Theorem clim 15203

Description: Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. This means that for any real 𝑥, no matter how small, there always exists an integer 𝑗 such that the absolute difference of any later complex number in the sequence and the limit is less than 𝑥. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Hypotheses

Ref	Expression
clim.1	⊢ (𝜑 → 𝐹 ∈ 𝑉)
clim.3	⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = 𝐵)

Assertion

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

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

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

Proof of Theorem clim

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

Step	Hyp	Ref	Expression
1		climrel 15201	. . . . 5 ⊢ Rel ⇝
2	1	brrelex2i 5644	. . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ V)
3	2	a1i 11	. . 3 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 → 𝐴 ∈ V))
4		elex 3450	. . . . 5 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ V)
5	4	adantr 481	. . . 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		clim.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
8		simpr 485	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
9	8	eleq1d 2823	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (𝑦 ∈ ℂ ↔ 𝐴 ∈ ℂ))
10		fveq1 6773	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑘) = (𝐹‘𝑘))
11	10	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (𝑓‘𝑘) = (𝐹‘𝑘))
12	11	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → ((𝑓‘𝑘) ∈ ℂ ↔ (𝐹‘𝑘) ∈ ℂ))
13		oveq12 7284	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑘) = (𝐹‘𝑘) ∧ 𝑦 = 𝐴) → ((𝑓‘𝑘) − 𝑦) = ((𝐹‘𝑘) − 𝐴))
14	10, 13	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → ((𝑓‘𝑘) − 𝑦) = ((𝐹‘𝑘) − 𝐴))
15	14	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (abs‘((𝑓‘𝑘) − 𝑦)) = (abs‘((𝐹‘𝑘) − 𝐴)))
16	15	breq1d 5084	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → ((abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
17	12, 16	anbi12d 631	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
18	17	ralbidv 3112	. . . . . . . . 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)))
19	18	rexbidv 3226	. . . . . . . 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)))
20	19	ralbidv 3112	. . . . . . 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)))
21	9, 20	anbi12d 631	. . . . . 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))))
22		df-clim 15197	. . . . . 6 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
23	21, 22	brabga 5447	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
24	23	ex 413	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐴 ∈ V → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))))
25	7, 24	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)))))
26	3, 6, 25	pm5.21ndd 381	. 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
27		eluzelz 12592	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑗) → 𝑘 ∈ ℤ)
28		clim.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = 𝐵)
29	28	eleq1d 2823	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((𝐹‘𝑘) ∈ ℂ ↔ 𝐵 ∈ ℂ))
30	28	fvoveq1d 7297	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(𝐵 − 𝐴)))
31	30	breq1d 5084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝑥))
32	29, 31	anbi12d 631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))
33	27, 32	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))
34	33	ralbidva 3111	. . . . 5 ⊢ (𝜑 → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))
35	34	rexbidv 3226	. . . 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)))
36	35	ralbidv 3112	. . 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)))
37	36	anbi2d 629	. 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))))
38	26, 37	bitrd 278	1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  ℂcc 10869   < clt 11009   − cmin 11205  ℤcz 12319  ℤ_≥cuz 12582  ℝ⁺crp 12730  abscabs 14945   ⇝ cli 15193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-cnex 10927  ax-resscn 10928

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-neg 11208  df-z 12320  df-uz 12583  df-clim 15197

This theorem is referenced by:  climcl  15208  clim2  15213  climshftlem  15283  climsuse  43149  0cnv  43283  climuzlem  43284  climisp  43287  climrescn  43289  climxrrelem  43290  climxrre  43291  ioodvbdlimc1lem2  43473  ioodvbdlimc2lem  43475