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Theorem climuz 40614
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 40613 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦)))
5 breq2 4813 . . . . . . . 8 (𝑦 = 𝑥 → ((abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ (abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
65ralbidv 3133 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
76rexbidv 3199 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
8 fveq2 6375 . . . . . . . . . 10 (𝑖 = 𝑗 → (ℤ𝑖) = (ℤ𝑗))
98raleqdv 3292 . . . . . . . . 9 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑗)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥))
10 nfcv 2907 . . . . . . . . . . . . 13 𝑘abs
11 climuz.k . . . . . . . . . . . . . . 15 𝑘𝐹
12 nfcv 2907 . . . . . . . . . . . . . . 15 𝑘𝑙
1311, 12nffv 6385 . . . . . . . . . . . . . 14 𝑘(𝐹𝑙)
14 nfcv 2907 . . . . . . . . . . . . . 14 𝑘
15 nfcv 2907 . . . . . . . . . . . . . 14 𝑘𝐴
1613, 14, 15nfov 6872 . . . . . . . . . . . . 13 𝑘((𝐹𝑙) − 𝐴)
1710, 16nffv 6385 . . . . . . . . . . . 12 𝑘(abs‘((𝐹𝑙) − 𝐴))
18 nfcv 2907 . . . . . . . . . . . 12 𝑘 <
19 nfcv 2907 . . . . . . . . . . . 12 𝑘𝑥
2017, 18, 19nfbr 4856 . . . . . . . . . . 11 𝑘(abs‘((𝐹𝑙) − 𝐴)) < 𝑥
21 nfv 2009 . . . . . . . . . . 11 𝑙(abs‘((𝐹𝑘) − 𝐴)) < 𝑥
22 fveq2 6375 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → (𝐹𝑙) = (𝐹𝑘))
2322fvoveq1d 6864 . . . . . . . . . . . 12 (𝑙 = 𝑘 → (abs‘((𝐹𝑙) − 𝐴)) = (abs‘((𝐹𝑘) − 𝐴)))
2423breq1d 4819 . . . . . . . . . . 11 (𝑙 = 𝑘 → ((abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
2520, 21, 24cbvral 3315 . . . . . . . . . 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 270 . . . . . . . 8 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
2827cbvrexv 3320 . . . . . . 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 270 . . . . 5 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
3130cbvralv 3319 . . . 4 (∀𝑦 ∈ ℝ+𝑖𝑍𝑙 ∈ (ℤ𝑖)(abs‘((𝐹𝑙) − 𝐴)) < 𝑦 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)
3231anbi2i 616 . . 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 270 1 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wnfc 2894  wral 3055  wrex 3056   class class class wbr 4809  wf 6064  cfv 6068  (class class class)co 6842  cc 10187   < clt 10328  cmin 10520  cz 11624  cuz 11886  +crp 12028  abscabs 14259  cli 14500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-pre-lttri 10263  ax-pre-lttrn 10264
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-po 5198  df-so 5199  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-neg 10523  df-z 11625  df-uz 11887  df-clim 14504
This theorem is referenced by:  liminflimsupclim  40677  climxlim2lem  40709
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