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Theorem clim 9775
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 9774 . . . . 5 Rel ⇝
21brrelex2i 4385 . . . 4 (𝐹𝐴𝐴 ∈ V)
32a1i 9 . . 3 (𝜑 → (𝐹𝐴𝐴 ∈ V))
4 elex 2566 . . . . 5 (𝐴 ∈ ℂ → 𝐴 ∈ V)
54adantr 261 . . . 4 ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)) → 𝐴 ∈ V)
65a1i 9 . . 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 103 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐴) → 𝑦 = 𝐴)
98eleq1d 2106 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐴) → (𝑦 ∈ ℂ ↔ 𝐴 ∈ ℂ))
10 fveq1 5177 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
1110adantr 261 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐴) → (𝑓𝑘) = (𝐹𝑘))
1211eleq1d 2106 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐴) → ((𝑓𝑘) ∈ ℂ ↔ (𝐹𝑘) ∈ ℂ))
13 oveq12 5521 . . . . . . . . . . . . . 14 (((𝑓𝑘) = (𝐹𝑘) ∧ 𝑦 = 𝐴) → ((𝑓𝑘) − 𝑦) = ((𝐹𝑘) − 𝐴))
1410, 13sylan 267 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐴) → ((𝑓𝑘) − 𝑦) = ((𝐹𝑘) − 𝐴))
1514fveq2d 5182 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐴) → (abs‘((𝑓𝑘) − 𝑦)) = (abs‘((𝐹𝑘) − 𝐴)))
1615breq1d 3774 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐴) → ((abs‘((𝑓𝑘) − 𝑦)) < 𝑥 ↔ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
1712, 16anbi12d 442 . . . . . . . . . 10 ((𝑓 = 𝐹𝑦 = 𝐴) → (((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1817ralbidv 2326 . . . . . . . . 9 ((𝑓 = 𝐹𝑦 = 𝐴) → (∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1918rexbidv 2327 . . . . . . . 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 2326 . . . . . . 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 442 . . . . . 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 9773 . . . . . 6 ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥))}
2321, 22brabga 4001 . . . . 5 ((𝐹𝑉𝐴 ∈ V) → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
2423ex 108 . . . 4 (𝐹𝑉 → (𝐴 ∈ V → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))))
257, 24syl 14 . . 3 (𝜑 → (𝐴 ∈ V → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))))
263, 6, 25pm5.21ndd 621 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
27 eluzelz 8480 . . . . . . 7 (𝑘 ∈ (ℤ𝑗) → 𝑘 ∈ ℤ)
28 clim.3 . . . . . . . . 9 ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)
2928eleq1d 2106 . . . . . . . 8 ((𝜑𝑘 ∈ ℤ) → ((𝐹𝑘) ∈ ℂ ↔ 𝐵 ∈ ℂ))
3028oveq1d 5527 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℤ) → ((𝐹𝑘) − 𝐴) = (𝐵𝐴))
3130fveq2d 5182 . . . . . . . . 9 ((𝜑𝑘 ∈ ℤ) → (abs‘((𝐹𝑘) − 𝐴)) = (abs‘(𝐵𝐴)))
3231breq1d 3774 . . . . . . . 8 ((𝜑𝑘 ∈ ℤ) → ((abs‘((𝐹𝑘) − 𝐴)) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑥))
3329, 32anbi12d 442 . . . . . . 7 ((𝜑𝑘 ∈ ℤ) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3427, 33sylan2 270 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3534ralbidva 2322 . . . . 5 (𝜑 → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3635rexbidv 2327 . . . 4 (𝜑 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3736ralbidv 2326 . . 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)))
3837anbi2d 437 . 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))))
3926, 38bitrd 177 1 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  wral 2306  wrex 2307  Vcvv 2557   class class class wbr 3764  cfv 4902  (class class class)co 5512  cc 6885   < clt 7058  cmin 7180  cz 8243  cuz 8471  +crp 8581  abscabs 9569  cli 9772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-cnex 6973  ax-resscn 6974
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-ov 5515  df-neg 7183  df-z 8244  df-uz 8472  df-clim 9773
This theorem is referenced by:  climcl  9776  clim2  9777  climshftlemg  9796
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