	15	anbi2d 642	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
17		fveq2 6905	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
18		fveq2 6905	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗))
19	17, 18	eqeq12d 2779	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑗) = (𝐺‘𝑗)))
20	16, 19	imbi12d 347	. . 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 2280	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗))
23	1, 2, 3, 4, 22	climeq 15760	1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147  Ⅎwnfc 2911   class class class wbr 5106  ‘cfv 6555  ℤcz 12717  ℤ_≥cuz 12989   ⇝ cli 15677

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-pre-lttri 11298  ax-pre-lttrn 11299

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-po 5563  df-so 5564  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-neg 11568  df-z 12718  df-uz 12990  df-clim 15681

This theorem is used by:  climeqmpt  46910