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Theorem clim 15527
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 (𝜑𝐹𝑉)
clim.3 ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)
Assertion
Ref Expression
clim (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 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.
StepHypRef Expression
1 climrel 15525 . . . . 5 Rel ⇝
21brrelex2i 5746 . . . 4 (𝐹𝐴𝐴 ∈ V)
32a1i 11 . . 3 (𝜑 → (𝐹𝐴𝐴 ∈ V))
4 elex 3499 . . . . 5 (𝐴 ∈ ℂ → 𝐴 ∈ V)
54adantr 480 . . . 4 ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)) → 𝐴 ∈ V)
65a1i 11 . . 3 (𝜑 → ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)) → 𝐴 ∈ V))
7 clim.1 . . . 4 (𝜑𝐹𝑉)
8 simpr 484 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐴) → 𝑦 = 𝐴)
98eleq1d 2824 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐴) → (𝑦 ∈ ℂ ↔ 𝐴 ∈ ℂ))
10 fveq1 6906 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
1110adantr 480 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐴) → (𝑓𝑘) = (𝐹𝑘))
1211eleq1d 2824 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐴) → ((𝑓𝑘) ∈ ℂ ↔ (𝐹𝑘) ∈ ℂ))
13 oveq12 7440 . . . . . . . . . . . . . 14 (((𝑓𝑘) = (𝐹𝑘) ∧ 𝑦 = 𝐴) → ((𝑓𝑘) − 𝑦) = ((𝐹𝑘) − 𝐴))
1410, 13sylan 580 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐴) → ((𝑓𝑘) − 𝑦) = ((𝐹𝑘) − 𝐴))
1514fveq2d 6911 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐴) → (abs‘((𝑓𝑘) − 𝑦)) = (abs‘((𝐹𝑘) − 𝐴)))
1615breq1d 5158 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐴) → ((abs‘((𝑓𝑘) − 𝑦)) < 𝑥 ↔ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
1712, 16anbi12d 632 . . . . . . . . . 10 ((𝑓 = 𝐹𝑦 = 𝐴) → (((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1817ralbidv 3176 . . . . . . . . 9 ((𝑓 = 𝐹𝑦 = 𝐴) → (∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1918rexbidv 3177 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐴) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
2019ralbidv 3176 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐴) → (∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
219, 20anbi12d 632 . . . . . 6 ((𝑓 = 𝐹𝑦 = 𝐴) → ((𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥)) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
22 df-clim 15521 . . . . . 6 ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥))}
2321, 22brabga 5544 . . . . 5 ((𝐹𝑉𝐴 ∈ V) → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
2423ex 412 . . . 4 (𝐹𝑉 → (𝐴 ∈ V → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))))
257, 24syl 17 . . 3 (𝜑 → (𝐴 ∈ V → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))))
263, 6, 25pm5.21ndd 379 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
27 eluzelz 12886 . . . . . . 7 (𝑘 ∈ (ℤ𝑗) → 𝑘 ∈ ℤ)
28 clim.3 . . . . . . . . 9 ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)
2928eleq1d 2824 . . . . . . . 8 ((𝜑𝑘 ∈ ℤ) → ((𝐹𝑘) ∈ ℂ ↔ 𝐵 ∈ ℂ))
3028fvoveq1d 7453 . . . . . . . . 9 ((𝜑𝑘 ∈ ℤ) → (abs‘((𝐹𝑘) − 𝐴)) = (abs‘(𝐵𝐴)))
3130breq1d 5158 . . . . . . . 8 ((𝜑𝑘 ∈ ℤ) → ((abs‘((𝐹𝑘) − 𝐴)) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑥))
3229, 31anbi12d 632 . . . . . . 7 ((𝜑𝑘 ∈ ℤ) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3327, 32sylan2 593 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3433ralbidva 3174 . . . . 5 (𝜑 → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3534rexbidv 3177 . . . 4 (𝜑 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3635ralbidv 3176 . . 3 (𝜑 → (∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3736anbi2d 630 . 2 (𝜑 → ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
3826, 37bitrd 279 1 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478   class class class wbr 5148  cfv 6563  (class class class)co 7431  cc 11151   < clt 11293  cmin 11490  cz 12611  cuz 12876  +crp 13032  abscabs 15270  cli 15517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-neg 11493  df-z 12612  df-uz 12877  df-clim 15521
This theorem is referenced by:  climcl  15532  clim2  15537  climshftlem  15607  climsuse  45564  0cnv  45698  climuzlem  45699  climisp  45702  climrescn  45704  climxrrelem  45705  climxrre  45706  ioodvbdlimc1lem2  45888  ioodvbdlimc2lem  45890
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