Theorem climuz 42386

Description: Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
climuz.k	⊢ Ⅎ𝑘𝐹
climuz.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
climuz.z	⊢ 𝑍 = (ℤ≥‘𝑀)
climuz.f	⊢ (𝜑 → 𝐹:𝑍⟶ℂ)

Assertion

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

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

Allowed substitution hints:   𝜑(𝑥,𝑗,𝑘)   𝐹(𝑘)   𝑀(𝑥,𝑗,𝑘)   𝑍(𝑘)

Proof of Theorem climuz

Dummy variables 𝑖 𝑙 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climuz.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		climuz.z	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑀)
3		climuz.f	. . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
4	1, 2, 3	climuzlem 42385	. 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦)))
5		breq2 5034	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦 ↔ (abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥))
6	5	ralbidv 3162	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦 ↔ ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥))
7	6	rexbidv 3256	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦 ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥))
8		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗))
9	8	raleqdv 3364	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥))
10		nfcv 2955	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘abs
11		climuz.k	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐹
12		nfcv 2955	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑙
13	11, 12	nffv 6655	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘𝑙)
14		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 −
15		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘𝐴
16	13, 14, 15	nfov 7165	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝐹‘𝑙) − 𝐴)
17	10, 16	nffv 6655	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(abs‘((𝐹‘𝑙) − 𝐴))
18		nfcv 2955	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 <
19		nfcv 2955	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑥
20	17, 18, 19	nfbr 5077	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥
21		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥
22		fveq2 6645	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘))
23	22	fvoveq1d 7157	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (abs‘((𝐹‘𝑙) − 𝐴)) = (abs‘((𝐹‘𝑘) − 𝐴)))
24	23	breq1d 5040	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
25	20, 21, 24	cbvralw 3387	. . . . . . . . . 10 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
26	25	a1i 11	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
27	9, 26	bitrd 282	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
28	27	cbvrexvw 3397	. . . . . . 7 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
29	28	a1i 11	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
30	7, 29	bitrd 282	. . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
31	30	cbvralvw 3396	. . . 4 ⊢ (∀𝑦 ∈ ℝ+ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
32	31	anbi2i 625	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
33	32	a1i 11	. 2 ⊢ (𝜑 → ((𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
34	4, 33	bitrd 282	1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  ∀wral 3106  ∃wrex 3107   class class class wbr 5030  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ℂcc 10524   < clt 10664   − cmin 10859  ℤcz 11969  ℤ_≥cuz 12231  ℝ⁺crp 12377  abscabs 14585   ⇝ cli 14833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-pre-lttri 10600  ax-pre-lttrn 10601

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-neg 10862  df-z 11970  df-uz 12232  df-clim 14837

This theorem is referenced by:  liminflimsupclim  42449  climxlim2lem  42487