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Theorem limsupval2 14419
 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 14413 . . 3 (𝐹𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
41, 3syl 17 . 2 (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
5 imassrn 5618 . . . . 5 (𝐺𝐴) ⊆ ran 𝐺
62limsupgf 14414 . . . . . . 7 𝐺:ℝ⟶ℝ*
7 frn 6193 . . . . . . 7 (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
86, 7ax-mp 5 . . . . . 6 ran 𝐺 ⊆ ℝ*
9 infxrlb 12369 . . . . . . 7 ((ran 𝐺 ⊆ ℝ*𝑥 ∈ ran 𝐺) → inf(ran 𝐺, ℝ*, < ) ≤ 𝑥)
109ralrimiva 3115 . . . . . 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 3815 . . . . 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 3761 . . . . 5 (𝐺𝐴) ⊆ ℝ*
15 infxrcl 12368 . . . . . 6 (ran 𝐺 ⊆ ℝ* → inf(ran 𝐺, ℝ*, < ) ∈ ℝ*)
168, 15ax-mp 5 . . . . 5 inf(ran 𝐺, ℝ*, < ) ∈ ℝ*
17 infxrgelb 12370 . . . . 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 672 . . . 4 (inf(ran 𝐺, ℝ*, < ) ≤ inf((𝐺𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ (𝐺𝐴)inf(ran 𝐺, ℝ*, < ) ≤ 𝑥)
1913, 18sylibr 224 . . 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 10285 . . . . . . . . 9 ℝ ⊆ ℝ*
2321, 22syl6ss 3764 . . . . . . . 8 (𝜑𝐴 ⊆ ℝ*)
24 supxrunb1 12354 . . . . . . . 8 (𝐴 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
2523, 24syl 17 . . . . . . 7 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
2620, 25mpbird 247 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℝ ∃𝑥𝐴 𝑛𝑥)
27 infxrcl 12368 . . . . . . . . . 10 ((𝐺𝐴) ⊆ ℝ* → inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*)
2814, 27mp1i 13 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*)
2921sselda 3752 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥 ∈ ℝ)
3029ad2ant2r 741 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥 ∈ ℝ)
316ffvelrni 6501 . . . . . . . . . 10 (𝑥 ∈ ℝ → (𝐺𝑥) ∈ ℝ*)
3230, 31syl 17 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ ℝ*)
336ffvelrni 6501 . . . . . . . . . 10 (𝑛 ∈ ℝ → (𝐺𝑛) ∈ ℝ*)
3433ad2antlr 706 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) ∈ ℝ*)
35 ffn 6185 . . . . . . . . . . . 12 (𝐺:ℝ⟶ℝ*𝐺 Fn ℝ)
366, 35mp1i 13 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝐺 Fn ℝ)
3721ad2antrr 705 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝐴 ⊆ ℝ)
38 simprl 754 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥𝐴)
39 fnfvima 6639 . . . . . . . . . . 11 ((𝐺 Fn ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
4036, 37, 38, 39syl3anc 1476 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ (𝐺𝐴))
41 infxrlb 12369 . . . . . . . . . 10 (((𝐺𝐴) ⊆ ℝ* ∧ (𝐺𝑥) ∈ (𝐺𝐴)) → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑥))
4214, 40, 41sylancr 575 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑥))
43 simplr 752 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛 ∈ ℝ)
44 simprr 756 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑛𝑥)
45 limsupgord 14411 . . . . . . . . . . 11 ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛𝑥) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
4643, 30, 44, 45syl3anc 1476 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
472limsupgval 14415 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝐺𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4830, 47syl 17 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
492limsupgval 14415 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (𝐺𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
5049ad2antlr 706 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
5146, 48, 503brtr4d 4818 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ≤ (𝐺𝑛))
5228, 32, 34, 42, 51xrletrd 12198 . . . . . . . 8 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛))
5352rexlimdvaa 3180 . . . . . . 7 ((𝜑𝑛 ∈ ℝ) → (∃𝑥𝐴 𝑛𝑥 → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛)))
5453ralimdva 3111 . . . . . 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 4790 . . . . . . 7 (𝑥 = (𝐺𝑛) → (inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥 ↔ inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛)))
5857ralrn 6505 . . . . . 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 224 . . . 4 (𝜑 → ∀𝑥 ∈ ran 𝐺inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥)
6114, 27ax-mp 5 . . . . 5 inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*
62 infxrgelb 12370 . . . . 5 ((ran 𝐺 ⊆ ℝ* ∧ inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*) → (inf((𝐺𝐴), ℝ*, < ) ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥))
638, 61, 62mp2an 672 . . . 4 (inf((𝐺𝐴), ℝ*, < ) ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥)
6460, 63sylibr 224 . . 3 (𝜑 → inf((𝐺𝐴), ℝ*, < ) ≤ inf(ran 𝐺, ℝ*, < ))
65 xrletri3 12190 . . . 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 672 . . 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 572 . 2 (𝜑 → inf(ran 𝐺, ℝ*, < ) = inf((𝐺𝐴), ℝ*, < ))
684, 67eqtrd 2805 1 (𝜑 → (lim sup‘𝐹) = inf((𝐺𝐴), ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062   ∩ cin 3722   ⊆ wss 3723   class class class wbr 4786   ↦ cmpt 4863  ran crn 5250   “ cima 5252   Fn wfn 6026  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793  supcsup 8502  infcinf 8503  ℝcr 10137  +∞cpnf 10273  ℝ*cxr 10275   < clt 10276   ≤ cle 10277  [,)cico 12382  lim supclsp 14409 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8504  df-inf 8505  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-ico 12386  df-limsup 14410 This theorem is referenced by:  mbflimsup  23653  limsupresico  40450  limsupvaluz  40458
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