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Theorem limsupvaluz 40258
Description: The superior limit, when the domain of the function is a set of upper integers (the first condition is needed, otherwise the l.h.s. would be -∞ and the r.h.s. would be +∞). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupvaluz.m (𝜑𝑀 ∈ ℤ)
limsupvaluz.z 𝑍 = (ℤ𝑀)
limsupvaluz.f (𝜑𝐹:𝑍⟶ℝ*)
Assertion
Ref Expression
limsupvaluz (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ))
Distinct variable groups:   𝑘,𝐹   𝑘,𝑍
Allowed substitution hints:   𝜑(𝑘)   𝑀(𝑘)

Proof of Theorem limsupvaluz
Dummy variables 𝑖 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . 3 (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ))
2 limsupvaluz.f . . . . 5 (𝜑𝐹:𝑍⟶ℝ*)
3 limsupvaluz.z . . . . . . 7 𝑍 = (ℤ𝑀)
4 fvex 6239 . . . . . . 7 (ℤ𝑀) ∈ V
53, 4eqeltri 2726 . . . . . 6 𝑍 ∈ V
65a1i 11 . . . . 5 (𝜑𝑍 ∈ V)
72, 6fexd 39610 . . . 4 (𝜑𝐹 ∈ V)
87elexd 3245 . . 3 (𝜑𝐹 ∈ V)
9 uzssre 39933 . . . . 5 (ℤ𝑀) ⊆ ℝ
103, 9eqsstri 3668 . . . 4 𝑍 ⊆ ℝ
1110a1i 11 . . 3 (𝜑𝑍 ⊆ ℝ)
12 limsupvaluz.m . . . 4 (𝜑𝑀 ∈ ℤ)
133uzsup 12702 . . . 4 (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)
1412, 13syl 17 . . 3 (𝜑 → sup(𝑍, ℝ*, < ) = +∞)
151, 8, 11, 14limsupval2 14255 . 2 (𝜑 → (lim sup‘𝐹) = inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ))
1611mptima2 39771 . . . 4 (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )))
17 oveq1 6697 . . . . . . . . . . 11 (𝑖 = 𝑛 → (𝑖[,)+∞) = (𝑛[,)+∞))
1817imaeq2d 5501 . . . . . . . . . 10 (𝑖 = 𝑛 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑛[,)+∞)))
1918ineq1d 3846 . . . . . . . . 9 (𝑖 = 𝑛 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*))
2019supeq1d 8393 . . . . . . . 8 (𝑖 = 𝑛 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
2120cbvmptv 4783 . . . . . . 7 (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
2221a1i 11 . . . . . 6 (𝜑 → (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )))
23 fimass 6119 . . . . . . . . . . . 12 (𝐹:𝑍⟶ℝ* → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
242, 23syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
25 df-ss 3621 . . . . . . . . . . . 12 ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
2625biimpi 206 . . . . . . . . . . 11 ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ* → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
2724, 26syl 17 . . . . . . . . . 10 (𝜑 → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
2827adantr 480 . . . . . . . . 9 ((𝜑𝑛𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
29 df-ima 5156 . . . . . . . . . 10 (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞))
3029a1i 11 . . . . . . . . 9 ((𝜑𝑛𝑍) → (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞)))
312freld 39739 . . . . . . . . . . . . 13 (𝜑 → Rel 𝐹)
32 resindm 5479 . . . . . . . . . . . . 13 (Rel 𝐹 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
3331, 32syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
3433adantr 480 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
35 incom 3838 . . . . . . . . . . . . . . 15 ((𝑛[,)+∞) ∩ 𝑍) = (𝑍 ∩ (𝑛[,)+∞))
363ineq1i 3843 . . . . . . . . . . . . . . 15 (𝑍 ∩ (𝑛[,)+∞)) = ((ℤ𝑀) ∩ (𝑛[,)+∞))
3735, 36eqtri 2673 . . . . . . . . . . . . . 14 ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ𝑀) ∩ (𝑛[,)+∞))
3837a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ𝑀) ∩ (𝑛[,)+∞)))
392fdmd 39734 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝐹 = 𝑍)
4039ineq2d 3847 . . . . . . . . . . . . . 14 (𝜑 → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
4140adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
423eleq2i 2722 . . . . . . . . . . . . . . . 16 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
4342biimpi 206 . . . . . . . . . . . . . . 15 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
4443adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → 𝑛 ∈ (ℤ𝑀))
4544uzinico2 40107 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → (ℤ𝑛) = ((ℤ𝑀) ∩ (𝑛[,)+∞)))
4638, 41, 453eqtr4d 2695 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = (ℤ𝑛))
4746reseq2d 5428 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (ℤ𝑛)))
4834, 47eqtr3d 2687 . . . . . . . . . 10 ((𝜑𝑛𝑍) → (𝐹 ↾ (𝑛[,)+∞)) = (𝐹 ↾ (ℤ𝑛)))
4948rneqd 5385 . . . . . . . . 9 ((𝜑𝑛𝑍) → ran (𝐹 ↾ (𝑛[,)+∞)) = ran (𝐹 ↾ (ℤ𝑛)))
5028, 30, 493eqtrd 2689 . . . . . . . 8 ((𝜑𝑛𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = ran (𝐹 ↾ (ℤ𝑛)))
5150supeq1d 8393 . . . . . . 7 ((𝜑𝑛𝑍) → sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
5251mpteq2dva 4777 . . . . . 6 (𝜑 → (𝑛𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5322, 52eqtrd 2685 . . . . 5 (𝜑 → (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5453rneqd 5385 . . . 4 (𝜑 → ran (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5516, 54eqtrd 2685 . . 3 (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5655infeq1d 8424 . 2 (𝜑 → inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ))
57 fveq2 6229 . . . . . . . . 9 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
5857reseq2d 5428 . . . . . . . 8 (𝑛 = 𝑘 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑘)))
5958rneqd 5385 . . . . . . 7 (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑘)))
6059supeq1d 8393 . . . . . 6 (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
6160cbvmptv 4783 . . . . 5 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
6261rneqi 5384 . . . 4 ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
6362infeq1i 8425 . . 3 inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < )
6463a1i 11 . 2 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ))
6515, 56, 643eqtrd 2689 1 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  wss 3607  cmpt 4762  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  supcsup 8387  infcinf 8388  cr 9973  +∞cpnf 10109  *cxr 10111   < clt 10112  cz 11415  cuz 11725  [,)cico 12215  lim supclsp 14245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-ico 12219  df-fl 12633  df-limsup 14246
This theorem is referenced by:  limsupvaluzmpt  40267  limsupvaluz2  40288  limsupgtlem  40327
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