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Theorem limsupreuz 41894
Description: Given a function on the 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 greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupreuz.1 𝑗𝐹
limsupreuz.2 (𝜑𝑀 ∈ ℤ)
limsupreuz.3 𝑍 = (ℤ𝑀)
limsupreuz.4 (𝜑𝐹:𝑍⟶ℝ)
Assertion
Ref Expression
limsupreuz (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)))
Distinct variable groups:   𝑘,𝐹,𝑥   𝑗,𝑍,𝑘,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑗,𝑘)   𝐹(𝑗)   𝑀(𝑥,𝑗,𝑘)

Proof of Theorem limsupreuz
Dummy variables 𝑖 𝑙 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2974 . . . 4 𝑙𝐹
2 limsupreuz.2 . . . 4 (𝜑𝑀 ∈ ℤ)
3 limsupreuz.3 . . . 4 𝑍 = (ℤ𝑀)
4 limsupreuz.4 . . . . 5 (𝜑𝐹:𝑍⟶ℝ)
54frexr 41531 . . . 4 (𝜑𝐹:𝑍⟶ℝ*)
61, 2, 3, 5limsupre3uzlem 41892 . . 3 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦)))
7 breq1 5060 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
87rexbidv 3294 . . . . . . . 8 (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
98ralbidv 3194 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
10 fveq2 6663 . . . . . . . . . . 11 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
1110rexeqdv 3414 . . . . . . . . . 10 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙)))
12 nfcv 2974 . . . . . . . . . . . . 13 𝑗𝑥
13 nfcv 2974 . . . . . . . . . . . . 13 𝑗
14 limsupreuz.1 . . . . . . . . . . . . . 14 𝑗𝐹
15 nfcv 2974 . . . . . . . . . . . . . 14 𝑗𝑙
1614, 15nffv 6673 . . . . . . . . . . . . 13 𝑗(𝐹𝑙)
1712, 13, 16nfbr 5104 . . . . . . . . . . . 12 𝑗 𝑥 ≤ (𝐹𝑙)
18 nfv 1906 . . . . . . . . . . . 12 𝑙 𝑥 ≤ (𝐹𝑗)
19 fveq2 6663 . . . . . . . . . . . . 13 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
2019breq2d 5069 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
2117, 18, 20cbvrexw 3440 . . . . . . . . . . 11 (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2221a1i 11 . . . . . . . . . 10 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2311, 22bitrd 280 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2423cbvralvw 3447 . . . . . . . 8 (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2524a1i 11 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
269, 25bitrd 280 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2726cbvrexvw 3448 . . . . 5 (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
28 breq2 5061 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
2928ralbidv 3194 . . . . . . . 8 (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
3029rexbidv 3294 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
3110raleqdv 3413 . . . . . . . . . 10 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥))
3216, 13, 12nfbr 5104 . . . . . . . . . . . 12 𝑗(𝐹𝑙) ≤ 𝑥
33 nfv 1906 . . . . . . . . . . . 12 𝑙(𝐹𝑗) ≤ 𝑥
3419breq1d 5067 . . . . . . . . . . . 12 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
3532, 33, 34cbvralw 3439 . . . . . . . . . . 11 (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3635a1i 11 . . . . . . . . . 10 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3731, 36bitrd 280 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3837cbvrexvw 3448 . . . . . . . 8 (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3938a1i 11 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4030, 39bitrd 280 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4140cbvrexvw 3448 . . . . 5 (∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
4227, 41anbi12i 626 . . . 4 ((∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4342a1i 11 . . 3 (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)))
446, 43bitrd 280 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))
45 nfv 1906 . . . . . . . 8 𝑖(𝐹𝑗) ≤ 𝑥
46 nfcv 2974 . . . . . . . . . 10 𝑗𝑖
4714, 46nffv 6673 . . . . . . . . 9 𝑗(𝐹𝑖)
4847, 13, 12nfbr 5104 . . . . . . . 8 𝑗(𝐹𝑖) ≤ 𝑥
49 fveq2 6663 . . . . . . . . 9 (𝑗 = 𝑖 → (𝐹𝑗) = (𝐹𝑖))
5049breq1d 5067 . . . . . . . 8 (𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥))
5145, 48, 50cbvralw 3439 . . . . . . 7 (∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥)
5251rexbii 3244 . . . . . 6 (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥)
5352rexbii 3244 . . . . 5 (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥)
5453a1i 11 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥))
55 nfv 1906 . . . . 5 𝑖𝜑
564adantr 481 . . . . . 6 ((𝜑𝑖𝑍) → 𝐹:𝑍⟶ℝ)
57 simpr 485 . . . . . 6 ((𝜑𝑖𝑍) → 𝑖𝑍)
5856, 57ffvelrnd 6844 . . . . 5 ((𝜑𝑖𝑍) → (𝐹𝑖) ∈ ℝ)
5955, 2, 3, 58uzub 41581 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥))
60 eqcom 2825 . . . . . . . . . 10 (𝑗 = 𝑖𝑖 = 𝑗)
6160imbi1i 351 . . . . . . . . 9 ((𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)))
62 bicom 223 . . . . . . . . . 10 (((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥) ↔ ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
6362imbi2i 337 . . . . . . . . 9 ((𝑖 = 𝑗 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥)))
6461, 63bitri 276 . . . . . . . 8 ((𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥)))
6550, 64mpbi 231 . . . . . . 7 (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
6648, 45, 65cbvralw 3439 . . . . . 6 (∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥 ↔ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
6766rexbii 3244 . . . . 5 (∃𝑥 ∈ ℝ ∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
6867a1i 11 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥))
6954, 59, 683bitrd 306 . . 3 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥))
7069anbi2d 628 . 2 (𝜑 → ((∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)))
7144, 70bitrd 280 1 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wnfc 2958  wral 3135  wrex 3136   class class class wbr 5057  wf 6344  cfv 6348  cr 10524  cle 10664  cz 11969  cuz 12231  lim supclsp 14815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-ico 12732  df-fz 12881  df-fzo 13022  df-fl 13150  df-ceil 13151  df-limsup 14816
This theorem is referenced by:  limsupreuzmpt  41896  limsupgtlem  41934
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