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Theorem climuz 41909
 Description: Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
climuz.k 𝑘𝐹
climuz.m (𝜑𝑀 ∈ ℤ)
climuz.z 𝑍 = (ℤ𝑀)
climuz.f (𝜑𝐹:𝑍⟶ℂ)
Assertion
Ref Expression
climuz (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
Distinct variable groups:   𝐴,𝑗,𝑘,𝑥   𝑗,𝐹,𝑥   𝑗,𝑍,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑗,𝑘)   𝐹(𝑘)   𝑀(𝑥,𝑗,𝑘)   𝑍(𝑘)

Proof of Theorem climuz
Dummy variables 𝑖 𝑙 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climuz.m . . 3 (𝜑𝑀 ∈ ℤ)
2 climuz.z . . 3 𝑍 = (ℤ𝑀)
3 climuz.f . . 3 (𝜑𝐹:𝑍⟶ℂ)
41, 2, 3climuzlem 41908 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦)))
5 breq2 5067 . . . . . . . 8 (𝑦 = 𝑥 → ((abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ (abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
65ralbidv 3202 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
76rexbidv 3302 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
8 fveq2 6669 . . . . . . . . . 10 (𝑖 = 𝑗 → (ℤ𝑖) = (ℤ𝑗))
98raleqdv 3421 . . . . . . . . 9 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑗)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
10 nfcv 2982 . . . . . . . . . . . . 13 𝑘abs
11 climuz.k . . . . . . . . . . . . . . 15 𝑘𝐹
12 nfcv 2982 . . . . . . . . . . . . . . 15 𝑘𝑙
1311, 12nffv 6679 . . . . . . . . . . . . . 14 𝑘(𝐹𝑙)
14 nfcv 2982 . . . . . . . . . . . . . 14 𝑘
15 nfcv 2982 . . . . . . . . . . . . . 14 𝑘𝐴
1613, 14, 15nfov 7180 . . . . . . . . . . . . 13 𝑘((𝐹𝑙) − 𝐴)
1710, 16nffv 6679 . . . . . . . . . . . 12 𝑘(abs‘((𝐹𝑙) − 𝐴))
18 nfcv 2982 . . . . . . . . . . . 12 𝑘 <
19 nfcv 2982 . . . . . . . . . . . 12 𝑘𝑥
2017, 18, 19nfbr 5110 . . . . . . . . . . 11 𝑘(abs‘((𝐹𝑙) − 𝐴)) < 𝑥
21 nfv 1908 . . . . . . . . . . 11 𝑙(abs‘((𝐹𝑘) − 𝐴)) < 𝑥
22 fveq2 6669 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → (𝐹𝑙) = (𝐹𝑘))
2322fvoveq1d 7172 . . . . . . . . . . . 12 (𝑙 = 𝑘 → (abs‘((𝐹𝑙) − 𝐴)) = (abs‘((𝐹𝑘) − 𝐴)))
2423breq1d 5073 . . . . . . . . . . 11 (𝑙 = 𝑘 → ((abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
2520, 21, 24cbvral 3451 . . . . . . . . . 10 (∀𝑙 ∈ (ℤ𝑗)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)
2625a1i 11 . . . . . . . . 9 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑗)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
279, 26bitrd 280 . . . . . . . 8 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
2827cbvrexv 3459 . . . . . . 7 (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)
2928a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
307, 29bitrd 280 . . . . 5 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
3130cbvralv 3458 . . . 4 (∀𝑦 ∈ ℝ+𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)
3231anbi2i 622 . . 3 ((𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
3332a1i 11 . 2 (𝜑 → ((𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
344, 33bitrd 280 1 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1530   ∈ wcel 2107  Ⅎwnfc 2966  ∀wral 3143  ∃wrex 3144   class class class wbr 5063  ⟶wf 6350  ‘cfv 6354  (class class class)co 7150  ℂcc 10529   < clt 10669   − cmin 10864  ℤcz 11975  ℤ≥cuz 12237  ℝ+crp 12384  abscabs 14588   ⇝ cli 14836 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-pre-lttri 10605  ax-pre-lttrn 10606 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-neg 10867  df-z 11976  df-uz 12238  df-clim 14840 This theorem is referenced by:  liminflimsupclim  41972  climxlim2lem  42010
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