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Theorem liminfval2 45294
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 7426 . . . . . . . . 9 (𝑘 = 𝑗 → (𝑘[,)+∞) = (𝑗[,)+∞))
43imaeq2d 6064 . . . . . . . 8 (𝑘 = 𝑗 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑗[,)+∞)))
54ineq1d 4209 . . . . . . 7 (𝑘 = 𝑗 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*))
65infeq1d 9502 . . . . . 6 (𝑘 = 𝑗 → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
76cbvmptv 5262 . . . . 5 (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
82, 7eqtri 2753 . . . 4 𝐺 = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
98liminfval 45285 . . 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 44952 . . . . . . . 8 (𝜑𝐴 ⊆ ℝ*)
14 supxrunb1 13333 . . . . . . . 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 45284 . . . . . . . . . . 11 𝐺:ℝ⟶ℝ*
1817ffvelcdmi 7092 . . . . . . . . . 10 (𝑛 ∈ ℝ → (𝐺𝑛) ∈ ℝ*)
1918ad2antlr 725 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ∈ ℝ*)
20 simpll 765 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝜑)
21 simprl 769 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥𝐴)
2212sselda 3976 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥 ∈ ℝ)
2317ffvelcdmi 7092 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝐺𝑥) ∈ ℝ*)
2422, 23syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ ℝ*)
2520, 21, 24syl2anc 582 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ ℝ*)
26 imassrn 6075 . . . . . . . . . . . 12 (𝐺𝐴) ⊆ ran 𝐺
27 frn 6730 . . . . . . . . . . . . 13 (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
2817, 27ax-mp 5 . . . . . . . . . . . 12 ran 𝐺 ⊆ ℝ*
2926, 28sstri 3986 . . . . . . . . . . 11 (𝐺𝐴) ⊆ ℝ*
30 supxrcl 13329 . . . . . . . . . . 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 767 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛 ∈ ℝ)
3420, 21, 22syl2anc 582 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥 ∈ ℝ)
35 simprr 771 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛𝑥)
36 liminfgord 45280 . . . . . . . . . . 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 45288 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ → (𝐺𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
3938ad2antlr 725 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → (𝐺𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
408liminfgval 45288 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4122, 40syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4241adantlr 713 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → (𝐺𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4339, 42breq12d 5162 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ 𝑥𝐴) → ((𝐺𝑛) ≤ (𝐺𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
4443adantrr 715 . . . . . . . . . 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 1909 . . . . . . . . . . . . . 14 𝑗𝜑
48 inss2 4228 . . . . . . . . . . . . . . . 16 ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ⊆ ℝ*
49 infxrcl 13347 . . . . . . . . . . . . . . . 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 6697 . . . . . . . . . . . . 13 (𝜑𝐺 Fn ℝ)
5352adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐺 Fn ℝ)
54 simpr 483 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑥𝐴)
5553, 22, 54fnfvimad 7246 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
56 supxrub 13338 . . . . . . . . . . 11 (((𝐺𝐴) ⊆ ℝ* ∧ (𝐺𝑥) ∈ (𝐺𝐴)) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5746, 55, 56syl2anc 582 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5820, 21, 57syl2anc 582 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ≤ sup((𝐺𝐴), ℝ*, < ))
5919, 25, 32, 45, 58xrletrd 13176 . . . . . . . 8 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < ))
6059rexlimdvaa 3145 . . . . . . 7 ((𝜑𝑛 ∈ ℝ) → (∃𝑥𝐴 𝑛𝑥 → (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < )))
6160ralimdva 3156 . . . . . 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 13155 . . . . . . . . 9 < Or ℝ*
6463infex 9518 . . . . . . . 8 inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
6564rgenw 3054 . . . . . . 7 𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
668fnmpt 6696 . . . . . . 7 (∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V → 𝐺 Fn ℝ)
6765, 66ax-mp 5 . . . . . 6 𝐺 Fn ℝ
68 breq1 5152 . . . . . . 7 (𝑥 = (𝐺𝑛) → (𝑥 ≤ sup((𝐺𝐴), ℝ*, < ) ↔ (𝐺𝑛) ≤ sup((𝐺𝐴), ℝ*, < )))
6968ralrn 7097 . . . . . 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 13340 . . . . 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 690 . . . 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 13346 . . . 4 (((𝐺𝐴) ⊆ ran 𝐺 ∧ ran 𝐺 ⊆ ℝ*) → sup((𝐺𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
7875, 76, 77syl2anc 582 . . 3 (𝜑 → sup((𝐺𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
79 supxrcl 13329 . . . . 5 (ran 𝐺 ⊆ ℝ* → sup(ran 𝐺, ℝ*, < ) ∈ ℝ*)
8028, 79ax-mp 5 . . . 4 sup(ran 𝐺, ℝ*, < ) ∈ ℝ*
81 xrletri3 13168 . . . 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 690 . . 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 581 . 2 (𝜑 → sup(ran 𝐺, ℝ*, < ) = sup((𝐺𝐴), ℝ*, < ))
8410, 83eqtrd 2765 1 (𝜑 → (lim inf‘𝐹) = sup((𝐺𝐴), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  Vcvv 3461  cin 3943  wss 3944   class class class wbr 5149  cmpt 5232  ran crn 5679  cima 5681   Fn wfn 6544  wf 6545  cfv 6549  (class class class)co 7419  supcsup 9465  infcinf 9466  cr 11139  +∞cpnf 11277  *cxr 11279   < clt 11280  cle 11281  [,)cico 13361  lim infclsi 45277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-ico 13365  df-liminf 45278
This theorem is referenced by:  liminfresico  45297
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