Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  limsupre3 Structured version   Visualization version   GIF version

Theorem limsupre3 40483
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 smaller or equal than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger or equal than 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 2913 . . 3 𝑙𝐹
2 limsupre3.2 . . 3 (𝜑𝐴 ⊆ ℝ)
3 limsupre3.3 . . 3 (𝜑𝐹:𝐴⟶ℝ*)
41, 2, 3limsupre3lem 40482 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦))))
5 breq1 4789 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
65anbi2d 614 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ (𝑖𝑙𝑥 ≤ (𝐹𝑙))))
76rexbidv 3200 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙))))
87ralbidv 3135 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙))))
9 breq1 4789 . . . . . . . . . . 11 (𝑖 = 𝑘 → (𝑖𝑙𝑘𝑙))
109anbi1d 615 . . . . . . . . . 10 (𝑖 = 𝑘 → ((𝑖𝑙𝑥 ≤ (𝐹𝑙)) ↔ (𝑘𝑙𝑥 ≤ (𝐹𝑙))))
1110rexbidv 3200 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑙𝐴 (𝑘𝑙𝑥 ≤ (𝐹𝑙))))
12 nfv 1995 . . . . . . . . . . . 12 𝑗 𝑘𝑙
13 nfcv 2913 . . . . . . . . . . . . 13 𝑗𝑥
14 nfcv 2913 . . . . . . . . . . . . 13 𝑗
15 limsupre3.1 . . . . . . . . . . . . . 14 𝑗𝐹
16 nfcv 2913 . . . . . . . . . . . . . 14 𝑗𝑙
1715, 16nffv 6339 . . . . . . . . . . . . 13 𝑗(𝐹𝑙)
1813, 14, 17nfbr 4833 . . . . . . . . . . . 12 𝑗 𝑥 ≤ (𝐹𝑙)
1912, 18nfan 1980 . . . . . . . . . . 11 𝑗(𝑘𝑙𝑥 ≤ (𝐹𝑙))
20 nfv 1995 . . . . . . . . . . 11 𝑙(𝑘𝑗𝑥 ≤ (𝐹𝑗))
21 breq2 4790 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝑘𝑙𝑘𝑗))
22 fveq2 6332 . . . . . . . . . . . . 13 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
2322breq2d 4798 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
2421, 23anbi12d 616 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝑘𝑙𝑥 ≤ (𝐹𝑙)) ↔ (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
2519, 20, 24cbvrex 3317 . . . . . . . . . 10 (∃𝑙𝐴 (𝑘𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
2625a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙𝐴 (𝑘𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
2711, 26bitrd 268 . . . . . . . 8 (𝑖 = 𝑘 → (∃𝑙𝐴 (𝑖𝑙𝑥 ≤ (𝐹𝑙)) ↔ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
2827cbvralv 3320 . . . . . . 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 268 . . . . 5 (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
3130cbvrexv 3321 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙𝐴 (𝑖𝑙𝑦 ≤ (𝐹𝑙)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
32 breq2 4790 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
3332imbi2d 329 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
3433ralbidv 3135 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
3534rexbidv 3200 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
369imbi1d 330 . . . . . . . . . 10 (𝑖 = 𝑘 → ((𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)))
3736ralbidv 3135 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)))
3817, 14, 13nfbr 4833 . . . . . . . . . . . 12 𝑗(𝐹𝑙) ≤ 𝑥
3912, 38nfim 1977 . . . . . . . . . . 11 𝑗(𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)
40 nfv 1995 . . . . . . . . . . 11 𝑙(𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)
4122breq1d 4796 . . . . . . . . . . . 12 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
4221, 41imbi12d 333 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4339, 40, 42cbvral 3316 . . . . . . . . . 10 (∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
4443a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4537, 44bitrd 268 . . . . . . . 8 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4645cbvrexv 3321 . . . . . . 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 268 . . . . 5 (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
4948cbvrexv 3321 . . . 4 (∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
5031, 49anbi12i 612 . . 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 268 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 196  wa 382   = wceq 1631  wcel 2145  wnfc 2900  wral 3061  wrex 3062  wss 3723   class class class wbr 4786  wf 6027  cfv 6031  cr 10137  *cxr 10275  cle 10277  lim supclsp 14409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8504  df-inf 8505  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-ico 12386  df-limsup 14410
This theorem is referenced by:  limsupre3mpt  40484  limsupre3uzlem  40485
  Copyright terms: Public domain W3C validator