MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  limsupval2 Structured version   Visualization version   GIF version

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  cr 10137  +∞cpnf 10273  *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
  Copyright terms: Public domain W3C validator