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Theorem limsupval2 15516
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 15510 . . 3 (𝐹𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
41, 3syl 17 . 2 (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
5 imassrn 6089 . . . . 5 (𝐺𝐴) ⊆ ran 𝐺
62limsupgf 15511 . . . . . . 7 𝐺:ℝ⟶ℝ*
7 frn 6743 . . . . . . 7 (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
86, 7ax-mp 5 . . . . . 6 ran 𝐺 ⊆ ℝ*
9 infxrlb 13376 . . . . . . 7 ((ran 𝐺 ⊆ ℝ*𝑥 ∈ ran 𝐺) → inf(ran 𝐺, ℝ*, < ) ≤ 𝑥)
109ralrimiva 3146 . . . . . 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 4052 . . . . 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 3993 . . . . 5 (𝐺𝐴) ⊆ ℝ*
15 infxrcl 13375 . . . . . 6 (ran 𝐺 ⊆ ℝ* → inf(ran 𝐺, ℝ*, < ) ∈ ℝ*)
168, 15ax-mp 5 . . . . 5 inf(ran 𝐺, ℝ*, < ) ∈ ℝ*
17 infxrgelb 13377 . . . . 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 692 . . . 4 (inf(ran 𝐺, ℝ*, < ) ≤ inf((𝐺𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ (𝐺𝐴)inf(ran 𝐺, ℝ*, < ) ≤ 𝑥)
1913, 18sylibr 234 . . 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 11305 . . . . . . . . 9 ℝ ⊆ ℝ*
2321, 22sstrdi 3996 . . . . . . . 8 (𝜑𝐴 ⊆ ℝ*)
24 supxrunb1 13361 . . . . . . . 8 (𝐴 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
2523, 24syl 17 . . . . . . 7 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
2620, 25mpbird 257 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥)
27 infxrcl 13375 . . . . . . . . . 10 ((𝐺𝐴) ⊆ ℝ* → inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*)
2814, 27mp1i 13 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*)
2921sselda 3983 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥 ∈ ℝ)
3029ad2ant2r 747 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥 ∈ ℝ)
316ffvelcdmi 7103 . . . . . . . . . 10 (𝑥 ∈ ℝ → (𝐺𝑥) ∈ ℝ*)
3230, 31syl 17 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ ℝ*)
336ffvelcdmi 7103 . . . . . . . . . 10 (𝑛 ∈ ℝ → (𝐺𝑛) ∈ ℝ*)
3433ad2antlr 727 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ∈ ℝ*)
35 ffn 6736 . . . . . . . . . . . 12 (𝐺:ℝ⟶ℝ*𝐺 Fn ℝ)
366, 35mp1i 13 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝐺 Fn ℝ)
3721ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝐴 ⊆ ℝ)
38 simprl 771 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥𝐴)
39 fnfvima 7253 . . . . . . . . . . 11 ((𝐺 Fn ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
4036, 37, 38, 39syl3anc 1373 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ (𝐺𝐴))
41 infxrlb 13376 . . . . . . . . . 10 (((𝐺𝐴) ⊆ ℝ* ∧ (𝐺𝑥) ∈ (𝐺𝐴)) → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑥))
4214, 40, 41sylancr 587 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑥))
43 simplr 769 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛 ∈ ℝ)
44 simprr 773 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛𝑥)
45 limsupgord 15508 . . . . . . . . . . 11 ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛𝑥) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
4643, 30, 44, 45syl3anc 1373 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
472limsupgval 15512 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝐺𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4830, 47syl 17 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
492limsupgval 15512 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (𝐺𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
5049ad2antlr 727 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
5146, 48, 503brtr4d 5175 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ≤ (𝐺𝑛))
5228, 32, 34, 42, 51xrletrd 13204 . . . . . . . 8 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛))
5352rexlimdvaa 3156 . . . . . . 7 ((𝜑𝑛 ∈ ℝ) → (∃𝑥𝐴 𝑛𝑥 → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛)))
5453ralimdva 3167 . . . . . 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 5147 . . . . . . 7 (𝑥 = (𝐺𝑛) → (inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥 ↔ inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛)))
5857ralrn 7108 . . . . . 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 234 . . . 4 (𝜑 → ∀𝑥 ∈ ran 𝐺inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥)
6114, 27ax-mp 5 . . . . 5 inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*
62 infxrgelb 13377 . . . . 5 ((ran 𝐺 ⊆ ℝ* ∧ inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*) → (inf((𝐺𝐴), ℝ*, < ) ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥))
638, 61, 62mp2an 692 . . . 4 (inf((𝐺𝐴), ℝ*, < ) ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥)
6460, 63sylibr 234 . . 3 (𝜑 → inf((𝐺𝐴), ℝ*, < ) ≤ inf(ran 𝐺, ℝ*, < ))
65 xrletri3 13196 . . . 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 692 . . 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 2777 1 (𝜑 → (lim sup‘𝐹) = inf((𝐺𝐴), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951   class class class wbr 5143  cmpt 5225  ran crn 5686  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  supcsup 9480  infcinf 9481  cr 11154  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  [,)cico 13389  lim supclsp 15506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-ico 13393  df-limsup 15507
This theorem is referenced by:  mbflimsup  25701  limsupresico  45715  limsupvaluz  45723
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