Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clim2d Structured version   Visualization version   GIF version

Theorem clim2d 41960
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 14852 . . . . . . 7 Rel ⇝
87a1i 11 . . . . . 6 (𝜑 → Rel ⇝ )
9 brrelex1 5608 . . . . . 6 ((Rel ⇝ ∧ 𝐹𝐴) → 𝐹 ∈ V)
108, 2, 9syl2anc 586 . . . . 5 (𝜑𝐹 ∈ V)
11 clim2d.b . . . . 5 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)
123, 4, 5, 6, 10, 11clim2f2 41957 . . . 4 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
132, 12mpbid 234 . . 3 (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
1413simprd 498 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))
15 breq2 5073 . . . . . 6 (𝑥 = 𝑋 → ((abs‘(𝐵𝐴)) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑋))
1615anbi2d 630 . . . . 5 (𝑥 = 𝑋 → ((𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋)))
1716ralbidv 3200 . . . 4 (𝑥 = 𝑋 → (∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋)))
1817rexbidv 3300 . . 3 (𝑥 = 𝑋 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋)))
1918rspcva 3624 . 2 ((𝑋 ∈ ℝ+ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋))
201, 14, 19syl2anc 586 1 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wnf 1783  wcel 2113  wnfc 2964  wral 3141  wrex 3142  Vcvv 3497   class class class wbr 5069  Rel wrel 5563  cfv 6358  (class class class)co 7159  cc 10538   < clt 10678  cmin 10873  cz 11984  cuz 12246  +crp 12392  abscabs 14596  cli 14844
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-pre-lttri 10614  ax-pre-lttrn 10615
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-po 5477  df-so 5478  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-neg 10876  df-z 11985  df-uz 12247  df-clim 14848
This theorem is referenced by:  climleltrp  41963
  Copyright terms: Public domain W3C validator