Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  liminfval2 Structured version   Visualization version   GIF version

Theorem liminfval2 42403
Description: The superior limit, relativized to an unbounded set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
liminfval2.1 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
liminfval2.2 (𝜑𝐹𝑉)
liminfval2.3 (𝜑𝐴 ⊆ ℝ)
liminfval2.4 (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
Assertion
Ref Expression
liminfval2 (𝜑 → (lim inf‘𝐹) = sup((𝐺𝐴), ℝ*, < ))
Distinct variable group:   𝑘,𝐹
Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐺(𝑘)   𝑉(𝑘)

Proof of Theorem liminfval2
Dummy variables 𝑛 𝑥 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 liminfval2.2 . . 3 (𝜑𝐹𝑉)
2 liminfval2.1 . . . . 5 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
3 oveq1 7146 . . . . . . . . 9 (𝑘 = 𝑗 → (𝑘[,)+∞) = (𝑗[,)+∞))
43imaeq2d 5900 . . . . . . . 8 (𝑘 = 𝑗 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑗[,)+∞)))
54ineq1d 4141 . . . . . . 7 (𝑘 = 𝑗 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*))
65infeq1d 8929 . . . . . 6 (𝑘 = 𝑗 → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
76cbvmptv 5136 . . . . 5 (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
82, 7eqtri 2824 . . . 4 𝐺 = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
98liminfval 42394 . . 3 (𝐹𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
101, 9syl 17 . 2 (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
11 liminfval2.4 . . . . . . 7 (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
12 liminfval2.3 . . . . . . . . 9 (𝜑𝐴 ⊆ ℝ)
1312ssrexr 42062 . . . . . . . 8 (𝜑𝐴 ⊆ ℝ*)
14 supxrunb1 12704 . . . . . . . 8 (𝐴 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
1513, 14syl 17 . . . . . . 7 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
1611, 15mpbird 260 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥)
178liminfgf 42393 . . . . . . . . . . 11 𝐺:ℝ⟶ℝ*
1817ffvelrni 6831 . . . . . . . . . 10 (𝑛 ∈ ℝ → (𝐺𝑛) ∈ ℝ*)
1918ad2antlr 726 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ∈ ℝ*)
20 simpll 766 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝜑)
21 simprl 770 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥𝐴)
2212sselda 3918 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥 ∈ ℝ)
2317ffvelrni 6831 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝐺𝑥) ∈ ℝ*)
2422, 23syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ ℝ*)
2520, 21, 24syl2anc 587 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ ℝ*)
26 imassrn 5911 . . . . . . . . . . . 12 (𝐺𝐴) ⊆ ran 𝐺
27 frn 6497 . . . . . . . . . . . . 13 (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
2817, 27ax-mp 5 . . . . . . . . . . . 12 ran 𝐺 ⊆ ℝ*
2926, 28sstri 3927 . . . . . . . . . . 11 (𝐺𝐴) ⊆ ℝ*
30 supxrcl 12700 . . . . . . . . . . 11 ((𝐺𝐴) ⊆ ℝ* → sup((𝐺𝐴), ℝ*, < ) ∈ ℝ*)
3129, 30ax-mp 5 . . . . . . . . . 10 sup((𝐺𝐴), ℝ*, < ) ∈ ℝ*
3231a1i 11 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → sup((𝐺𝐴), ℝ*, < ) ∈ ℝ*)
33 simplr 768 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛 ∈ ℝ)
3420, 21, 22syl2anc 587 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥 ∈ ℝ)
35 simprr 772 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛𝑥)
36 liminfgord 42389 . . . . . . . . . . 11 ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛𝑥) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
3733, 34, 35, 36syl3anc 1368 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
388liminfgval 42397 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ → (𝐺𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
3938ad2antlr 726 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → (𝐺𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
408liminfgval 42397 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4122, 40syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4241adantlr 714 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4339, 42breq12d 5046 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → ((𝐺𝑛) ≤ (𝐺𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
4443adantrr 716 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → ((𝐺𝑛) ≤ (𝐺𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
4537, 44mpbird 260 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ≤ (𝐺𝑥))
4629a1i 11 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐺𝐴) ⊆ ℝ*)
47 nfv 1915 . . . . . . . . . . . . . 14 𝑗𝜑
48 inss2 4159 . . . . . . . . . . . . . . . 16 ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ⊆ ℝ*
49 infxrcl 12718 . . . . . . . . . . . . . . . 16 (((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ⊆ ℝ* → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*)
5048, 49ax-mp 5 . . . . . . . . . . . . . . 15 inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*
5150a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℝ) → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*)
5247, 51, 8fnmptd 6465 . . . . . . . . . . . . 13 (𝜑𝐺 Fn ℝ)
5352adantr 484 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐺 Fn ℝ)
54 simpr 488 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑥𝐴)
5553, 22, 54fnfvimad 6978 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
56 supxrub 12709 . . . . . . . . . . 11 (((𝐺𝐴) ⊆ ℝ* ∧ (𝐺𝑥) ∈ (𝐺𝐴)) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5746, 55, 56syl2anc 587 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5820, 21, 57syl2anc 587 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5919, 25, 32, 45, 58xrletrd 12547 . . . . . . . 8 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < ))
6059rexlimdvaa 3247 . . . . . . 7 ((𝜑𝑛 ∈ ℝ) → (∃𝑥𝐴 𝑛𝑥 → (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < )))
6160ralimdva 3147 . . . . . 6 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 → ∀𝑛 ∈ ℝ (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < )))
6216, 61mpd 15 . . . . 5 (𝜑 → ∀𝑛 ∈ ℝ (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < ))
63 xrltso 12526 . . . . . . . . 9 < Or ℝ*
6463infex 8945 . . . . . . . 8 inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
6564rgenw 3121 . . . . . . 7 𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
668fnmpt 6464 . . . . . . 7 (∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V → 𝐺 Fn ℝ)
6765, 66ax-mp 5 . . . . . 6 𝐺 Fn ℝ
68 breq1 5036 . . . . . . 7 (𝑥 = (𝐺𝑛) → (𝑥 ≤ sup((𝐺𝐴), ℝ*, < ) ↔ (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < )))
6968ralrn 6835 . . . . . 6 (𝐺 Fn ℝ → (∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺𝐴), ℝ*, < ) ↔ ∀𝑛 ∈ ℝ (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < )))
7067, 69ax-mp 5 . . . . 5 (∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺𝐴), ℝ*, < ) ↔ ∀𝑛 ∈ ℝ (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < ))
7162, 70sylibr 237 . . . 4 (𝜑 → ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺𝐴), ℝ*, < ))
72 supxrleub 12711 . . . . 5 ((ran 𝐺 ⊆ ℝ* ∧ sup((𝐺𝐴), ℝ*, < ) ∈ ℝ*) → (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺𝐴), ℝ*, < )))
7328, 31, 72mp2an 691 . . . 4 (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺𝐴), ℝ*, < ))
7471, 73sylibr 237 . . 3 (𝜑 → sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺𝐴), ℝ*, < ))
7526a1i 11 . . . 4 (𝜑 → (𝐺𝐴) ⊆ ran 𝐺)
7628a1i 11 . . . 4 (𝜑 → ran 𝐺 ⊆ ℝ*)
77 supxrss 12717 . . . 4 (((𝐺𝐴) ⊆ ran 𝐺 ∧ ran 𝐺 ⊆ ℝ*) → sup((𝐺𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
7875, 76, 77syl2anc 587 . . 3 (𝜑 → sup((𝐺𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
79 supxrcl 12700 . . . . 5 (ran 𝐺 ⊆ ℝ* → sup(ran 𝐺, ℝ*, < ) ∈ ℝ*)
8028, 79ax-mp 5 . . . 4 sup(ran 𝐺, ℝ*, < ) ∈ ℝ*
81 xrletri3 12539 . . . 4 ((sup(ran 𝐺, ℝ*, < ) ∈ ℝ* ∧ sup((𝐺𝐴), ℝ*, < ) ∈ ℝ*) → (sup(ran 𝐺, ℝ*, < ) = sup((𝐺𝐴), ℝ*, < ) ↔ (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺𝐴), ℝ*, < ) ∧ sup((𝐺𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))))
8280, 31, 81mp2an 691 . . 3 (sup(ran 𝐺, ℝ*, < ) = sup((𝐺𝐴), ℝ*, < ) ↔ (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺𝐴), ℝ*, < ) ∧ sup((𝐺𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < )))
8374, 78, 82sylanbrc 586 . 2 (𝜑 → sup(ran 𝐺, ℝ*, < ) = sup((𝐺𝐴), ℝ*, < ))
8410, 83eqtrd 2836 1 (𝜑 → (lim inf‘𝐹) = sup((𝐺𝐴), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  wrex 3110  Vcvv 3444  cin 3883  wss 3884   class class class wbr 5033  cmpt 5113  ran crn 5524  cima 5526   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  supcsup 8892  infcinf 8893  cr 10529  +∞cpnf 10665  *cxr 10667   < clt 10668  cle 10669  [,)cico 12732  lim infclsi 42386
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-sup 8894  df-inf 8895  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-ico 12736  df-liminf 42387
This theorem is referenced by:  liminfresico  42406
  Copyright terms: Public domain W3C validator