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Theorem clim 10032
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 10031 . . . . 5 Rel ⇝
21brrelex2i 4412 . . . 4 (𝐹𝐴𝐴 ∈ V)
32a1i 9 . . 3 (𝜑 → (𝐹𝐴𝐴 ∈ V))
4 elex 2583 . . . . 5 (𝐴 ∈ ℂ → 𝐴 ∈ V)
54adantr 265 . . . 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 107 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐴) → 𝑦 = 𝐴)
98eleq1d 2122 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐴) → (𝑦 ∈ ℂ ↔ 𝐴 ∈ ℂ))
10 fveq1 5204 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
1110adantr 265 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐴) → (𝑓𝑘) = (𝐹𝑘))
1211eleq1d 2122 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐴) → ((𝑓𝑘) ∈ ℂ ↔ (𝐹𝑘) ∈ ℂ))
13 oveq12 5548 . . . . . . . . . . . . . 14 (((𝑓𝑘) = (𝐹𝑘) ∧ 𝑦 = 𝐴) → ((𝑓𝑘) − 𝑦) = ((𝐹𝑘) − 𝐴))
1410, 13sylan 271 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐴) → ((𝑓𝑘) − 𝑦) = ((𝐹𝑘) − 𝐴))
1514fveq2d 5209 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐴) → (abs‘((𝑓𝑘) − 𝑦)) = (abs‘((𝐹𝑘) − 𝐴)))
1615breq1d 3801 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐴) → ((abs‘((𝑓𝑘) − 𝑦)) < 𝑥 ↔ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
1712, 16anbi12d 450 . . . . . . . . . 10 ((𝑓 = 𝐹𝑦 = 𝐴) → (((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1817ralbidv 2343 . . . . . . . . 9 ((𝑓 = 𝐹𝑦 = 𝐴) → (∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
1918rexbidv 2344 . . . . . . . 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 2343 . . . . . . 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 450 . . . . . 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 10030 . . . . . 6 ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥))}
2321, 22brabga 4028 . . . . 5 ((𝐹𝑉𝐴 ∈ V) → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
2423ex 112 . . . 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 631 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
27 eluzelz 8577 . . . . . . 7 (𝑘 ∈ (ℤ𝑗) → 𝑘 ∈ ℤ)
28 clim.3 . . . . . . . . 9 ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)
2928eleq1d 2122 . . . . . . . 8 ((𝜑𝑘 ∈ ℤ) → ((𝐹𝑘) ∈ ℂ ↔ 𝐵 ∈ ℂ))
3028oveq1d 5554 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℤ) → ((𝐹𝑘) − 𝐴) = (𝐵𝐴))
3130fveq2d 5209 . . . . . . . . 9 ((𝜑𝑘 ∈ ℤ) → (abs‘((𝐹𝑘) − 𝐴)) = (abs‘(𝐵𝐴)))
3231breq1d 3801 . . . . . . . 8 ((𝜑𝑘 ∈ ℤ) → ((abs‘((𝐹𝑘) − 𝐴)) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑥))
3329, 32anbi12d 450 . . . . . . 7 ((𝜑𝑘 ∈ ℤ) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3427, 33sylan2 274 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3534ralbidva 2339 . . . . 5 (𝜑 → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3635rexbidv 2344 . . . 4 (𝜑 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
3736ralbidv 2343 . . 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 445 . 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 181 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 101  wb 102   = wceq 1259  wcel 1409  wral 2323  wrex 2324  Vcvv 2574   class class class wbr 3791  cfv 4929  (class class class)co 5539  cc 6944   < clt 7118  cmin 7244  cz 8301  cuz 8568  +crp 8680  abscabs 9823  cli 10029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-cnex 7032  ax-resscn 7033
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fv 4937  df-ov 5542  df-neg 7247  df-z 8302  df-uz 8569  df-clim 10030
This theorem is referenced by:  climcl  10033  clim2  10034  climshftlemg  10053
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