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Theorem limsupval2 15512
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 15506 . . 3 (𝐹𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
41, 3syl 17 . 2 (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
5 imassrn 6090 . . . . 5 (𝐺𝐴) ⊆ ran 𝐺
62limsupgf 15507 . . . . . . 7 𝐺:ℝ⟶ℝ*
7 frn 6743 . . . . . . 7 (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
86, 7ax-mp 5 . . . . . 6 ran 𝐺 ⊆ ℝ*
9 infxrlb 13372 . . . . . . 7 ((ran 𝐺 ⊆ ℝ*𝑥 ∈ ran 𝐺) → inf(ran 𝐺, ℝ*, < ) ≤ 𝑥)
109ralrimiva 3143 . . . . . 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 4063 . . . . 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 4004 . . . . 5 (𝐺𝐴) ⊆ ℝ*
15 infxrcl 13371 . . . . . 6 (ran 𝐺 ⊆ ℝ* → inf(ran 𝐺, ℝ*, < ) ∈ ℝ*)
168, 15ax-mp 5 . . . . 5 inf(ran 𝐺, ℝ*, < ) ∈ ℝ*
17 infxrgelb 13373 . . . . 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 11302 . . . . . . . . 9 ℝ ⊆ ℝ*
2321, 22sstrdi 4007 . . . . . . . 8 (𝜑𝐴 ⊆ ℝ*)
24 supxrunb1 13357 . . . . . . . 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 13371 . . . . . . . . . 10 ((𝐺𝐴) ⊆ ℝ* → inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*)
2814, 27mp1i 13 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf((𝐺𝐴), ℝ*, < ) ∈ ℝ*)
2921sselda 3994 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥 ∈ ℝ)
3029ad2ant2r 747 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → 𝑥 ∈ ℝ)
316ffvelcdmi 7102 . . . . . . . . . 10 (𝑥 ∈ ℝ → (𝐺𝑥) ∈ ℝ*)
3230, 31syl 17 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ ℝ*)
336ffvelcdmi 7102 . . . . . . . . . 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 7252 . . . . . . . . . . 11 ((𝐺 Fn ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
4036, 37, 38, 39syl3anc 1370 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ∈ (𝐺𝐴))
41 infxrlb 13372 . . . . . . . . . 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 15504 . . . . . . . . . . 11 ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛𝑥) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
4643, 30, 44, 45syl3anc 1370 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
472limsupgval 15508 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝐺𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
4830, 47syl 17 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
492limsupgval 15508 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (𝐺𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
5049ad2antlr 727 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
5146, 48, 503brtr4d 5179 . . . . . . . . 9 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → (𝐺𝑥) ≤ (𝐺𝑛))
5228, 32, 34, 42, 51xrletrd 13200 . . . . . . . 8 (((𝜑𝑛 ∈ ℝ) ∧ (𝑥𝐴𝑛𝑥)) → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛))
5352rexlimdvaa 3153 . . . . . . 7 ((𝜑𝑛 ∈ ℝ) → (∃𝑥𝐴 𝑛𝑥 → inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛)))
5453ralimdva 3164 . . . . . 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 5151 . . . . . . 7 (𝑥 = (𝐺𝑛) → (inf((𝐺𝐴), ℝ*, < ) ≤ 𝑥 ↔ inf((𝐺𝐴), ℝ*, < ) ≤ (𝐺𝑛)))
5857ralrn 7107 . . . . . 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 13373 . . . . 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 13192 . . . 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 2774 1 (𝜑 → (lim sup‘𝐹) = inf((𝐺𝐴), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  cin 3961  wss 3962   class class class wbr 5147  cmpt 5230  ran crn 5689  cima 5691   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  supcsup 9477  infcinf 9478  cr 11151  +∞cpnf 11289  *cxr 11291   < clt 11292  cle 11293  [,)cico 13385  lim supclsp 15502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-ico 13389  df-limsup 15503
This theorem is referenced by:  mbflimsup  25714  limsupresico  45655  limsupvaluz  45663
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