Theorem clim 14225

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 14223	. . . . 5 ⊢ Rel ⇝
2	1	brrelex2i 5159	. . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ V)
3	2	a1i 11	. . 3 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 → 𝐴 ∈ V))
4		elex 3212	. . . . 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 477	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
9	8	eleq1d 2686	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (𝑦 ∈ ℂ ↔ 𝐴 ∈ ℂ))
10		fveq1 6190	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑘) = (𝐹‘𝑘))
11	10	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (𝑓‘𝑘) = (𝐹‘𝑘))
12	11	eleq1d 2686	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → ((𝑓‘𝑘) ∈ ℂ ↔ (𝐹‘𝑘) ∈ ℂ))
13		oveq12 6659	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑘) = (𝐹‘𝑘) ∧ 𝑦 = 𝐴) → ((𝑓‘𝑘) − 𝑦) = ((𝐹‘𝑘) − 𝐴))
14	10, 13	sylan 488	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → ((𝑓‘𝑘) − 𝑦) = ((𝐹‘𝑘) − 𝐴))
15	14	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (abs‘((𝑓‘𝑘) − 𝑦)) = (abs‘((𝐹‘𝑘) − 𝐴)))
16	15	breq1d 4663	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → ((abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
17	12, 16	anbi12d 747	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐴) → (((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
18	17	ralbidv 2986	. . . . . . . . 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 3052	. . . . . . . 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 2986	. . . . . . 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 747	. . . . . 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 14219	. . . . . 6 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
23	21, 22	brabga 4989	. . . . 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 450	. . . 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 369	. 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
27		eluzelz 11697	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑗) → 𝑘 ∈ ℤ)
28		clim.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = 𝐵)
29	28	eleq1d 2686	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((𝐹‘𝑘) ∈ ℂ ↔ 𝐵 ∈ ℂ))
30	28	oveq1d 6665	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((𝐹‘𝑘) − 𝐴) = (𝐵 − 𝐴))
31	30	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(𝐵 − 𝐴)))
32	31	breq1d 4663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝑥))
33	29, 32	anbi12d 747	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))
34	27, 33	sylan2 491	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))
35	34	ralbidva 2985	. . . . 5 ⊢ (𝜑 → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))
36	35	rexbidv 3052	. . . 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)))
37	36	ralbidv 2986	. . 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)))
38	37	anbi2d 740	. 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))))
39	26, 38	bitrd 268	1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℂcc 9934   < clt 10074   − cmin 10266  ℤcz 11377  ℤ_≥cuz 11687  ℝ⁺crp 11832  abscabs 13974   ⇝ cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-neg 10269  df-z 11378  df-uz 11688  df-clim 14219

This theorem is referenced by:  climcl  14230  clim2  14235  climshftlem  14305  climsuse  39840  0cnv  39974  climuzlem  39975  climisp  39978  climrescn  39980  climxrrelem  39981  climxrre  39982  ioodvbdlimc1lem2  40147  ioodvbdlimc2lem  40149