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Theorem limsupre3 42418
 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, at some point, in any upper part of the reals; 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
limsupre3.1 𝑗𝐹
limsupre3.2 (𝜑𝐴 ⊆ ℝ)
limsupre3.3 (𝜑𝐹:𝐴⟶ℝ*)
Assertion
Ref Expression
limsupre3 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))))
Distinct variable groups:   𝐴,𝑗,𝑘,𝑥   𝑘,𝐹,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑗,𝑘)   𝐹(𝑗)

Proof of Theorem limsupre3
Dummy variables 𝑖 𝑙 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2955 . . 3 𝑙𝐹
2 limsupre3.2 . . 3 (𝜑𝐴 ⊆ ℝ)
3 limsupre3.3 . . 3 (𝜑𝐹:𝐴⟶ℝ*)
41, 2, 3limsupre3lem 42417 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦))))
5 breq1 5034 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
65anbi2d 631 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ (𝑖𝑙𝑥 ≤ (𝐹𝑙))))
76rexbidv 3256 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙))))
87ralbidv 3162 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙))))
9 breq1 5034 . . . . . . . . . . 11 (𝑖 = 𝑘 → (𝑖𝑙𝑘𝑙))
109anbi1d 632 . . . . . . . . . 10 (𝑖 = 𝑘 → ((𝑖𝑙𝑥 ≤ (𝐹𝑙)) ↔ (𝑘𝑙𝑥 ≤ (𝐹𝑙))))
1110rexbidv 3256 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑙𝐴 (𝑘𝑙𝑥 ≤ (𝐹𝑙))))
12 nfv 1915 . . . . . . . . . . . 12 𝑗 𝑘𝑙
13 nfcv 2955 . . . . . . . . . . . . 13 𝑗𝑥
14 nfcv 2955 . . . . . . . . . . . . 13 𝑗
15 limsupre3.1 . . . . . . . . . . . . . 14 𝑗𝐹
16 nfcv 2955 . . . . . . . . . . . . . 14 𝑗𝑙
1715, 16nffv 6656 . . . . . . . . . . . . 13 𝑗(𝐹𝑙)
1813, 14, 17nfbr 5078 . . . . . . . . . . . 12 𝑗 𝑥 ≤ (𝐹𝑙)
1912, 18nfan 1900 . . . . . . . . . . 11 𝑗(𝑘𝑙𝑥 ≤ (𝐹𝑙))
20 nfv 1915 . . . . . . . . . . 11 𝑙(𝑘𝑗𝑥 ≤ (𝐹𝑗))
21 breq2 5035 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝑘𝑙𝑘𝑗))
22 fveq2 6646 . . . . . . . . . . . . 13 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
2322breq2d 5043 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
2421, 23anbi12d 633 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝑘𝑙𝑥 ≤ (𝐹𝑙)) ↔ (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
2519, 20, 24cbvrexw 3388 . . . . . . . . . 10 (∃𝑙𝐴 (𝑘𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
2625a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙𝐴 (𝑘𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
2711, 26bitrd 282 . . . . . . . 8 (𝑖 = 𝑘 → (∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
2827cbvralvw 3396 . . . . . . 7 (∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
2928a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
308, 29bitrd 282 . . . . 5 (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
3130cbvrexvw 3397 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
32 breq2 5035 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
3332imbi2d 344 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
3433ralbidv 3162 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
3534rexbidv 3256 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
369imbi1d 345 . . . . . . . . . 10 (𝑖 = 𝑘 → ((𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)))
3736ralbidv 3162 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)))
3817, 14, 13nfbr 5078 . . . . . . . . . . . 12 𝑗(𝐹𝑙) ≤ 𝑥
3912, 38nfim 1897 . . . . . . . . . . 11 𝑗(𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)
40 nfv 1915 . . . . . . . . . . 11 𝑙(𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)
4122breq1d 5041 . . . . . . . . . . . 12 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
4221, 41imbi12d 348 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4339, 40, 42cbvralw 3387 . . . . . . . . . 10 (∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
4443a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4537, 44bitrd 282 . . . . . . . 8 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4645cbvrexvw 3397 . . . . . . 7 (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
4746a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4835, 47bitrd 282 . . . . 5 (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4948cbvrexvw 3397 . . . 4 (∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
5031, 49anbi12i 629 . . 3 ((∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
5150a1i 11 . 2 (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))))
524, 51bitrd 282 1 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (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   ⊆ wss 3881   class class class wbr 5031  ⟶wf 6321  ‘cfv 6325  ℝcr 10528  ℝ*cxr 10666   ≤ cle 10668  lim supclsp 14822 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 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607 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 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-po 5439  df-so 5440  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-sup 8893  df-inf 8894  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-ico 12735  df-limsup 14823 This theorem is referenced by:  limsupre3mpt  42419  limsupre3uzlem  42420
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