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 43006
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 42999 . 2 lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
2 imaeq1 5939 . . . . . . . 8 (𝑥 = 𝐹 → (𝑥 “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
32ineq1d 4141 . . . . . . 7 (𝑥 = 𝐹 → ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
43infeq1d 9118 . . . . . 6 (𝑥 = 𝐹 → inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
54mpteq2dv 5166 . . . . 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 2781 . . . 4 (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺)
98rneqd 5822 . . 3 (𝑥 = 𝐹 → ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺)
109supeq1d 9087 . 2 (𝑥 = 𝐹 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = sup(ran 𝐺, ℝ*, < ))
11 elex 3439 . 2 (𝐹𝑉𝐹 ∈ V)
12 xrltso 12756 . . . 4 < Or ℝ*
1312supex 9104 . . 3 sup(ran 𝐺, ℝ*, < ) ∈ V
1413a1i 11 . 2 (𝐹𝑉 → sup(ran 𝐺, ℝ*, < ) ∈ V)
151, 10, 11, 14fvmptd3 6860 1 (𝐹𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2111  Vcvv 3421  cin 3880  cmpt 5150  ran crn 5567  cima 5569  cfv 6398  (class class class)co 7232  supcsup 9081  infcinf 9082  cr 10753  +∞cpnf 10889  *cxr 10891   < clt 10892  [,)cico 12962  lim infclsi 42998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pow 5273  ax-pr 5337  ax-un 7542  ax-cnex 10810  ax-resscn 10811  ax-pre-lttri 10828  ax-pre-lttrn 10829
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3067  df-rex 3068  df-rmo 3070  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-mpt 5151  df-id 5470  df-po 5483  df-so 5484  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-er 8412  df-en 8648  df-dom 8649  df-sdom 8650  df-sup 9083  df-inf 9084  df-pnf 10894  df-mnf 10895  df-xr 10896  df-ltxr 10897  df-liminf 42999
This theorem is referenced by:  liminfcl  43010  liminfvald  43011  liminfval5  43012  liminfresxr  43014  liminfval2  43015  liminfvalxr  43030
  Copyright terms: Public domain W3C validator