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Theorem liminfval2 40318
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 6697 . . . . . . . . 9 (𝑘 = 𝑗 → (𝑘[,)+∞) = (𝑗[,)+∞))
43imaeq2d 5501 . . . . . . . 8 (𝑘 = 𝑗 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑗[,)+∞)))
54ineq1d 3846 . . . . . . 7 (𝑘 = 𝑗 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*))
65infeq1d 8424 . . . . . 6 (𝑘 = 𝑗 → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
76cbvmptv 4783 . . . . 5 (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
82, 7eqtri 2673 . . . 4 𝐺 = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
98liminfval 40309 . . 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 39972 . . . . . . . 8 (𝜑𝐴 ⊆ ℝ*)
14 supxrunb1 12187 . . . . . . . 8 (𝐴 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
1513, 14syl 17 . . . . . . 7 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
1611, 15mpbird 247 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥)
178liminfgf 40308 . . . . . . . . . . 11 𝐺:ℝ⟶ℝ*
1817ffvelrni 6398 . . . . . . . . . 10 (𝑛 ∈ ℝ → (𝐺𝑛) ∈ ℝ*)
1918ad2antlr 763 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ∈ ℝ*)
20 simpll 805 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝜑)
21 simprl 809 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥𝐴)
2212sselda 3636 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥 ∈ ℝ)
2317ffvelrni 6398 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝐺𝑥) ∈ ℝ*)
2422, 23syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ ℝ*)
2520, 21, 24syl2anc 694 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ ℝ*)
26 imassrn 5512 . . . . . . . . . . . 12 (𝐺𝐴) ⊆ ran 𝐺
27 frn 6091 . . . . . . . . . . . . 13 (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
2817, 27ax-mp 5 . . . . . . . . . . . 12 ran 𝐺 ⊆ ℝ*
2926, 28sstri 3645 . . . . . . . . . . 11 (𝐺𝐴) ⊆ ℝ*
30 supxrcl 12183 . . . . . . . . . . 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 807 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛 ∈ ℝ)
3420, 21, 22syl2anc 694 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥 ∈ ℝ)
35 simprr 811 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛𝑥)
36 liminfgord 40304 . . . . . . . . . . 11 ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛𝑥) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
3733, 34, 35, 36syl3anc 1366 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
388liminfgval 40312 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ → (𝐺𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
3938ad2antlr 763 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → (𝐺𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
408liminfgval 40312 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4122, 40syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4241adantlr 751 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4339, 42breq12d 4698 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → ((𝐺𝑛) ≤ (𝐺𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
4443adantrr 753 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → ((𝐺𝑛) ≤ (𝐺𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
4537, 44mpbird 247 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ≤ (𝐺𝑥))
4629a1i 11 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐺𝐴) ⊆ ℝ*)
47 nfv 1883 . . . . . . . . . . . . . 14 𝑗𝜑
48 inss2 3867 . . . . . . . . . . . . . . . 16 ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ⊆ ℝ*
49 infxrcl 12201 . . . . . . . . . . . . . . . 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 39748 . . . . . . . . . . . . 13 (𝜑𝐺 Fn ℝ)
5352adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐺 Fn ℝ)
54 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑥𝐴)
5553, 22, 54fnfvimad 39773 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
56 supxrub 12192 . . . . . . . . . . 11 (((𝐺𝐴) ⊆ ℝ* ∧ (𝐺𝑥) ∈ (𝐺𝐴)) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5746, 55, 56syl2anc 694 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5820, 21, 57syl2anc 694 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5919, 25, 32, 45, 58xrletrd 12031 . . . . . . . 8 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < ))
6059rexlimdvaa 3061 . . . . . . 7 ((𝜑𝑛 ∈ ℝ) → (∃𝑥𝐴 𝑛𝑥 → (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < )))
6160ralimdva 2991 . . . . . 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 12012 . . . . . . . . 9 < Or ℝ*
6463infex 8440 . . . . . . . 8 inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
6564rgenw 2953 . . . . . . 7 𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
668fnmpt 6058 . . . . . . 7 (∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V → 𝐺 Fn ℝ)
6765, 66ax-mp 5 . . . . . 6 𝐺 Fn ℝ
68 breq1 4688 . . . . . . 7 (𝑥 = (𝐺𝑛) → (𝑥 ≤ sup((𝐺𝐴), ℝ*, < ) ↔ (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < )))
6968ralrn 6402 . . . . . 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 224 . . . 4 (𝜑 → ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺𝐴), ℝ*, < ))
72 supxrleub 12194 . . . . 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 708 . . . 4 (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺𝐴), ℝ*, < ))
7471, 73sylibr 224 . . 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 12200 . . . 4 (((𝐺𝐴) ⊆ ran 𝐺 ∧ ran 𝐺 ⊆ ℝ*) → sup((𝐺𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
7875, 76, 77syl2anc 694 . . 3 (𝜑 → sup((𝐺𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
79 supxrcl 12183 . . . . 5 (ran 𝐺 ⊆ ℝ* → sup(ran 𝐺, ℝ*, < ) ∈ ℝ*)
8028, 79ax-mp 5 . . . 4 sup(ran 𝐺, ℝ*, < ) ∈ ℝ*
81 xrletri3 12023 . . . 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 708 . . 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 699 . 2 (𝜑 → sup(ran 𝐺, ℝ*, < ) = sup((𝐺𝐴), ℝ*, < ))
8410, 83eqtrd 2685 1 (𝜑 → (lim inf‘𝐹) = sup((𝐺𝐴), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607   class class class wbr 4685  cmpt 4762  ran crn 5144  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  supcsup 8387  infcinf 8388  cr 9973  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  [,)cico 12215  lim infclsi 40301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-ico 12219  df-liminf 40302
This theorem is referenced by:  liminfresico  40321
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