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Theorem clim 15440
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 15438 . . . . 5 Rel ⇝
21brrelex2i 5724 . . . 4 (𝐹𝐴𝐴 ∈ V)
32a1i 11 . . 3 (𝜑 → (𝐹𝐴𝐴 ∈ V))
4 elex 3485 . . . . 5 (𝐴 ∈ ℂ → 𝐴 ∈ V)
54adantr 480 . . . 4 ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)) → 𝐴 ∈ V)
65a1i 11 . . 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 484 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐴) → 𝑦 = 𝐴)
98eleq1d 2810 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐴) → (𝑦 ∈ ℂ ↔ 𝐴 ∈ ℂ))
10 fveq1 6881 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
1110adantr 480 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐴) → (𝑓𝑘) = (𝐹𝑘))
1211eleq1d 2810 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐴) → ((𝑓𝑘) ∈ ℂ ↔ (𝐹𝑘) ∈ ℂ))
13 oveq12 7411 . . . . . . . . . . . . . 14 (((𝑓𝑘) = (𝐹𝑘) ∧ 𝑦 = 𝐴) → ((𝑓𝑘) − 𝑦) = ((𝐹𝑘) − 𝐴))
1410, 13sylan 579 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐴) → ((𝑓𝑘) − 𝑦) = ((𝐹𝑘) − 𝐴))
1514fveq2d 6886 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐴) → (abs‘((𝑓𝑘) − 𝑦)) = (abs‘((𝐹𝑘) − 𝐴)))
1615breq1d 5149 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐴) → ((abs‘((𝑓𝑘) − 𝑦)) < 𝑥 ↔ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
1712, 16anbi12d 630 . . . . . . . . . 10 ((𝑓 = 𝐹𝑦 = 𝐴) → (((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1817ralbidv 3169 . . . . . . . . 9 ((𝑓 = 𝐹𝑦 = 𝐴) → (∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1918rexbidv 3170 . . . . . . . 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 3169 . . . . . . 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 630 . . . . . 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 15434 . . . . . 6 ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥))}
2321, 22brabga 5525 . . . . 5 ((𝐹𝑉𝐴 ∈ V) → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
2423ex 412 . . . 4 (𝐹𝑉 → (𝐴 ∈ V → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))))
257, 24syl 17 . . 3 (𝜑 → (𝐴 ∈ V → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))))
263, 6, 25pm5.21ndd 379 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
27 eluzelz 12831 . . . . . . 7 (𝑘 ∈ (ℤ𝑗) → 𝑘 ∈ ℤ)
28 clim.3 . . . . . . . . 9 ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)
2928eleq1d 2810 . . . . . . . 8 ((𝜑𝑘 ∈ ℤ) → ((𝐹𝑘) ∈ ℂ ↔ 𝐵 ∈ ℂ))
3028fvoveq1d 7424 . . . . . . . . 9 ((𝜑𝑘 ∈ ℤ) → (abs‘((𝐹𝑘) − 𝐴)) = (abs‘(𝐵𝐴)))
3130breq1d 5149 . . . . . . . 8 ((𝜑𝑘 ∈ ℤ) → ((abs‘((𝐹𝑘) − 𝐴)) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑥))
3229, 31anbi12d 630 . . . . . . 7 ((𝜑𝑘 ∈ ℤ) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3327, 32sylan2 592 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3433ralbidva 3167 . . . . 5 (𝜑 → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3534rexbidv 3170 . . . 4 (𝜑 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3635ralbidv 3169 . . 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)))
3736anbi2d 628 . 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))))
3826, 37bitrd 279 1 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3053  wrex 3062  Vcvv 3466   class class class wbr 5139  cfv 6534  (class class class)co 7402  cc 11105   < clt 11247  cmin 11443  cz 12557  cuz 12821  +crp 12975  abscabs 15183  cli 15430
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 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-cnex 11163  ax-resscn 11164
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-neg 11446  df-z 12558  df-uz 12822  df-clim 15434
This theorem is referenced by:  climcl  15445  clim2  15450  climshftlem  15520  climsuse  44869  0cnv  45003  climuzlem  45004  climisp  45007  climrescn  45009  climxrrelem  45010  climxrre  45011  ioodvbdlimc1lem2  45193  ioodvbdlimc2lem  45195
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