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Theorem clim2d 39705
Description: The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
clim2d.k 𝑘𝜑
clim2d.f 𝑘𝐹
clim2d.m (𝜑𝑀 ∈ ℤ)
clim2d.z 𝑍 = (ℤ𝑀)
clim2d.c (𝜑𝐹𝐴)
clim2d.b ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)
clim2d.x (𝜑𝑋 ∈ ℝ+)
Assertion
Ref Expression
clim2d (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋))
Distinct variable groups:   𝐴,𝑗,𝑘   𝑗,𝐹   𝑗,𝑀   𝑗,𝑋,𝑘   𝑗,𝑍,𝑘   𝜑,𝑗
Allowed substitution hints:   𝜑(𝑘)   𝐵(𝑗,𝑘)   𝐹(𝑘)   𝑀(𝑘)

Proof of Theorem clim2d
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 clim2d.x . 2 (𝜑𝑋 ∈ ℝ+)
2 clim2d.c . . . 4 (𝜑𝐹𝐴)
3 clim2d.k . . . . 5 𝑘𝜑
4 clim2d.f . . . . 5 𝑘𝐹
5 clim2d.z . . . . 5 𝑍 = (ℤ𝑀)
6 clim2d.m . . . . 5 (𝜑𝑀 ∈ ℤ)
7 climrel 14204 . . . . . . 7 Rel ⇝
87a1i 11 . . . . . 6 (𝜑 → Rel ⇝ )
9 brrelex 5146 . . . . . 6 ((Rel ⇝ ∧ 𝐹𝐴) → 𝐹 ∈ V)
108, 2, 9syl2anc 692 . . . . 5 (𝜑𝐹 ∈ V)
11 clim2d.b . . . . 5 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)
123, 4, 5, 6, 10, 11clim2f2 39702 . . . 4 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
132, 12mpbid 222 . . 3 (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
1413simprd 479 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))
15 breq2 4648 . . . . . 6 (𝑥 = 𝑋 → ((abs‘(𝐵𝐴)) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑋))
1615anbi2d 739 . . . . 5 (𝑥 = 𝑋 → ((𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋)))
1716ralbidv 2983 . . . 4 (𝑥 = 𝑋 → (∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋)))
1817rexbidv 3048 . . 3 (𝑥 = 𝑋 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋)))
1918rspcva 3302 . 2 ((𝑋 ∈ ℝ+ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋))
201, 14, 19syl2anc 692 1 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wnf 1706  wcel 1988  wnfc 2749  wral 2909  wrex 2910  Vcvv 3195   class class class wbr 4644  Rel wrel 5109  cfv 5876  (class class class)co 6635  cc 9919   < clt 10059  cmin 10251  cz 11362  cuz 11672  +crp 11817  abscabs 13955  cli 14196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-pre-lttri 9995  ax-pre-lttrn 9996
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-po 5025  df-so 5026  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-neg 10254  df-z 11363  df-uz 11673  df-clim 14200
This theorem is referenced by:  climleltrp  39708
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