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Theorem limsupre3uz 43277
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 2907 . . 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 43276 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦)))
6 breq1 5077 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
76rexbidv 3226 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
87ralbidv 3112 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
9 fveq2 6774 . . . . . . . . . 10 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
109rexeqdv 3349 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙)))
11 nfcv 2907 . . . . . . . . . . . 12 𝑗𝑥
12 nfcv 2907 . . . . . . . . . . . 12 𝑗
13 limsupre3uz.1 . . . . . . . . . . . . 13 𝑗𝐹
14 nfcv 2907 . . . . . . . . . . . . 13 𝑗𝑙
1513, 14nffv 6784 . . . . . . . . . . . 12 𝑗(𝐹𝑙)
1611, 12, 15nfbr 5121 . . . . . . . . . . 11 𝑗 𝑥 ≤ (𝐹𝑙)
17 nfv 1917 . . . . . . . . . . 11 𝑙 𝑥 ≤ (𝐹𝑗)
18 fveq2 6774 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
1918breq2d 5086 . . . . . . . . . . 11 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
2016, 17, 19cbvrexw 3374 . . . . . . . . . 10 (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2120a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2210, 21bitrd 278 . . . . . . . 8 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2322cbvralvw 3383 . . . . . . 7 (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2423a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
258, 24bitrd 278 . . . . 5 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2625cbvrexvw 3384 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
27 breq2 5078 . . . . . . . 8 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
2827ralbidv 3112 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
2928rexbidv 3226 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
309raleqdv 3348 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥))
3115, 12, 11nfbr 5121 . . . . . . . . . . 11 𝑗(𝐹𝑙) ≤ 𝑥
32 nfv 1917 . . . . . . . . . . 11 𝑙(𝐹𝑗) ≤ 𝑥
3318breq1d 5084 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
3431, 32, 33cbvralw 3373 . . . . . . . . . 10 (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3534a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3630, 35bitrd 278 . . . . . . . 8 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3736cbvrexvw 3384 . . . . . . 7 (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3837a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3929, 38bitrd 278 . . . . 5 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4039cbvrexvw 3384 . . . 4 (∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
4126, 40anbi12i 627 . . 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 278 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 205  wa 396   = wceq 1539  wcel 2106  wnfc 2887  wral 3064  wrex 3065   class class class wbr 5074  wf 6429  cfv 6433  cr 10870  *cxr 11008  cle 11010  cz 12319  cuz 12582  lim supclsp 15179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-ico 13085  df-fl 13512  df-ceil 13513  df-limsup 15180
This theorem is referenced by:  limsupvaluz2  43279  supcnvlimsup  43281
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