Theorem climuz 45759

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 45758	. 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦)))
5		breq2 5147	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦 ↔ (abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥))
6	5	ralbidv 3178	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦 ↔ ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥))
7	6	rexbidv 3179	. . . . . 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 6906	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗))
9	8	raleqdv 3326	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥))
10		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘abs
11		climuz.k	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐹
12		nfcv 2905	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑙
13	11, 12	nffv 6916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘𝑙)
14		nfcv 2905	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 −
15		nfcv 2905	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘𝐴
16	13, 14, 15	nfov 7461	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝐹‘𝑙) − 𝐴)
17	10, 16	nffv 6916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(abs‘((𝐹‘𝑙) − 𝐴))
18		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 <
19		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑥
20	17, 18, 19	nfbr 5190	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥
21		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥
22		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘))
23	22	fvoveq1d 7453	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (abs‘((𝐹‘𝑙) − 𝐴)) = (abs‘((𝐹‘𝑘) − 𝐴)))
24	23	breq1d 5153	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
25	20, 21, 24	cbvralw 3306	. . . . . . . . . 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 279	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
28	27	cbvrexvw 3238	. . . . . . 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 279	. . . . 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 3237	. . . 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 623	. . 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 279	1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2890  ∀wral 3061  ∃wrex 3070   class class class wbr 5143  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℂcc 11153   < clt 11295   − cmin 11492  ℤcz 12613  ℤ_≥cuz 12878  ℝ⁺crp 13034  abscabs 15273   ⇝ cli 15520

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-neg 11495  df-z 12614  df-uz 12879  df-clim 15524

This theorem is referenced by:  liminflimsupclim  45822  climxlim2lem  45860