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Theorem liminfval2 43653
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 7344 . . . . . . . . 9 (𝑘 = 𝑗 → (𝑘[,)+∞) = (𝑗[,)+∞))
43imaeq2d 5999 . . . . . . . 8 (𝑘 = 𝑗 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑗[,)+∞)))
54ineq1d 4158 . . . . . . 7 (𝑘 = 𝑗 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*))
65infeq1d 9334 . . . . . 6 (𝑘 = 𝑗 → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
76cbvmptv 5205 . . . . 5 (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
82, 7eqtri 2764 . . . 4 𝐺 = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
98liminfval 43644 . . 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 43315 . . . . . . . 8 (𝜑𝐴 ⊆ ℝ*)
14 supxrunb1 13154 . . . . . . . 8 (𝐴 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
1513, 14syl 17 . . . . . . 7 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
1611, 15mpbird 256 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥)
178liminfgf 43643 . . . . . . . . . . 11 𝐺:ℝ⟶ℝ*
1817ffvelcdmi 7016 . . . . . . . . . 10 (𝑛 ∈ ℝ → (𝐺𝑛) ∈ ℝ*)
1918ad2antlr 724 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ∈ ℝ*)
20 simpll 764 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝜑)
21 simprl 768 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥𝐴)
2212sselda 3932 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥 ∈ ℝ)
2317ffvelcdmi 7016 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝐺𝑥) ∈ ℝ*)
2422, 23syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ ℝ*)
2520, 21, 24syl2anc 584 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ ℝ*)
26 imassrn 6010 . . . . . . . . . . . 12 (𝐺𝐴) ⊆ ran 𝐺
27 frn 6658 . . . . . . . . . . . . 13 (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
2817, 27ax-mp 5 . . . . . . . . . . . 12 ran 𝐺 ⊆ ℝ*
2926, 28sstri 3941 . . . . . . . . . . 11 (𝐺𝐴) ⊆ ℝ*
30 supxrcl 13150 . . . . . . . . . . 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 766 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛 ∈ ℝ)
3420, 21, 22syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥 ∈ ℝ)
35 simprr 770 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛𝑥)
36 liminfgord 43639 . . . . . . . . . . 11 ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛𝑥) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
3733, 34, 35, 36syl3anc 1370 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
388liminfgval 43647 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ → (𝐺𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
3938ad2antlr 724 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → (𝐺𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
408liminfgval 43647 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4122, 40syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4241adantlr 712 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4339, 42breq12d 5105 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → ((𝐺𝑛) ≤ (𝐺𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
4443adantrr 714 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → ((𝐺𝑛) ≤ (𝐺𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
4537, 44mpbird 256 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ≤ (𝐺𝑥))
4629a1i 11 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐺𝐴) ⊆ ℝ*)
47 nfv 1916 . . . . . . . . . . . . . 14 𝑗𝜑
48 inss2 4176 . . . . . . . . . . . . . . . 16 ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ⊆ ℝ*
49 infxrcl 13168 . . . . . . . . . . . . . . . 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 6625 . . . . . . . . . . . . 13 (𝜑𝐺 Fn ℝ)
5352adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐺 Fn ℝ)
54 simpr 485 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑥𝐴)
5553, 22, 54fnfvimad 7166 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
56 supxrub 13159 . . . . . . . . . . 11 (((𝐺𝐴) ⊆ ℝ* ∧ (𝐺𝑥) ∈ (𝐺𝐴)) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5746, 55, 56syl2anc 584 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5820, 21, 57syl2anc 584 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5919, 25, 32, 45, 58xrletrd 12997 . . . . . . . 8 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < ))
6059rexlimdvaa 3149 . . . . . . 7 ((𝜑𝑛 ∈ ℝ) → (∃𝑥𝐴 𝑛𝑥 → (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < )))
6160ralimdva 3160 . . . . . 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 12976 . . . . . . . . 9 < Or ℝ*
6463infex 9350 . . . . . . . 8 inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
6564rgenw 3065 . . . . . . 7 𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
668fnmpt 6624 . . . . . . 7 (∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V → 𝐺 Fn ℝ)
6765, 66ax-mp 5 . . . . . 6 𝐺 Fn ℝ
68 breq1 5095 . . . . . . 7 (𝑥 = (𝐺𝑛) → (𝑥 ≤ sup((𝐺𝐴), ℝ*, < ) ↔ (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < )))
6968ralrn 7020 . . . . . 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 233 . . . 4 (𝜑 → ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺𝐴), ℝ*, < ))
72 supxrleub 13161 . . . . 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 689 . . . 4 (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺𝐴), ℝ*, < ))
7471, 73sylibr 233 . . 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 13167 . . . 4 (((𝐺𝐴) ⊆ ran 𝐺 ∧ ran 𝐺 ⊆ ℝ*) → sup((𝐺𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
7875, 76, 77syl2anc 584 . . 3 (𝜑 → sup((𝐺𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
79 supxrcl 13150 . . . . 5 (ran 𝐺 ⊆ ℝ* → sup(ran 𝐺, ℝ*, < ) ∈ ℝ*)
8028, 79ax-mp 5 . . . 4 sup(ran 𝐺, ℝ*, < ) ∈ ℝ*
81 xrletri3 12989 . . . 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 689 . . 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 583 . 2 (𝜑 → sup(ran 𝐺, ℝ*, < ) = sup((𝐺𝐴), ℝ*, < ))
8410, 83eqtrd 2776 1 (𝜑 → (lim inf‘𝐹) = sup((𝐺𝐴), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wral 3061  wrex 3070  Vcvv 3441  cin 3897  wss 3898   class class class wbr 5092  cmpt 5175  ran crn 5621  cima 5623   Fn wfn 6474  wf 6475  cfv 6479  (class class class)co 7337  supcsup 9297  infcinf 9298  cr 10971  +∞cpnf 11107  *cxr 11109   < clt 11110  cle 11111  [,)cico 13182  lim infclsi 43636
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-po 5532  df-so 5533  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-sup 9299  df-inf 9300  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-ico 13186  df-liminf 43637
This theorem is referenced by:  liminfresico  43656
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