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Theorem limsupval3 41980
Description: The superior limit of an infinite sequence 𝐹 of extended real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupval3.1 𝑘𝜑
limsupval3.2 (𝜑𝐴𝑉)
limsupval3.3 (𝜑𝐹:𝐴⟶ℝ*)
limsupval3.4 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))
Assertion
Ref Expression
limsupval3 (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
Distinct variable group:   𝑘,𝐹
Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐺(𝑘)   𝑉(𝑘)

Proof of Theorem limsupval3
StepHypRef Expression
1 limsupval3.3 . . . 4 (𝜑𝐹:𝐴⟶ℝ*)
2 limsupval3.2 . . . 4 (𝜑𝐴𝑉)
31, 2fexd 41386 . . 3 (𝜑𝐹 ∈ V)
4 eqid 2823 . . . 4 (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
54limsupval 14833 . . 3 (𝐹 ∈ V → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
63, 5syl 17 . 2 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
7 limsupval3.4 . . . . . 6 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))
87a1i 11 . . . . 5 (𝜑𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )))
9 limsupval3.1 . . . . . 6 𝑘𝜑
101fimassd 41505 . . . . . . . . . 10 (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
11 df-ss 3954 . . . . . . . . . 10 ((𝐹 “ (𝑘[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞)))
1210, 11sylib 220 . . . . . . . . 9 (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞)))
1312eqcomd 2829 . . . . . . . 8 (𝜑 → (𝐹 “ (𝑘[,)+∞)) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
1413supeq1d 8912 . . . . . . 7 (𝜑 → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
1514adantr 483 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
169, 15mpteq2da 5162 . . . . 5 (𝜑 → (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )))
178, 16eqtr2d 2859 . . . 4 (𝜑 → (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺)
1817rneqd 5810 . . 3 (𝜑 → ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺)
1918infeq1d 8943 . 2 (𝜑 → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = inf(ran 𝐺, ℝ*, < ))
206, 19eqtrd 2858 1 (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wnf 1784  wcel 2114  Vcvv 3496  cin 3937  wss 3938  cmpt 5148  ran crn 5558  cima 5560  wf 6353  cfv 6357  (class class class)co 7158  supcsup 8906  infcinf 8907  cr 10538  +∞cpnf 10674  *cxr 10676   < clt 10677  [,)cico 12743  lim supclsp 14829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-pre-lttri 10613  ax-pre-lttrn 10614
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-po 5476  df-so 5477  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-sup 8908  df-inf 8909  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-limsup 14830
This theorem is referenced by:  limsupmnflem  42008  limsup10ex  42061
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