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Theorem limsupmnf 40471
Description: The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to an upper interval of real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupmnf.j 𝑗𝐹
limsupmnf.a (𝜑𝐴 ⊆ ℝ)
limsupmnf.f (𝜑𝐹:𝐴⟶ℝ*)
Assertion
Ref Expression
limsupmnf (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀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 limsupmnf
Dummy variables 𝑖 𝑙 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limsupmnf.a . . 3 (𝜑𝐴 ⊆ ℝ)
2 limsupmnf.f . . 3 (𝜑𝐹:𝐴⟶ℝ*)
3 eqid 2771 . . 3 (𝑖 ∈ ℝ ↦ sup((𝐹 “ (𝑖[,)+∞)), ℝ*, < )) = (𝑖 ∈ ℝ ↦ sup((𝐹 “ (𝑖[,)+∞)), ℝ*, < ))
41, 2, 3limsupmnflem 40470 . 2 (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦)))
5 breq2 4790 . . . . . . . 8 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
65imbi2d 329 . . . . . . 7 (𝑦 = 𝑥 → ((𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
76ralbidv 3135 . . . . . 6 (𝑦 = 𝑥 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
87rexbidv 3200 . . . . 5 (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥)))
9 breq1 4789 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝑖𝑙𝑘𝑙))
109imbi1d 330 . . . . . . . . 9 (𝑖 = 𝑘 → ((𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)))
1110ralbidv 3135 . . . . . . . 8 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)))
12 nfv 1995 . . . . . . . . . . 11 𝑗 𝑘𝑙
13 limsupmnf.j . . . . . . . . . . . . 13 𝑗𝐹
14 nfcv 2913 . . . . . . . . . . . . 13 𝑗𝑙
1513, 14nffv 6339 . . . . . . . . . . . 12 𝑗(𝐹𝑙)
16 nfcv 2913 . . . . . . . . . . . 12 𝑗
17 nfcv 2913 . . . . . . . . . . . 12 𝑗𝑥
1815, 16, 17nfbr 4833 . . . . . . . . . . 11 𝑗(𝐹𝑙) ≤ 𝑥
1912, 18nfim 1977 . . . . . . . . . 10 𝑗(𝑘𝑙 → (𝐹𝑙) ≤ 𝑥)
20 nfv 1995 . . . . . . . . . 10 𝑙(𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)
21 breq2 4790 . . . . . . . . . . 11 (𝑙 = 𝑗 → (𝑘𝑙𝑘𝑗))
22 fveq2 6332 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
2322breq1d 4796 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
2421, 23imbi12d 333 . . . . . . . . . 10 (𝑙 = 𝑗 → ((𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
2519, 20, 24cbvral 3316 . . . . . . . . 9 (∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
2625a1i 11 . . . . . . . 8 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
2711, 26bitrd 268 . . . . . . 7 (𝑖 = 𝑘 → (∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
2827cbvrexv 3321 . . . . . 6 (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
2928a1i 11 . . . . 5 (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
308, 29bitrd 268 . . . 4 (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
3130cbvralv 3320 . . 3 (∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
3231a1i 11 . 2 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙𝐴 (𝑖𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
334, 32bitrd 268 1 (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wnfc 2900  wral 3061  wrex 3062  wss 3723   class class class wbr 4786  cmpt 4863  cima 5252  wf 6027  cfv 6031  (class class class)co 6793  supcsup 8502  cr 10137  +∞cpnf 10273  -∞cmnf 10274  *cxr 10275   < clt 10276  cle 10277  [,)cico 12382  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:  limsupre2lem  40474  limsupmnfuzlem  40476
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