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Theorem limsupval2 14825
Description: The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
Hypotheses
Ref Expression
limsupval.1 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
limsupval2.1 (𝜑𝐹𝑉)
limsupval2.2 (𝜑𝐴 ⊆ ℝ)
limsupval2.3 (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
Assertion
Ref Expression
limsupval2 (𝜑 → (lim sup‘𝐹) = inf((𝐺𝐴), ℝ*, < ))
Distinct variable groups:   𝑘,𝐹   𝐴,𝑘
Allowed substitution hints:   𝜑(𝑘)   𝐺(𝑘)   𝑉(𝑘)

Proof of Theorem limsupval2
Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limsupval2.1 . . 3 (𝜑𝐹𝑉)
2 limsupval.1 . . . 4 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
32limsupval 14819 . . 3 (𝐹𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
41, 3syl 17 . 2 (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
5 imassrn 5933 . . . . 5 (𝐺𝐴) ⊆ ran 𝐺
62limsupgf 14820 . . . . . . 7 𝐺:ℝ⟶ℝ*
7 frn 6513 . . . . . . 7 (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
86, 7ax-mp 5 . . . . . 6 ran 𝐺 ⊆ ℝ*
9 infxrlb 12715 . . . . . . 7 ((ran 𝐺 ⊆ ℝ*𝑥 ∈ ran 𝐺) → inf(ran 𝐺, ℝ*, < ) ≤ 𝑥)
109ralrimiva 3179 . . . . . 6 (ran 𝐺 ⊆ ℝ* → ∀𝑥 ∈ ran 𝐺inf(ran 𝐺, ℝ*, < ) ≤ 𝑥)
118, 10mp1i 13 . . . . 5 (𝜑 → ∀𝑥 ∈ ran 𝐺inf(ran 𝐺, ℝ*, < ) ≤ 𝑥)
12 ssralv 4030 . . . . 5 ((𝐺𝐴) ⊆ ran 𝐺 → (∀𝑥 ∈ ran 𝐺inf(ran 𝐺, ℝ*, < ) ≤ 𝑥 → ∀𝑥 ∈ (𝐺𝐴)inf(ran 𝐺, ℝ*, < ) ≤ 𝑥))
135, 11, 12mpsyl 68 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐺𝐴)inf(ran 𝐺, ℝ*, < ) ≤ 𝑥)
145, 8sstri 3973 . . . . 5 (𝐺𝐴) ⊆ ℝ*
15 infxrcl 12714 . . . . . 6 (ran 𝐺 ⊆ ℝ* → inf(ran 𝐺, ℝ*, < ) ∈ ℝ*)
168, 15ax-mp 5 . . . . 5 inf(ran 𝐺, ℝ*, < ) ∈ ℝ*
17 infxrgelb 12716 . . . . 5 (((𝐺𝐴) ⊆ ℝ* ∧ inf(ran 𝐺, ℝ*, < ) ∈ ℝ*) → (inf(ran 𝐺, ℝ*, < ) ≤ inf((𝐺𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ (𝐺𝐴)inf(ran 𝐺, ℝ*, < ) ≤ 𝑥))
1814, 16, 17mp2an 688 . . . 4 (inf(ran 𝐺, ℝ*, < ) ≤ inf((𝐺𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ (𝐺𝐴)inf(ran 𝐺, ℝ*, < ) ≤ 𝑥)
1913, 18sylibr 235 . . 3 (𝜑 → inf(ran 𝐺, ℝ*, < ) ≤ inf((𝐺𝐴), ℝ*, < ))
20 limsupval2.3 . . . . . . 7 (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
21 limsupval2.2 . . . . . . . . 9 (𝜑𝐴 ⊆ ℝ)
22 ressxr 10673 . . . . . . . . 9 ℝ ⊆ ℝ*
2321, 22sstrdi 3976 . . . . . . . 8 (𝜑𝐴 ⊆ ℝ*)
24 supxrunb1 12700 . . . . . . . 8 (𝐴 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
2523, 24syl 17 . . . . . . 7 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
2620, 25mpbird 258 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥)
27 infxrcl 12714 . . . . . . . . . 10 ((𝐺𝐴) ⊆ ℝ* → inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*)
2814, 27mp1i 13 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*)
2921sselda 3964 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥 ∈ ℝ)
3029ad2ant2r 743 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥 ∈ ℝ)
316ffvelrni 6842 . . . . . . . . . 10 (𝑥 ∈ ℝ → (𝐺𝑥) ∈ ℝ*)
3230, 31syl 17 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ ℝ*)
336ffvelrni 6842 . . . . . . . . . 10 (𝑛 ∈ ℝ → (𝐺𝑛) ∈ ℝ*)
3433ad2antlr 723 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ∈ ℝ*)
35 ffn 6507 . . . . . . . . . . . 12 (𝐺:ℝ⟶ℝ*𝐺 Fn ℝ)
366, 35mp1i 13 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝐺 Fn ℝ)
3721ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝐴 ⊆ ℝ)
38 simprl 767 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥𝐴)
39 fnfvima 6986 . . . . . . . . . . 11 ((𝐺 Fn ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
4036, 37, 38, 39syl3anc 1363 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ (𝐺𝐴))
41 infxrlb 12715 . . . . . . . . . 10 (((𝐺𝐴) ⊆ ℝ* ∧ (𝐺𝑥) ∈ (𝐺𝐴)) → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑥))
4214, 40, 41sylancr 587 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑥))
43 simplr 765 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛 ∈ ℝ)
44 simprr 769 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛𝑥)
45 limsupgord 14817 . . . . . . . . . . 11 ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛𝑥) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
4643, 30, 44, 45syl3anc 1363 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
472limsupgval 14821 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝐺𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4830, 47syl 17 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
492limsupgval 14821 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (𝐺𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
5049ad2antlr 723 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
5146, 48, 503brtr4d 5089 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ≤ (𝐺𝑛))
5228, 32, 34, 42, 51xrletrd 12543 . . . . . . . 8 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛))
5352rexlimdvaa 3282 . . . . . . 7 ((𝜑𝑛 ∈ ℝ) → (∃𝑥𝐴 𝑛𝑥 → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛)))
5453ralimdva 3174 . . . . . 6 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 → ∀𝑛 ∈ ℝ inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛)))
5526, 54mpd 15 . . . . 5 (𝜑 → ∀𝑛 ∈ ℝ inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛))
566, 35ax-mp 5 . . . . . 6 𝐺 Fn ℝ
57 breq2 5061 . . . . . . 7 (𝑥 = (𝐺𝑛) → (inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥 ↔ inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛)))
5857ralrn 6846 . . . . . 6 (𝐺 Fn ℝ → (∀𝑥 ∈ ran 𝐺inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ℝ inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛)))
5956, 58ax-mp 5 . . . . 5 (∀𝑥 ∈ ran 𝐺inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ℝ inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛))
6055, 59sylibr 235 . . . 4 (𝜑 → ∀𝑥 ∈ ran 𝐺inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥)
6114, 27ax-mp 5 . . . . 5 inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*
62 infxrgelb 12716 . . . . 5 ((ran 𝐺 ⊆ ℝ* ∧ inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*) → (inf((𝐺𝐴), ℝ*, < ) ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥))
638, 61, 62mp2an 688 . . . 4 (inf((𝐺𝐴), ℝ*, < ) ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥)
6460, 63sylibr 235 . . 3 (𝜑 → inf((𝐺𝐴), ℝ*, < ) ≤ inf(ran 𝐺, ℝ*, < ))
65 xrletri3 12535 . . . 4 ((inf(ran 𝐺, ℝ*, < ) ∈ ℝ* ∧ inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*) → (inf(ran 𝐺, ℝ*, < ) = inf((𝐺𝐴), ℝ*, < ) ↔ (inf(ran 𝐺, ℝ*, < ) ≤ inf((𝐺𝐴), ℝ*, < ) ∧ inf((𝐺𝐴), ℝ*, < ) ≤ inf(ran 𝐺, ℝ*, < ))))
6616, 61, 65mp2an 688 . . 3 (inf(ran 𝐺, ℝ*, < ) = inf((𝐺𝐴), ℝ*, < ) ↔ (inf(ran 𝐺, ℝ*, < ) ≤ inf((𝐺𝐴), ℝ*, < ) ∧ inf((𝐺𝐴), ℝ*, < ) ≤ inf(ran 𝐺, ℝ*, < )))
6719, 64, 66sylanbrc 583 . 2 (𝜑 → inf(ran 𝐺, ℝ*, < ) = inf((𝐺𝐴), ℝ*, < ))
684, 67eqtrd 2853 1 (𝜑 → (lim sup‘𝐹) = inf((𝐺𝐴), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  cin 3932  wss 3933   class class class wbr 5057  cmpt 5137  ran crn 5549  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  supcsup 8892  infcinf 8893  cr 10524  +∞cpnf 10660  *cxr 10662   < clt 10663  cle 10664  [,)cico 12728  lim supclsp 14815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-ico 12732  df-limsup 14816
This theorem is referenced by:  mbflimsup  24194  limsupresico  41857  limsupvaluz  41865
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