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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  ℝcr 9973  +∞cpnf 10109  ℝ*cxr 10111   < clt 10112  ℤcz 11415  ℤ≥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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