Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  liminfval Structured version   Visualization version   GIF version

Theorem liminfval 42046
Description: The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
liminfval.1 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
Assertion
Ref Expression
liminfval (𝐹𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
Distinct variable group:   𝑘,𝐹
Allowed substitution hints:   𝐺(𝑘)   𝑉(𝑘)

Proof of Theorem liminfval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-liminf 42039 . 2 lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
2 imaeq1 5927 . . . . . . . 8 (𝑥 = 𝐹 → (𝑥 “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
32ineq1d 4191 . . . . . . 7 (𝑥 = 𝐹 → ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
43infeq1d 8944 . . . . . 6 (𝑥 = 𝐹 → inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
54mpteq2dv 5165 . . . . 5 (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )))
6 liminfval.1 . . . . . 6 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
76a1i 11 . . . . 5 (𝑥 = 𝐹𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )))
85, 7eqtr4d 2862 . . . 4 (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺)
98rneqd 5811 . . 3 (𝑥 = 𝐹 → ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺)
109supeq1d 8913 . 2 (𝑥 = 𝐹 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = sup(ran 𝐺, ℝ*, < ))
11 elex 3515 . 2 (𝐹𝑉𝐹 ∈ V)
12 xrltso 12537 . . . 4 < Or ℝ*
1312supex 8930 . . 3 sup(ran 𝐺, ℝ*, < ) ∈ V
1413a1i 11 . 2 (𝐹𝑉 → sup(ran 𝐺, ℝ*, < ) ∈ V)
151, 10, 11, 14fvmptd3 6794 1 (𝐹𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2113  Vcvv 3497  cin 3938  cmpt 5149  ran crn 5559  cima 5561  cfv 6358  (class class class)co 7159  supcsup 8907  infcinf 8908  cr 10539  +∞cpnf 10675  *cxr 10677   < clt 10678  [,)cico 12743  lim infclsi 42038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-pre-lttri 10614  ax-pre-lttrn 10615
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-po 5477  df-so 5478  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-sup 8909  df-inf 8910  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-liminf 42039
This theorem is referenced by:  liminfcl  42050  liminfvald  42051  liminfval5  42052  liminfresxr  42054  liminfval2  42055  liminfvalxr  42070
  Copyright terms: Public domain W3C validator