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Theorem limsupreuz 45693
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 2903 . . . 4 𝑙𝐹
2 limsupreuz.2 . . . 4 (𝜑𝑀 ∈ ℤ)
3 limsupreuz.3 . . . 4 𝑍 = (ℤ𝑀)
4 limsupreuz.4 . . . . 5 (𝜑𝐹:𝑍⟶ℝ)
54frexr 45335 . . . 4 (𝜑𝐹:𝑍⟶ℝ*)
61, 2, 3, 5limsupre3uzlem 45691 . . 3 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦)))
7 breq1 5151 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
87rexbidv 3177 . . . . . . . 8 (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
98ralbidv 3176 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
10 fveq2 6907 . . . . . . . . . . 11 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
1110rexeqdv 3325 . . . . . . . . . 10 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙)))
12 nfcv 2903 . . . . . . . . . . . . 13 𝑗𝑥
13 nfcv 2903 . . . . . . . . . . . . 13 𝑗
14 limsupreuz.1 . . . . . . . . . . . . . 14 𝑗𝐹
15 nfcv 2903 . . . . . . . . . . . . . 14 𝑗𝑙
1614, 15nffv 6917 . . . . . . . . . . . . 13 𝑗(𝐹𝑙)
1712, 13, 16nfbr 5195 . . . . . . . . . . . 12 𝑗 𝑥 ≤ (𝐹𝑙)
18 nfv 1912 . . . . . . . . . . . 12 𝑙 𝑥 ≤ (𝐹𝑗)
19 fveq2 6907 . . . . . . . . . . . . 13 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
2019breq2d 5160 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
2117, 18, 20cbvrexw 3305 . . . . . . . . . . 11 (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2221a1i 11 . . . . . . . . . 10 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2311, 22bitrd 279 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2423cbvralvw 3235 . . . . . . . 8 (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2524a1i 11 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
269, 25bitrd 279 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2726cbvrexvw 3236 . . . . 5 (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
28 breq2 5152 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
2928ralbidv 3176 . . . . . . . 8 (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
3029rexbidv 3177 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
3110raleqdv 3324 . . . . . . . . . 10 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥))
3216, 13, 12nfbr 5195 . . . . . . . . . . . 12 𝑗(𝐹𝑙) ≤ 𝑥
33 nfv 1912 . . . . . . . . . . . 12 𝑙(𝐹𝑗) ≤ 𝑥
3419breq1d 5158 . . . . . . . . . . . 12 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
3532, 33, 34cbvralw 3304 . . . . . . . . . . 11 (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3635a1i 11 . . . . . . . . . 10 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3731, 36bitrd 279 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3837cbvrexvw 3236 . . . . . . . 8 (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3938a1i 11 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4030, 39bitrd 279 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4140cbvrexvw 3236 . . . . 5 (∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
4227, 41anbi12i 628 . . . 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 279 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))
45 nfv 1912 . . . . . . . 8 𝑖(𝐹𝑗) ≤ 𝑥
46 nfcv 2903 . . . . . . . . . 10 𝑗𝑖
4714, 46nffv 6917 . . . . . . . . 9 𝑗(𝐹𝑖)
4847, 13, 12nfbr 5195 . . . . . . . 8 𝑗(𝐹𝑖) ≤ 𝑥
49 fveq2 6907 . . . . . . . . 9 (𝑗 = 𝑖 → (𝐹𝑗) = (𝐹𝑖))
5049breq1d 5158 . . . . . . . 8 (𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥))
5145, 48, 50cbvralw 3304 . . . . . . 7 (∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥)
5251rexbii 3092 . . . . . 6 (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥)
5352rexbii 3092 . . . . 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 1912 . . . . 5 𝑖𝜑
564adantr 480 . . . . . 6 ((𝜑𝑖𝑍) → 𝐹:𝑍⟶ℝ)
57 simpr 484 . . . . . 6 ((𝜑𝑖𝑍) → 𝑖𝑍)
5856, 57ffvelcdmd 7105 . . . . 5 ((𝜑𝑖𝑍) → (𝐹𝑖) ∈ ℝ)
5955, 2, 3, 58uzub 45381 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥))
60 eqcom 2742 . . . . . . . . . 10 (𝑗 = 𝑖𝑖 = 𝑗)
6160imbi1i 349 . . . . . . . . 9 ((𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)))
62 bicom 222 . . . . . . . . . 10 (((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥) ↔ ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
6362imbi2i 336 . . . . . . . . 9 ((𝑖 = 𝑗 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥)))
6461, 63bitri 275 . . . . . . . 8 ((𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥)))
6550, 64mpbi 230 . . . . . . 7 (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
6648, 45, 65cbvralw 3304 . . . . . 6 (∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥 ↔ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
6766rexbii 3092 . . . . 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 305 . . 3 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥))
7069anbi2d 630 . 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 279 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 206  wa 395   = wceq 1537  wcel 2106  wnfc 2888  wral 3059  wrex 3068   class class class wbr 5148  wf 6559  cfv 6563  cr 11152  cle 11294  cz 12611  cuz 12876  lim supclsp 15503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-ico 13390  df-fz 13545  df-fzo 13692  df-fl 13829  df-ceil 13830  df-limsup 15504
This theorem is referenced by:  limsupreuzmpt  45695  limsupgtlem  45733
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