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Theorem limsupref 40435
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 4790 . . . . . . . 8 (𝑏 = 𝑦 → ((abs‘(𝐹𝑗)) ≤ 𝑏 ↔ (abs‘(𝐹𝑗)) ≤ 𝑦))
65imbi2d 329 . . . . . . 7 (𝑏 = 𝑦 → ((𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
76ralbidv 3135 . . . . . 6 (𝑏 = 𝑦 → (∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
87rexbidv 3200 . . . . 5 (𝑏 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
9 breq1 4789 . . . . . . . . . 10 (𝑘 = 𝑖 → (𝑘𝑗𝑖𝑗))
109imbi1d 330 . . . . . . . . 9 (𝑘 = 𝑖 → ((𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ (𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
1110ralbidv 3135 . . . . . . . 8 (𝑘 = 𝑖 → (∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∀𝑗𝐴 (𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
12 nfv 1995 . . . . . . . . . 10 𝑥(𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)
13 nfv 1995 . . . . . . . . . . 11 𝑗 𝑖𝑥
14 nfcv 2913 . . . . . . . . . . . . 13 𝑗abs
15 limsupref.j . . . . . . . . . . . . . 14 𝑗𝐹
16 nfcv 2913 . . . . . . . . . . . . . 14 𝑗𝑥
1715, 16nffv 6339 . . . . . . . . . . . . 13 𝑗(𝐹𝑥)
1814, 17nffv 6339 . . . . . . . . . . . 12 𝑗(abs‘(𝐹𝑥))
19 nfcv 2913 . . . . . . . . . . . 12 𝑗
20 nfcv 2913 . . . . . . . . . . . 12 𝑗𝑦
2118, 19, 20nfbr 4833 . . . . . . . . . . 11 𝑗(abs‘(𝐹𝑥)) ≤ 𝑦
2213, 21nfim 1977 . . . . . . . . . 10 𝑗(𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)
23 breq2 4790 . . . . . . . . . . 11 (𝑗 = 𝑥 → (𝑖𝑗𝑖𝑥))
24 fveq2 6332 . . . . . . . . . . . . 13 (𝑗 = 𝑥 → (𝐹𝑗) = (𝐹𝑥))
2524fveq2d 6336 . . . . . . . . . . . 12 (𝑗 = 𝑥 → (abs‘(𝐹𝑗)) = (abs‘(𝐹𝑥)))
2625breq1d 4796 . . . . . . . . . . 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 3316 . . . . . . . . 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 3321 . . . . . 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 3321 . . 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 40391 1 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  wnfc 2900  wral 3061  wrex 3062  wss 3723   class class class wbr 4786  wf 6027  cfv 6031  supcsup 8502  cr 10137  +∞cpnf 10273  *cxr 10275   < clt 10276  cle 10277  abscabs 14182  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-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  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-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  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-om 7213  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  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-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-rp 12036  df-ico 12386  df-seq 13009  df-exp 13068  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-limsup 14410
This theorem is referenced by: (None)
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