Theorem climf2 45662

Description: Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 15515, but without the disjoint var constraint 𝜑𝑘 and 𝐹𝑘. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
climf2.1	⊢ Ⅎ𝑘𝜑
climf2.nf	⊢ Ⅎ𝑘𝐹
climf2.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
climf2.fv	⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = 𝐵)

Assertion

Ref	Expression
climf2	⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))))

Distinct variable groups:   𝐴,𝑗,𝑘,𝑥   𝑗,𝐹,𝑥   𝜑,𝑗,𝑥

Allowed substitution hints:   𝜑(𝑘)   𝐵(𝑥,𝑗,𝑘)   𝐹(𝑘)   𝑉(𝑥,𝑗,𝑘)

Proof of Theorem climf2

Dummy variables 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climrel 15513	. . . . 5 ⊢ Rel ⇝
2	1	brrelex2i 5716	. . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ V)
3	2	a1i 11	. . 3 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 → 𝐴 ∈ V))
4		elex 3485	. . . . 5 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ V)
5	4	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)) → 𝐴 ∈ V)
6	5	a1i 11	. . 3 ⊢ (𝜑 → ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)) → 𝐴 ∈ V))
7		climf2.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
8		simpr 484	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
9	8	eleq1d 2820	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (𝑦 ∈ ℂ ↔ 𝐴 ∈ ℂ))
10		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑓 = 𝐹 ∧ 𝑦 = 𝐴)
11		climf2.nf	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝐹
12	11	nfeq2 2917	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑓 = 𝐹
13		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑦 = 𝐴
14	12, 13	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑓 = 𝐹 ∧ 𝑦 = 𝐴)
15		fveq1 6880	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑘) = (𝐹‘𝑘))
16	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (𝑓‘𝑘) = (𝐹‘𝑘))
17	16	eleq1d 2820	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → ((𝑓‘𝑘) ∈ ℂ ↔ (𝐹‘𝑘) ∈ ℂ))
18		oveq12 7419	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑘) = (𝐹‘𝑘) ∧ 𝑦 = 𝐴) → ((𝑓‘𝑘) − 𝑦) = ((𝐹‘𝑘) − 𝐴))
19	15, 18	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → ((𝑓‘𝑘) − 𝑦) = ((𝐹‘𝑘) − 𝐴))
20	19	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (abs‘((𝑓‘𝑘) − 𝑦)) = (abs‘((𝐹‘𝑘) − 𝐴)))
21	20	breq1d 5134	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → ((abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
22	17, 21	anbi12d 632	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
23	14, 22	ralbid 3259	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
24	23	rexbidv 3165	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
25	10, 24	ralbid 3259	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
26	9, 25	anbi12d 632	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → ((𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥)) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
27		df-clim 15509	. . . . . 6 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
28	26, 27	brabga 5514	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
29	28	ex 412	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐴 ∈ V → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))))
30	7, 29	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ V → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))))
31	3, 6, 30	pm5.21ndd 379	. 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
32		climf2.1	. . . . . 6 ⊢ Ⅎ𝑘𝜑
33		eluzelz 12867	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑗) → 𝑘 ∈ ℤ)
34		climf2.fv	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = 𝐵)
35	34	eleq1d 2820	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((𝐹‘𝑘) ∈ ℂ ↔ 𝐵 ∈ ℂ))
36	34	fvoveq1d 7432	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(𝐵 − 𝐴)))
37	36	breq1d 5134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝑥))
38	35, 37	anbi12d 632	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))
39	33, 38	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))
40	32, 39	ralbida 3257	. . . . 5 ⊢ (𝜑 → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))
41	40	rexbidv 3165	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))
42	41	ralbidv 3164	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))
43	42	anbi2d 630	. 2 ⊢ (𝜑 → ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))))
44	31, 43	bitrd 279	1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2884  ∀wral 3052  ∃wrex 3061  Vcvv 3464   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  ℂcc 11132   < clt 11274   − cmin 11471  ℤcz 12593  ℤ_≥cuz 12857  ℝ⁺crp 13013  abscabs 15258   ⇝ cli 15505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-cnex 11190  ax-resscn 11191

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-neg 11474  df-z 12594  df-uz 12858  df-clim 15509

This theorem is referenced by:  clim2f2  45666