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Theorem limsupref 40235
Description: If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupref.j 𝑗𝐹
limsupref.a (𝜑𝐴 ⊆ ℝ)
limsupref.s (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
limsupref.f (𝜑𝐹:𝐴⟶ℝ)
limsupref.b (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
Assertion
Ref Expression
limsupref (𝜑 → (lim sup‘𝐹) ∈ ℝ)
Distinct variable groups:   𝐴,𝑏,𝑗,𝑘   𝐹,𝑏,𝑘
Allowed substitution hints:   𝜑(𝑗,𝑘,𝑏)   𝐹(𝑗)

Proof of Theorem limsupref
Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limsupref.a . 2 (𝜑𝐴 ⊆ ℝ)
2 limsupref.s . 2 (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
3 limsupref.f . 2 (𝜑𝐹:𝐴⟶ℝ)
4 limsupref.b . . 3 (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
5 breq2 4689 . . . . . . . 8 (𝑏 = 𝑦 → ((abs‘(𝐹𝑗)) ≤ 𝑏 ↔ (abs‘(𝐹𝑗)) ≤ 𝑦))
65imbi2d 329 . . . . . . 7 (𝑏 = 𝑦 → ((𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
76ralbidv 3015 . . . . . 6 (𝑏 = 𝑦 → (∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
87rexbidv 3081 . . . . 5 (𝑏 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
9 breq1 4688 . . . . . . . . . 10 (𝑘 = 𝑖 → (𝑘𝑗𝑖𝑗))
109imbi1d 330 . . . . . . . . 9 (𝑘 = 𝑖 → ((𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ (𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
1110ralbidv 3015 . . . . . . . 8 (𝑘 = 𝑖 → (∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∀𝑗𝐴 (𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
12 nfv 1883 . . . . . . . . . 10 𝑥(𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)
13 nfv 1883 . . . . . . . . . . 11 𝑗 𝑖𝑥
14 nfcv 2793 . . . . . . . . . . . . 13 𝑗abs
15 limsupref.j . . . . . . . . . . . . . 14 𝑗𝐹
16 nfcv 2793 . . . . . . . . . . . . . 14 𝑗𝑥
1715, 16nffv 6236 . . . . . . . . . . . . 13 𝑗(𝐹𝑥)
1814, 17nffv 6236 . . . . . . . . . . . 12 𝑗(abs‘(𝐹𝑥))
19 nfcv 2793 . . . . . . . . . . . 12 𝑗
20 nfcv 2793 . . . . . . . . . . . 12 𝑗𝑦
2118, 19, 20nfbr 4732 . . . . . . . . . . 11 𝑗(abs‘(𝐹𝑥)) ≤ 𝑦
2213, 21nfim 1865 . . . . . . . . . 10 𝑗(𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)
23 breq2 4689 . . . . . . . . . . 11 (𝑗 = 𝑥 → (𝑖𝑗𝑖𝑥))
24 fveq2 6229 . . . . . . . . . . . . 13 (𝑗 = 𝑥 → (𝐹𝑗) = (𝐹𝑥))
2524fveq2d 6233 . . . . . . . . . . . 12 (𝑗 = 𝑥 → (abs‘(𝐹𝑗)) = (abs‘(𝐹𝑥)))
2625breq1d 4695 . . . . . . . . . . 11 (𝑗 = 𝑥 → ((abs‘(𝐹𝑗)) ≤ 𝑦 ↔ (abs‘(𝐹𝑥)) ≤ 𝑦))
2723, 26imbi12d 333 . . . . . . . . . 10 (𝑗 = 𝑥 → ((𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)))
2812, 22, 27cbvral 3197 . . . . . . . . 9 (∀𝑗𝐴 (𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦))
2928a1i 11 . . . . . . . 8 (𝑘 = 𝑖 → (∀𝑗𝐴 (𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)))
3011, 29bitrd 268 . . . . . . 7 (𝑘 = 𝑖 → (∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)))
3130cbvrexv 3202 . . . . . 6 (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦))
3231a1i 11 . . . . 5 (𝑏 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)))
338, 32bitrd 268 . . . 4 (𝑏 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ ∃𝑖 ∈ ℝ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)))
3433cbvrexv 3202 . . 3 (∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦))
354, 34sylib 208 . 2 (𝜑 → ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦))
361, 2, 3, 35limsupre 40191 1 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wnfc 2780  wral 2941  wrex 2942  wss 3607   class class class wbr 4685  wf 5922  cfv 5926  supcsup 8387  cr 9973  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  abscabs 14018  lim supclsp 14245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246
This theorem is referenced by: (None)
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