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Theorem limsupre3uz 40858
Description: Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, infinitely often; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupre3uz.1 𝑗𝐹
limsupre3uz.2 (𝜑𝑀 ∈ ℤ)
limsupre3uz.3 𝑍 = (ℤ𝑀)
limsupre3uz.4 (𝜑𝐹:𝑍⟶ℝ*)
Assertion
Ref Expression
limsupre3uz (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))
Distinct variable groups:   𝑘,𝐹,𝑥   𝑘,𝑍,𝑥   𝑗,𝑘,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑗,𝑘)   𝐹(𝑗)   𝑀(𝑥,𝑗,𝑘)   𝑍(𝑗)

Proof of Theorem limsupre3uz
Dummy variables 𝑖 𝑙 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2933 . . 3 𝑙𝐹
2 limsupre3uz.2 . . 3 (𝜑𝑀 ∈ ℤ)
3 limsupre3uz.3 . . 3 𝑍 = (ℤ𝑀)
4 limsupre3uz.4 . . 3 (𝜑𝐹:𝑍⟶ℝ*)
51, 2, 3, 4limsupre3uzlem 40857 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦)))
6 breq1 4889 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
76rexbidv 3236 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
87ralbidv 3167 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
9 fveq2 6446 . . . . . . . . . 10 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
109rexeqdv 3340 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙)))
11 nfcv 2933 . . . . . . . . . . . 12 𝑗𝑥
12 nfcv 2933 . . . . . . . . . . . 12 𝑗
13 limsupre3uz.1 . . . . . . . . . . . . 13 𝑗𝐹
14 nfcv 2933 . . . . . . . . . . . . 13 𝑗𝑙
1513, 14nffv 6456 . . . . . . . . . . . 12 𝑗(𝐹𝑙)
1611, 12, 15nfbr 4933 . . . . . . . . . . 11 𝑗 𝑥 ≤ (𝐹𝑙)
17 nfv 1957 . . . . . . . . . . 11 𝑙 𝑥 ≤ (𝐹𝑗)
18 fveq2 6446 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
1918breq2d 4898 . . . . . . . . . . 11 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
2016, 17, 19cbvrex 3363 . . . . . . . . . 10 (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2120a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2210, 21bitrd 271 . . . . . . . 8 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2322cbvralv 3366 . . . . . . 7 (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2423a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
258, 24bitrd 271 . . . . 5 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2625cbvrexv 3367 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
27 breq2 4890 . . . . . . . 8 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
2827ralbidv 3167 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
2928rexbidv 3236 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
309raleqdv 3339 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥))
3115, 12, 11nfbr 4933 . . . . . . . . . . 11 𝑗(𝐹𝑙) ≤ 𝑥
32 nfv 1957 . . . . . . . . . . 11 𝑙(𝐹𝑗) ≤ 𝑥
3318breq1d 4896 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
3431, 32, 33cbvral 3362 . . . . . . . . . 10 (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3534a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3630, 35bitrd 271 . . . . . . . 8 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3736cbvrexv 3367 . . . . . . 7 (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3837a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3929, 38bitrd 271 . . . . 5 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4039cbvrexv 3367 . . . 4 (∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
4126, 40anbi12i 620 . . 3 ((∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4241a1i 11 . 2 (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))
435, 42bitrd 271 1 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2106  wnfc 2918  wral 3089  wrex 3090   class class class wbr 4886  wf 6131  cfv 6135  cr 10271  *cxr 10410  cle 10412  cz 11728  cuz 11992  lim supclsp 14609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-sup 8636  df-inf 8637  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-z 11729  df-uz 11993  df-ico 12493  df-fl 12912  df-ceil 12913  df-limsup 14610
This theorem is referenced by:  limsupvaluz2  40860  supcnvlimsup  40862
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