Theorem climuz 46441

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 46440	. 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦)))
5		breq2 5114	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦 ↔ (abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥))
6	5	ralbidv 3188	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑦 ↔ ∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥))
7	6	rexbidv 3189	. . . . . 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 6883	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗))
9	8	raleqdv 3323	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥))
10		nfcv 2925	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘abs
11		climuz.k	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐹
12		nfcv 2925	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑙
13	11, 12	nffv 6893	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘𝑙)
14		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 −
15		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘𝐴
16	13, 14, 15	nfov 7442	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝐹‘𝑙) − 𝐴)
17	10, 16	nffv 6893	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(abs‘((𝐹‘𝑙) − 𝐴))
18		nfcv 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 <
19		nfcv 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑥
20	17, 18, 19	nfbr 5159	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥
21		nfv 1944	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥
22		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘))
23	22	fvoveq1d 7434	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (abs‘((𝐹‘𝑙) − 𝐴)) = (abs‘((𝐹‘𝑘) − 𝐴)))
24	23	breq1d 5120	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((abs‘((𝐹‘𝑙) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
25	20, 21, 24	cbvralw 3307	. . . . . . . . . 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 3244	. . . . . . 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 3243	. . . 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 634	. . 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 400   = wceq 1570   ∈ wcel 2143  Ⅎwnfc 2910  ∀wral 3079  ∃wrex 3089   class class class wbr 5110  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  ℂcc 11099   < clt 11244   − cmin 11442  ℤcz 12592  ℤ_≥cuz 12863  ℝ⁺crp 13017  abscabs 15287   ⇝ cli 15537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-pre-lttri 11175  ax-pre-lttrn 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-po 5571  df-so 5572  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7415  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-neg 11445  df-z 12593  df-uz 12864  df-clim 15541

This theorem is referenced by:  liminflimsupclim  46504  climxlim2lem  46542