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Theorem limsupvald 45281
Description: The superior limit of a sequence 𝐹 of extended real numbers is the infimum of the set of suprema of all restrictions of 𝐹 to an upperset of reals . (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
limsupvald.1 (𝜑𝐹𝑉)
limsupvald.2 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
Assertion
Ref Expression
limsupvald (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
Distinct variable group:   𝑘,𝐹
Allowed substitution hints:   𝜑(𝑘)   𝐺(𝑘)   𝑉(𝑘)

Proof of Theorem limsupvald
StepHypRef Expression
1 limsupvald.1 . 2 (𝜑𝐹𝑉)
2 limsupvald.2 . . 3 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
32limsupval 15454 . 2 (𝐹𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
41, 3syl 17 1 (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cin 3943  cmpt 5232  ran crn 5679  cima 5681  cfv 6549  (class class class)co 7419  supcsup 9465  infcinf 9466  cr 11139  +∞cpnf 11277  *cxr 11279   < clt 11280  [,)cico 13361  lim supclsp 15450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-pre-lttri 11214  ax-pre-lttrn 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-limsup 15451
This theorem is referenced by:  liminflelimsuplem  45301
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