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Theorem limsupvaluz 43594
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 2736 . . 3 (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ))
2 limsupvaluz.f . . . 4 (𝜑𝐹:𝑍⟶ℝ*)
3 limsupvaluz.z . . . . . 6 𝑍 = (ℤ𝑀)
43fvexi 6839 . . . . 5 𝑍 ∈ V
54a1i 11 . . . 4 (𝜑𝑍 ∈ V)
62, 5fexd 7159 . . 3 (𝜑𝐹 ∈ V)
7 uzssre 12705 . . . . 5 (ℤ𝑀) ⊆ ℝ
83, 7eqsstri 3966 . . . 4 𝑍 ⊆ ℝ
98a1i 11 . . 3 (𝜑𝑍 ⊆ ℝ)
10 limsupvaluz.m . . . 4 (𝜑𝑀 ∈ ℤ)
113uzsup 13684 . . . 4 (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)
1210, 11syl 17 . . 3 (𝜑 → sup(𝑍, ℝ*, < ) = +∞)
131, 6, 9, 12limsupval2 15288 . 2 (𝜑 → (lim sup‘𝐹) = inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ))
149mptima2 43128 . . . 4 (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )))
15 oveq1 7344 . . . . . . . . . . 11 (𝑖 = 𝑛 → (𝑖[,)+∞) = (𝑛[,)+∞))
1615imaeq2d 5999 . . . . . . . . . 10 (𝑖 = 𝑛 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑛[,)+∞)))
1716ineq1d 4158 . . . . . . . . 9 (𝑖 = 𝑛 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*))
1817supeq1d 9303 . . . . . . . 8 (𝑖 = 𝑛 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
1918cbvmptv 5205 . . . . . . 7 (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
2019a1i 11 . . . . . 6 (𝜑 → (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )))
21 fimass 6672 . . . . . . . . . . . 12 (𝐹:𝑍⟶ℝ* → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
222, 21syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
23 df-ss 3915 . . . . . . . . . . . 12 ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
2423biimpi 215 . . . . . . . . . . 11 ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ* → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
2522, 24syl 17 . . . . . . . . . 10 (𝜑 → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
2625adantr 481 . . . . . . . . 9 ((𝜑𝑛𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
27 df-ima 5633 . . . . . . . . . 10 (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞))
2827a1i 11 . . . . . . . . 9 ((𝜑𝑛𝑍) → (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞)))
292freld 6657 . . . . . . . . . . . . 13 (𝜑 → Rel 𝐹)
30 resindm 5972 . . . . . . . . . . . . 13 (Rel 𝐹 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
3129, 30syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
3231adantr 481 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
33 incom 4148 . . . . . . . . . . . . . . 15 ((𝑛[,)+∞) ∩ 𝑍) = (𝑍 ∩ (𝑛[,)+∞))
343ineq1i 4155 . . . . . . . . . . . . . . 15 (𝑍 ∩ (𝑛[,)+∞)) = ((ℤ𝑀) ∩ (𝑛[,)+∞))
3533, 34eqtri 2764 . . . . . . . . . . . . . 14 ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ𝑀) ∩ (𝑛[,)+∞))
3635a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ𝑀) ∩ (𝑛[,)+∞)))
372fdmd 6662 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝐹 = 𝑍)
3837ineq2d 4159 . . . . . . . . . . . . . 14 (𝜑 → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
3938adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
403eleq2i 2828 . . . . . . . . . . . . . . . 16 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
4140biimpi 215 . . . . . . . . . . . . . . 15 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
4241adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → 𝑛 ∈ (ℤ𝑀))
4342uzinico2 43445 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → (ℤ𝑛) = ((ℤ𝑀) ∩ (𝑛[,)+∞)))
4436, 39, 433eqtr4d 2786 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = (ℤ𝑛))
4544reseq2d 5923 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (ℤ𝑛)))
4632, 45eqtr3d 2778 . . . . . . . . . 10 ((𝜑𝑛𝑍) → (𝐹 ↾ (𝑛[,)+∞)) = (𝐹 ↾ (ℤ𝑛)))
4746rneqd 5879 . . . . . . . . 9 ((𝜑𝑛𝑍) → ran (𝐹 ↾ (𝑛[,)+∞)) = ran (𝐹 ↾ (ℤ𝑛)))
4826, 28, 473eqtrd 2780 . . . . . . . 8 ((𝜑𝑛𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = ran (𝐹 ↾ (ℤ𝑛)))
4948supeq1d 9303 . . . . . . 7 ((𝜑𝑛𝑍) → sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
5049mpteq2dva 5192 . . . . . 6 (𝜑 → (𝑛𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5120, 50eqtrd 2776 . . . . 5 (𝜑 → (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5251rneqd 5879 . . . 4 (𝜑 → ran (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5314, 52eqtrd 2776 . . 3 (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5453infeq1d 9334 . 2 (𝜑 → inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ))
55 fveq2 6825 . . . . . . . . 9 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
5655reseq2d 5923 . . . . . . . 8 (𝑛 = 𝑘 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑘)))
5756rneqd 5879 . . . . . . 7 (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑘)))
5857supeq1d 9303 . . . . . 6 (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
5958cbvmptv 5205 . . . . 5 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
6059rneqi 5878 . . . 4 ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
6160infeq1i 9335 . . 3 inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < )
6261a1i 11 . 2 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ))
6313, 54, 623eqtrd 2780 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 396   = wceq 1540  wcel 2105  Vcvv 3441  cin 3897  wss 3898  cmpt 5175  dom cdm 5620  ran crn 5621  cres 5622  cima 5623  Rel wrel 5625  wf 6475  cfv 6479  (class class class)co 7337  supcsup 9297  infcinf 9298  cr 10971  +∞cpnf 11107  *cxr 11109   < clt 11110  cz 12420  cuz 12683  [,)cico 13182  lim supclsp 15278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-sup 9299  df-inf 9300  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-n0 12335  df-z 12421  df-uz 12684  df-ico 13186  df-fl 13613  df-limsup 15279
This theorem is referenced by:  limsupvaluzmpt  43603  limsupvaluz2  43624  limsupgtlem  43663
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