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Theorem clim2f 44905
Description: Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 15442. Similar to clim2 15452, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
nf 𝑘𝐹
clim2f.z 𝑍 = (ℤ𝑀)
clim2f.m (𝜑𝑀 ∈ ℤ)
clim2f.f (𝜑𝐹𝑉)
clim2f.b ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)
Assertion
Ref Expression
clim2f (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
Distinct variable groups:   𝐴,𝑗,𝑘,𝑥   𝑗,𝐹,𝑥   𝑗,𝑀   𝑗,𝑍,𝑘   𝜑,𝑗,𝑘,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑗,𝑘)   𝐹(𝑘)   𝑀(𝑥,𝑘)   𝑉(𝑥,𝑗,𝑘)   𝑍(𝑥)

Proof of Theorem clim2f
StepHypRef Expression
1 nf . . 3 𝑘𝐹
2 clim2f.f . . 3 (𝜑𝐹𝑉)
3 eqidd 2727 . . 3 ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = (𝐹𝑘))
41, 2, 3climf 44891 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
5 clim2f.z . . . . . . . . . 10 𝑍 = (ℤ𝑀)
65uztrn2 12842 . . . . . . . . 9 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
7 clim2f.b . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)
87eleq1d 2812 . . . . . . . . . 10 ((𝜑𝑘𝑍) → ((𝐹𝑘) ∈ ℂ ↔ 𝐵 ∈ ℂ))
97fvoveq1d 7426 . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (abs‘((𝐹𝑘) − 𝐴)) = (abs‘(𝐵𝐴)))
109breq1d 5151 . . . . . . . . . 10 ((𝜑𝑘𝑍) → ((abs‘((𝐹𝑘) − 𝐴)) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑥))
118, 10anbi12d 630 . . . . . . . . 9 ((𝜑𝑘𝑍) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
126, 11sylan2 592 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
1312anassrs 467 . . . . . . 7 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
1413ralbidva 3169 . . . . . 6 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
1514rexbidva 3170 . . . . 5 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
16 clim2f.m . . . . . 6 (𝜑𝑀 ∈ ℤ)
175rexuz3 15299 . . . . . 6 (𝑀 ∈ ℤ → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1816, 17syl 17 . . . . 5 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1915, 18bitr3d 281 . . . 4 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
2019ralbidv 3171 . . 3 (𝜑 → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
2120anbi2d 628 . 2 (𝜑 → ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
224, 21bitr4d 282 1 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wnfc 2877  wral 3055  wrex 3064   class class class wbr 5141  cfv 6536  (class class class)co 7404  cc 11107   < clt 11249  cmin 11445  cz 12559  cuz 12823  +crp 12977  abscabs 15185  cli 15432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-pre-lttri 11183  ax-pre-lttrn 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-neg 11448  df-z 12560  df-uz 12824  df-clim 15436
This theorem is referenced by:  clim2cf  44919
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