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Theorem clim 10633
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 10632 . . . . 5 Rel ⇝
21brrelex2i 4471 . . . 4 (𝐹𝐴𝐴 ∈ V)
32a1i 9 . . 3 (𝜑 → (𝐹𝐴𝐴 ∈ V))
4 elex 2630 . . . . 5 (𝐴 ∈ ℂ → 𝐴 ∈ V)
54adantr 270 . . . 4 ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)) → 𝐴 ∈ V)
65a1i 9 . . 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 108 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐴) → 𝑦 = 𝐴)
98eleq1d 2156 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐴) → (𝑦 ∈ ℂ ↔ 𝐴 ∈ ℂ))
10 fveq1 5288 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
1110adantr 270 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐴) → (𝑓𝑘) = (𝐹𝑘))
1211eleq1d 2156 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐴) → ((𝑓𝑘) ∈ ℂ ↔ (𝐹𝑘) ∈ ℂ))
13 oveq12 5643 . . . . . . . . . . . . . 14 (((𝑓𝑘) = (𝐹𝑘) ∧ 𝑦 = 𝐴) → ((𝑓𝑘) − 𝑦) = ((𝐹𝑘) − 𝐴))
1410, 13sylan 277 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐴) → ((𝑓𝑘) − 𝑦) = ((𝐹𝑘) − 𝐴))
1514fveq2d 5293 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐴) → (abs‘((𝑓𝑘) − 𝑦)) = (abs‘((𝐹𝑘) − 𝐴)))
1615breq1d 3847 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐴) → ((abs‘((𝑓𝑘) − 𝑦)) < 𝑥 ↔ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
1712, 16anbi12d 457 . . . . . . . . . 10 ((𝑓 = 𝐹𝑦 = 𝐴) → (((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1817ralbidv 2380 . . . . . . . . 9 ((𝑓 = 𝐹𝑦 = 𝐴) → (∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1918rexbidv 2381 . . . . . . . 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 2380 . . . . . . 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 457 . . . . . 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 10631 . . . . . 6 ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥))}
2321, 22brabga 4082 . . . . 5 ((𝐹𝑉𝐴 ∈ V) → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
2423ex 113 . . . 4 (𝐹𝑉 → (𝐴 ∈ V → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))))
257, 24syl 14 . . 3 (𝜑 → (𝐴 ∈ V → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))))
263, 6, 25pm5.21ndd 656 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
27 eluzelz 8997 . . . . . . 7 (𝑘 ∈ (ℤ𝑗) → 𝑘 ∈ ℤ)
28 clim.3 . . . . . . . . 9 ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)
2928eleq1d 2156 . . . . . . . 8 ((𝜑𝑘 ∈ ℤ) → ((𝐹𝑘) ∈ ℂ ↔ 𝐵 ∈ ℂ))
3028oveq1d 5649 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℤ) → ((𝐹𝑘) − 𝐴) = (𝐵𝐴))
3130fveq2d 5293 . . . . . . . . 9 ((𝜑𝑘 ∈ ℤ) → (abs‘((𝐹𝑘) − 𝐴)) = (abs‘(𝐵𝐴)))
3231breq1d 3847 . . . . . . . 8 ((𝜑𝑘 ∈ ℤ) → ((abs‘((𝐹𝑘) − 𝐴)) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑥))
3329, 32anbi12d 457 . . . . . . 7 ((𝜑𝑘 ∈ ℤ) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3427, 33sylan2 280 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3534ralbidva 2376 . . . . 5 (𝜑 → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3635rexbidv 2381 . . . 4 (𝜑 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3736ralbidv 2380 . . 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)))
3837anbi2d 452 . 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))))
3926, 38bitrd 186 1 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359  wrex 2360  Vcvv 2619   class class class wbr 3837  cfv 5002  (class class class)co 5634  cc 7327   < clt 7501  cmin 7632  cz 8720  cuz 8988  +crp 9103  abscabs 10395  cli 10630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-cnex 7415  ax-resscn 7416
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-ov 5637  df-neg 7635  df-z 8721  df-uz 8989  df-clim 10631
This theorem is referenced by:  climcl  10634  clim2  10635  climshftlemg  10654
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