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Theorem limsupref 41427
 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 4930 . . . . . . . 8 (𝑏 = 𝑦 → ((abs‘(𝐹𝑗)) ≤ 𝑏 ↔ (abs‘(𝐹𝑗)) ≤ 𝑦))
65imbi2d 333 . . . . . . 7 (𝑏 = 𝑦 → ((𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
76ralbidv 3142 . . . . . 6 (𝑏 = 𝑦 → (∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
87rexbidv 3237 . . . . 5 (𝑏 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
9 breq1 4929 . . . . . . . . . 10 (𝑘 = 𝑖 → (𝑘𝑗𝑖𝑗))
109imbi1d 334 . . . . . . . . 9 (𝑘 = 𝑖 → ((𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ (𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
1110ralbidv 3142 . . . . . . . 8 (𝑘 = 𝑖 → (∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∀𝑗𝐴 (𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)))
12 nfv 1874 . . . . . . . . . 10 𝑥(𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦)
13 nfv 1874 . . . . . . . . . . 11 𝑗 𝑖𝑥
14 nfcv 2927 . . . . . . . . . . . . 13 𝑗abs
15 limsupref.j . . . . . . . . . . . . . 14 𝑗𝐹
16 nfcv 2927 . . . . . . . . . . . . . 14 𝑗𝑥
1715, 16nffv 6507 . . . . . . . . . . . . 13 𝑗(𝐹𝑥)
1814, 17nffv 6507 . . . . . . . . . . . 12 𝑗(abs‘(𝐹𝑥))
19 nfcv 2927 . . . . . . . . . . . 12 𝑗
20 nfcv 2927 . . . . . . . . . . . 12 𝑗𝑦
2118, 19, 20nfbr 4973 . . . . . . . . . . 11 𝑗(abs‘(𝐹𝑥)) ≤ 𝑦
2213, 21nfim 1860 . . . . . . . . . 10 𝑗(𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)
23 breq2 4930 . . . . . . . . . . 11 (𝑗 = 𝑥 → (𝑖𝑗𝑖𝑥))
24 2fveq3 6502 . . . . . . . . . . . 12 (𝑗 = 𝑥 → (abs‘(𝐹𝑗)) = (abs‘(𝐹𝑥)))
2524breq1d 4936 . . . . . . . . . . 11 (𝑗 = 𝑥 → ((abs‘(𝐹𝑗)) ≤ 𝑦 ↔ (abs‘(𝐹𝑥)) ≤ 𝑦))
2623, 25imbi12d 337 . . . . . . . . . 10 (𝑗 = 𝑥 → ((𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)))
2712, 22, 26cbvral 3374 . . . . . . . . 9 (∀𝑗𝐴 (𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦))
2827a1i 11 . . . . . . . 8 (𝑘 = 𝑖 → (∀𝑗𝐴 (𝑖𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)))
2911, 28bitrd 271 . . . . . . 7 (𝑘 = 𝑖 → (∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)))
3029cbvrexv 3379 . . . . . 6 (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦))
3130a1i 11 . . . . 5 (𝑏 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)))
328, 31bitrd 271 . . . 4 (𝑏 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ ∃𝑖 ∈ ℝ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦)))
3332cbvrexv 3379 . . 3 (∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ↔ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦))
344, 33sylib 210 . 2 (𝜑 → ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥𝐴 (𝑖𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑦))
351, 2, 3, 34limsupre 41383 1 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1508   ∈ wcel 2051  Ⅎwnfc 2911  ∀wral 3083  ∃wrex 3084   ⊆ wss 3824   class class class wbr 4926  ⟶wf 6182  ‘cfv 6186  supcsup 8698  ℝcr 10333  +∞cpnf 10470  ℝ*cxr 10472   < clt 10473   ≤ cle 10474  abscabs 14453  lim supclsp 14687 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411  ax-pre-sup 10412 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-2nd 7501  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-er 8088  df-en 8306  df-dom 8307  df-sdom 8308  df-sup 8700  df-inf 8701  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-div 11098  df-nn 11439  df-2 11502  df-3 11503  df-n0 11707  df-z 11793  df-uz 12058  df-rp 12204  df-ico 12559  df-seq 13184  df-exp 13244  df-cj 14318  df-re 14319  df-im 14320  df-sqrt 14454  df-abs 14455  df-limsup 14688 This theorem is referenced by: (None)
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