Theorem climeqf 46698

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 1946	. . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
7	5, 6	nfan 1931	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
8		climeqf.k	. . . . . 6 ⊢ Ⅎ𝑘𝐹
9		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑘𝑗
10	8, 9	nffv 6895	. . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗)
11		climeqf.n	. . . . . 6 ⊢ Ⅎ𝑘𝐺
12	11, 9	nffv 6895	. . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗)
13	10, 12	nfeq 2935	. . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗)
14	7, 13	nfim 1928	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗))
15		eleq1w 2843	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
16	15	anbi2d 642	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
17		fveq2 6885	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
18		fveq2 6885	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗))
19	17, 18	eqeq12d 2776	. . . 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 2276	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗))
23	1, 2, 3, 4, 22	climeq 15734	1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144  Ⅎwnfc 2907   class class class wbr 5102  ‘cfv 6538  ℤcz 12693  ℤ_≥cuz 12965   ⇝ cli 15651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-pre-lttri 11274  ax-pre-lttrn 11275

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-po 5558  df-so 5559  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-neg 11544  df-z 12694  df-uz 12966  df-clim 15655

This theorem is used by:  climeqmpt  46707