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Theorem climuz 45759
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 𝑘𝐹
climuz.m (𝜑𝑀 ∈ ℤ)
climuz.z 𝑍 = (ℤ𝑀)
climuz.f (𝜑𝐹:𝑍⟶ℂ)
Assertion
Ref Expression
climuz (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 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.
StepHypRef Expression
1 climuz.m . . 3 (𝜑𝑀 ∈ ℤ)
2 climuz.z . . 3 𝑍 = (ℤ𝑀)
3 climuz.f . . 3 (𝜑𝐹:𝑍⟶ℂ)
41, 2, 3climuzlem 45758 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦)))
5 breq2 5147 . . . . . . . 8 (𝑦 = 𝑥 → ((abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ (abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
65ralbidv 3178 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
76rexbidv 3179 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
8 fveq2 6906 . . . . . . . . . 10 (𝑖 = 𝑗 → (ℤ𝑖) = (ℤ𝑗))
98raleqdv 3326 . . . . . . . . 9 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑗)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
10 nfcv 2905 . . . . . . . . . . . . 13 𝑘abs
11 climuz.k . . . . . . . . . . . . . . 15 𝑘𝐹
12 nfcv 2905 . . . . . . . . . . . . . . 15 𝑘𝑙
1311, 12nffv 6916 . . . . . . . . . . . . . 14 𝑘(𝐹𝑙)
14 nfcv 2905 . . . . . . . . . . . . . 14 𝑘
15 nfcv 2905 . . . . . . . . . . . . . 14 𝑘𝐴
1613, 14, 15nfov 7461 . . . . . . . . . . . . 13 𝑘((𝐹𝑙) − 𝐴)
1710, 16nffv 6916 . . . . . . . . . . . 12 𝑘(abs‘((𝐹𝑙) − 𝐴))
18 nfcv 2905 . . . . . . . . . . . 12 𝑘 <
19 nfcv 2905 . . . . . . . . . . . 12 𝑘𝑥
2017, 18, 19nfbr 5190 . . . . . . . . . . 11 𝑘(abs‘((𝐹𝑙) − 𝐴)) < 𝑥
21 nfv 1914 . . . . . . . . . . 11 𝑙(abs‘((𝐹𝑘) − 𝐴)) < 𝑥
22 fveq2 6906 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → (𝐹𝑙) = (𝐹𝑘))
2322fvoveq1d 7453 . . . . . . . . . . . 12 (𝑙 = 𝑘 → (abs‘((𝐹𝑙) − 𝐴)) = (abs‘((𝐹𝑘) − 𝐴)))
2423breq1d 5153 . . . . . . . . . . 11 (𝑙 = 𝑘 → ((abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
2520, 21, 24cbvralw 3306 . . . . . . . . . 10 (∀𝑙 ∈ (ℤ𝑗)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)
2625a1i 11 . . . . . . . . 9 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑗)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
279, 26bitrd 279 . . . . . . . 8 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
2827cbvrexvw 3238 . . . . . . 7 (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)
2928a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
307, 29bitrd 279 . . . . 5 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
3130cbvralvw 3237 . . . 4 (∀𝑦 ∈ ℝ+𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)
3231anbi2i 623 . . 3 ((𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
3332a1i 11 . 2 (𝜑 → ((𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
344, 33bitrd 279 1 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wnfc 2890  wral 3061  wrex 3070   class class class wbr 5143  wf 6557  cfv 6561  (class class class)co 7431  cc 11153   < clt 11295  cmin 11492  cz 12613  cuz 12878  +crp 13034  abscabs 15273  cli 15520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-neg 11495  df-z 12614  df-uz 12879  df-clim 15524
This theorem is referenced by:  liminflimsupclim  45822  climxlim2lem  45860
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