Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  liminfval5 Structured version   Visualization version   GIF version

Theorem liminfval5 42264
 Description: The inferior limit of an infinite sequence 𝐹 of extended real numbers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
limsupval5.1 𝑘𝜑
limsupval5.2 (𝜑𝐴𝑉)
limsupval5.3 (𝜑𝐹:𝐴⟶ℝ*)
limsupval5.4 𝐺 = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))
Assertion
Ref Expression
liminfval5 (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
Distinct variable group:   𝑘,𝐹
Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐺(𝑘)   𝑉(𝑘)

Proof of Theorem liminfval5
StepHypRef Expression
1 limsupval5.3 . . . 4 (𝜑𝐹:𝐴⟶ℝ*)
2 limsupval5.2 . . . 4 (𝜑𝐴𝑉)
31, 2fexd 6973 . . 3 (𝜑𝐹 ∈ V)
4 eqid 2824 . . . 4 (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
54liminfval 42258 . . 3 (𝐹 ∈ V → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
63, 5syl 17 . 2 (𝜑 → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
7 limsupval5.4 . . . . . 6 𝐺 = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))
87a1i 11 . . . . 5 (𝜑𝐺 = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )))
9 limsupval5.1 . . . . . 6 𝑘𝜑
101fimassd 41720 . . . . . . . . . 10 (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
11 df-ss 3935 . . . . . . . . . 10 ((𝐹 “ (𝑘[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞)))
1210, 11sylib 221 . . . . . . . . 9 (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞)))
1312eqcomd 2830 . . . . . . . 8 (𝜑 → (𝐹 “ (𝑘[,)+∞)) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
1413adantr 484 . . . . . . 7 ((𝜑𝑘 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
1514infeq1d 8927 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
169, 15mpteq2da 5143 . . . . 5 (𝜑 → (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )))
178, 16eqtr2d 2860 . . . 4 (𝜑 → (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺)
1817rneqd 5791 . . 3 (𝜑 → ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺)
1918supeq1d 8896 . 2 (𝜑 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = sup(ran 𝐺, ℝ*, < ))
206, 19eqtrd 2859 1 (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2115  Vcvv 3479   ∩ cin 3917   ⊆ wss 3918   ↦ cmpt 5129  ran crn 5539   “ cima 5541  ⟶wf 6334  ‘cfv 6338  (class class class)co 7140  supcsup 8890  infcinf 8891  ℝcr 10523  +∞cpnf 10659  ℝ*cxr 10661   < clt 10662  [,)cico 12728  lim infclsi 42250 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-cnex 10580  ax-resscn 10581  ax-pre-lttri 10598  ax-pre-lttrn 10599 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-po 5457  df-so 5458  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-sup 8892  df-inf 8893  df-pnf 10664  df-mnf 10665  df-xr 10666  df-ltxr 10667  df-liminf 42251 This theorem is referenced by:  liminf10ex  42273
 Copyright terms: Public domain W3C validator