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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  ℝcr 9973  +∞cpnf 10109  ℝ*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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