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Theorem limsupre3 40489
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 2916 . . 3 𝑙𝐹
2 limsupre3.2 . . 3 (𝜑𝐴 ⊆ ℝ)
3 limsupre3.3 . . 3 (𝜑𝐹:𝐴⟶ℝ*)
41, 2, 3limsupre3lem 40488 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦))))
5 breq1 4800 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
65anbi2d 615 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ (𝑖𝑙𝑥 ≤ (𝐹𝑙))))
76rexbidv 3204 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙))))
87ralbidv 3138 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙))))
9 breq1 4800 . . . . . . . . . . 11 (𝑖 = 𝑘 → (𝑖𝑙𝑘𝑙))
109anbi1d 616 . . . . . . . . . 10 (𝑖 = 𝑘 → ((𝑖𝑙𝑥 ≤ (𝐹𝑙)) ↔ (𝑘𝑙𝑥 ≤ (𝐹𝑙))))
1110rexbidv 3204 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑙𝐴 (𝑘𝑙𝑥 ≤ (𝐹𝑙))))
12 nfv 1998 . . . . . . . . . . . 12 𝑗 𝑘𝑙
13 nfcv 2916 . . . . . . . . . . . . 13 𝑗𝑥
14 nfcv 2916 . . . . . . . . . . . . 13 𝑗
15 limsupre3.1 . . . . . . . . . . . . . 14 𝑗𝐹
16 nfcv 2916 . . . . . . . . . . . . . 14 𝑗𝑙
1715, 16nffv 6356 . . . . . . . . . . . . 13 𝑗(𝐹𝑙)
1813, 14, 17nfbr 4844 . . . . . . . . . . . 12 𝑗 𝑥 ≤ (𝐹𝑙)
1912, 18nfan 1983 . . . . . . . . . . 11 𝑗(𝑘𝑙𝑥 ≤ (𝐹𝑙))
20 nfv 1998 . . . . . . . . . . 11 𝑙(𝑘𝑗𝑥 ≤ (𝐹𝑗))
21 breq2 4801 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝑘𝑙𝑘𝑗))
22 fveq2 6348 . . . . . . . . . . . . 13 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
2322breq2d 4809 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
2421, 23anbi12d 617 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝑘𝑙𝑥 ≤ (𝐹𝑙)) ↔ (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
2519, 20, 24cbvrex 3321 . . . . . . . . . 10 (∃𝑙𝐴 (𝑘𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
2625a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙𝐴 (𝑘𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
2711, 26bitrd 269 . . . . . . . 8 (𝑖 = 𝑘 → (∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
2827cbvralv 3324 . . . . . . 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 269 . . . . 5 (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
3130cbvrexv 3325 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
32 breq2 4801 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
3332imbi2d 330 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
3433ralbidv 3138 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
3534rexbidv 3204 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
369imbi1d 331 . . . . . . . . . 10 (𝑖 = 𝑘 → ((𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)))
3736ralbidv 3138 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)))
3817, 14, 13nfbr 4844 . . . . . . . . . . . 12 𝑗(𝐹𝑙) ≤ 𝑥
3912, 38nfim 1980 . . . . . . . . . . 11 𝑗(𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)
40 nfv 1998 . . . . . . . . . . 11 𝑙(𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)
4122breq1d 4807 . . . . . . . . . . . 12 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
4221, 41imbi12d 334 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4339, 40, 42cbvral 3320 . . . . . . . . . 10 (∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
4443a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4537, 44bitrd 269 . . . . . . . 8 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4645cbvrexv 3325 . . . . . . 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 269 . . . . 5 (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4948cbvrexv 3325 . . . 4 (∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
5031, 49anbi12i 613 . . 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 269 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 197  wa 383   = wceq 1634  wcel 2148  wnfc 2903  wral 3064  wrex 3065  wss 3729   class class class wbr 4797  wf 6038  cfv 6042  cr 10158  *cxr 10296  cle 10298  lim supclsp 14431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-cnex 10215  ax-resscn 10216  ax-1cn 10217  ax-icn 10218  ax-addcl 10219  ax-addrcl 10220  ax-mulcl 10221  ax-mulrcl 10222  ax-mulcom 10223  ax-addass 10224  ax-mulass 10225  ax-distr 10226  ax-i2m1 10227  ax-1ne0 10228  ax-1rid 10229  ax-rnegex 10230  ax-rrecex 10231  ax-cnre 10232  ax-pre-lttri 10233  ax-pre-lttrn 10234  ax-pre-ltadd 10235  ax-pre-mulgt0 10236  ax-pre-sup 10237
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-po 5184  df-so 5185  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-er 7917  df-en 8131  df-dom 8132  df-sdom 8133  df-sup 8525  df-inf 8526  df-pnf 10299  df-mnf 10300  df-xr 10301  df-ltxr 10302  df-le 10303  df-sub 10491  df-neg 10492  df-ico 12405  df-limsup 14432
This theorem is referenced by:  limsupre3mpt  40490  limsupre3uzlem  40491
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