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Theorem limsupvaluz 42738
 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 2758 . . 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)
43fvexi 6672 . . . . . 6 𝑍 ∈ V
54a1i 11 . . . . 5 (𝜑𝑍 ∈ V)
62, 5fexd 6981 . . . 4 (𝜑𝐹 ∈ V)
76elexd 3430 . . 3 (𝜑𝐹 ∈ V)
8 uzssre 12304 . . . . 5 (ℤ𝑀) ⊆ ℝ
93, 8eqsstri 3926 . . . 4 𝑍 ⊆ ℝ
109a1i 11 . . 3 (𝜑𝑍 ⊆ ℝ)
11 limsupvaluz.m . . . 4 (𝜑𝑀 ∈ ℤ)
123uzsup 13280 . . . 4 (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)
1311, 12syl 17 . . 3 (𝜑 → sup(𝑍, ℝ*, < ) = +∞)
141, 7, 10, 13limsupval2 14885 . 2 (𝜑 → (lim sup‘𝐹) = inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ))
1510mptima2 42273 . . . 4 (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )))
16 oveq1 7157 . . . . . . . . . . 11 (𝑖 = 𝑛 → (𝑖[,)+∞) = (𝑛[,)+∞))
1716imaeq2d 5901 . . . . . . . . . 10 (𝑖 = 𝑛 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑛[,)+∞)))
1817ineq1d 4116 . . . . . . . . 9 (𝑖 = 𝑛 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*))
1918supeq1d 8943 . . . . . . . 8 (𝑖 = 𝑛 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
2019cbvmptv 5135 . . . . . . 7 (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
2120a1i 11 . . . . . 6 (𝜑 → (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )))
22 fimass 6540 . . . . . . . . . . . 12 (𝐹:𝑍⟶ℝ* → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
232, 22syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
24 df-ss 3875 . . . . . . . . . . . 12 ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
2524biimpi 219 . . . . . . . . . . 11 ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ* → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
2623, 25syl 17 . . . . . . . . . 10 (𝜑 → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
2726adantr 484 . . . . . . . . 9 ((𝜑𝑛𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
28 df-ima 5537 . . . . . . . . . 10 (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞))
2928a1i 11 . . . . . . . . 9 ((𝜑𝑛𝑍) → (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞)))
302freld 42249 . . . . . . . . . . . . 13 (𝜑 → Rel 𝐹)
31 resindm 5872 . . . . . . . . . . . . 13 (Rel 𝐹 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
3230, 31syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
3332adantr 484 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
34 incom 4106 . . . . . . . . . . . . . . 15 ((𝑛[,)+∞) ∩ 𝑍) = (𝑍 ∩ (𝑛[,)+∞))
353ineq1i 4113 . . . . . . . . . . . . . . 15 (𝑍 ∩ (𝑛[,)+∞)) = ((ℤ𝑀) ∩ (𝑛[,)+∞))
3634, 35eqtri 2781 . . . . . . . . . . . . . 14 ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ𝑀) ∩ (𝑛[,)+∞))
3736a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ𝑀) ∩ (𝑛[,)+∞)))
382fdmd 6508 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝐹 = 𝑍)
3938ineq2d 4117 . . . . . . . . . . . . . 14 (𝜑 → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
4039adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
413eleq2i 2843 . . . . . . . . . . . . . . . 16 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
4241biimpi 219 . . . . . . . . . . . . . . 15 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
4342adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → 𝑛 ∈ (ℤ𝑀))
4443uzinico2 42587 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → (ℤ𝑛) = ((ℤ𝑀) ∩ (𝑛[,)+∞)))
4537, 40, 443eqtr4d 2803 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = (ℤ𝑛))
4645reseq2d 5823 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (ℤ𝑛)))
4733, 46eqtr3d 2795 . . . . . . . . . 10 ((𝜑𝑛𝑍) → (𝐹 ↾ (𝑛[,)+∞)) = (𝐹 ↾ (ℤ𝑛)))
4847rneqd 5779 . . . . . . . . 9 ((𝜑𝑛𝑍) → ran (𝐹 ↾ (𝑛[,)+∞)) = ran (𝐹 ↾ (ℤ𝑛)))
4927, 29, 483eqtrd 2797 . . . . . . . 8 ((𝜑𝑛𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = ran (𝐹 ↾ (ℤ𝑛)))
5049supeq1d 8943 . . . . . . 7 ((𝜑𝑛𝑍) → sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ))
5150mpteq2dva 5127 . . . . . 6 (𝜑 → (𝑛𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5221, 51eqtrd 2793 . . . . 5 (𝜑 → (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5352rneqd 5779 . . . 4 (𝜑 → ran (𝑖𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5415, 53eqtrd 2793 . . 3 (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )))
5554infeq1d 8974 . 2 (𝜑 → inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ) = inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ))
56 fveq2 6658 . . . . . . . . 9 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
5756reseq2d 5823 . . . . . . . 8 (𝑛 = 𝑘 → (𝐹 ↾ (ℤ𝑛)) = (𝐹 ↾ (ℤ𝑘)))
5857rneqd 5779 . . . . . . 7 (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ𝑛)) = ran (𝐹 ↾ (ℤ𝑘)))
5958supeq1d 8943 . . . . . 6 (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
6059cbvmptv 5135 . . . . 5 (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
6160rneqi 5778 . . . 4 ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )) = ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < ))
6261infeq1i 8975 . . 3 inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < )
6362a1i 11 . 2 (𝜑 → inf(ran (𝑛𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ))
6414, 55, 633eqtrd 2797 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 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∩ cin 3857   ⊆ wss 3858   ↦ cmpt 5112  dom cdm 5524  ran crn 5525   ↾ cres 5526   “ cima 5527  Rel wrel 5529  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  supcsup 8937  infcinf 8938  ℝcr 10574  +∞cpnf 10710  ℝ*cxr 10712   < clt 10713  ℤcz 12020  ℤ≥cuz 12282  [,)cico 12781  lim supclsp 14875 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-sup 8939  df-inf 8940  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-n0 11935  df-z 12021  df-uz 12283  df-ico 12785  df-fl 13211  df-limsup 14876 This theorem is referenced by:  limsupvaluzmpt  42747  limsupvaluz2  42768  limsupgtlem  42807
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