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Theorem limsupreuz 42547
 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 2955 . . . 4 𝑙𝐹
2 limsupreuz.2 . . . 4 (𝜑𝑀 ∈ ℤ)
3 limsupreuz.3 . . . 4 𝑍 = (ℤ𝑀)
4 limsupreuz.4 . . . . 5 (𝜑𝐹:𝑍⟶ℝ)
54frexr 42187 . . . 4 (𝜑𝐹:𝑍⟶ℝ*)
61, 2, 3, 5limsupre3uzlem 42545 . . 3 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦)))
7 breq1 5037 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
87rexbidv 3257 . . . . . . . 8 (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
98ralbidv 3162 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
10 fveq2 6655 . . . . . . . . . . 11 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
1110rexeqdv 3366 . . . . . . . . . 10 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙)))
12 nfcv 2955 . . . . . . . . . . . . 13 𝑗𝑥
13 nfcv 2955 . . . . . . . . . . . . 13 𝑗
14 limsupreuz.1 . . . . . . . . . . . . . 14 𝑗𝐹
15 nfcv 2955 . . . . . . . . . . . . . 14 𝑗𝑙
1614, 15nffv 6665 . . . . . . . . . . . . 13 𝑗(𝐹𝑙)
1712, 13, 16nfbr 5081 . . . . . . . . . . . 12 𝑗 𝑥 ≤ (𝐹𝑙)
18 nfv 1915 . . . . . . . . . . . 12 𝑙 𝑥 ≤ (𝐹𝑗)
19 fveq2 6655 . . . . . . . . . . . . 13 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
2019breq2d 5046 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
2117, 18, 20cbvrexw 3389 . . . . . . . . . . 11 (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2221a1i 11 . . . . . . . . . 10 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2311, 22bitrd 282 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2423cbvralvw 3397 . . . . . . . 8 (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2524a1i 11 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
269, 25bitrd 282 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2726cbvrexvw 3398 . . . . 5 (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
28 breq2 5038 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
2928ralbidv 3162 . . . . . . . 8 (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
3029rexbidv 3257 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
3110raleqdv 3365 . . . . . . . . . 10 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥))
3216, 13, 12nfbr 5081 . . . . . . . . . . . 12 𝑗(𝐹𝑙) ≤ 𝑥
33 nfv 1915 . . . . . . . . . . . 12 𝑙(𝐹𝑗) ≤ 𝑥
3419breq1d 5044 . . . . . . . . . . . 12 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
3532, 33, 34cbvralw 3388 . . . . . . . . . . 11 (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3635a1i 11 . . . . . . . . . 10 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3731, 36bitrd 282 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3837cbvrexvw 3398 . . . . . . . 8 (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3938a1i 11 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4030, 39bitrd 282 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4140cbvrexvw 3398 . . . . 5 (∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
4227, 41anbi12i 629 . . . 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 282 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))
45 nfv 1915 . . . . . . . 8 𝑖(𝐹𝑗) ≤ 𝑥
46 nfcv 2955 . . . . . . . . . 10 𝑗𝑖
4714, 46nffv 6665 . . . . . . . . 9 𝑗(𝐹𝑖)
4847, 13, 12nfbr 5081 . . . . . . . 8 𝑗(𝐹𝑖) ≤ 𝑥
49 fveq2 6655 . . . . . . . . 9 (𝑗 = 𝑖 → (𝐹𝑗) = (𝐹𝑖))
5049breq1d 5044 . . . . . . . 8 (𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥))
5145, 48, 50cbvralw 3388 . . . . . . 7 (∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥)
5251rexbii 3211 . . . . . 6 (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥)
5352rexbii 3211 . . . . 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 1915 . . . . 5 𝑖𝜑
564adantr 484 . . . . . 6 ((𝜑𝑖𝑍) → 𝐹:𝑍⟶ℝ)
57 simpr 488 . . . . . 6 ((𝜑𝑖𝑍) → 𝑖𝑍)
5856, 57ffvelrnd 6839 . . . . 5 ((𝜑𝑖𝑍) → (𝐹𝑖) ∈ ℝ)
5955, 2, 3, 58uzub 42236 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥))
60 eqcom 2805 . . . . . . . . . 10 (𝑗 = 𝑖𝑖 = 𝑗)
6160imbi1i 353 . . . . . . . . 9 ((𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)))
62 bicom 225 . . . . . . . . . 10 (((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥) ↔ ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
6362imbi2i 339 . . . . . . . . 9 ((𝑖 = 𝑗 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥)))
6461, 63bitri 278 . . . . . . . 8 ((𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥)))
6550, 64mpbi 233 . . . . . . 7 (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
6648, 45, 65cbvralw 3388 . . . . . 6 (∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥 ↔ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
6766rexbii 3211 . . . . 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 308 . . 3 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥))
7069anbi2d 631 . 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 282 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  ∀wral 3106  ∃wrex 3107   class class class wbr 5034  ⟶wf 6328  ‘cfv 6332  ℝcr 10543   ≤ cle 10683  ℤcz 11989  ℤ≥cuz 12251  lim supclsp 14839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-pre-sup 10622 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-oadd 8107  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-sup 8908  df-inf 8909  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-n0 11904  df-z 11990  df-uz 12252  df-ico 12752  df-fz 12906  df-fzo 13049  df-fl 13177  df-ceil 13178  df-limsup 14840 This theorem is referenced by:  limsupreuzmpt  42549  limsupgtlem  42587
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