Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  limsupvaluzmpt Structured version   Visualization version   GIF version

Theorem limsupvaluzmpt 41990
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
limsupvaluzmpt.j 𝑗𝜑
limsupvaluzmpt.m (𝜑𝑀 ∈ ℤ)
limsupvaluzmpt.z 𝑍 = (ℤ𝑀)
limsupvaluzmpt.b ((𝜑𝑗𝑍) → 𝐵 ∈ ℝ*)
Assertion
Ref Expression
limsupvaluzmpt (𝜑 → (lim sup‘(𝑗𝑍𝐵)) = inf(ran (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < )), ℝ*, < ))
Distinct variable groups:   𝐵,𝑘   𝑗,𝑍,𝑘
Allowed substitution hints:   𝜑(𝑗,𝑘)   𝐵(𝑗)   𝑀(𝑗,𝑘)

Proof of Theorem limsupvaluzmpt
StepHypRef Expression
1 limsupvaluzmpt.m . . 3 (𝜑𝑀 ∈ ℤ)
2 limsupvaluzmpt.z . . 3 𝑍 = (ℤ𝑀)
3 limsupvaluzmpt.j . . . 4 𝑗𝜑
4 limsupvaluzmpt.b . . . 4 ((𝜑𝑗𝑍) → 𝐵 ∈ ℝ*)
53, 4fmptd2f 41497 . . 3 (𝜑 → (𝑗𝑍𝐵):𝑍⟶ℝ*)
61, 2, 5limsupvaluz 41981 . 2 (𝜑 → (lim sup‘(𝑗𝑍𝐵)) = inf(ran (𝑘𝑍 ↦ sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ))
72uzssd3 41692 . . . . . . . . 9 (𝑘𝑍 → (ℤ𝑘) ⊆ 𝑍)
87resmptd 5903 . . . . . . . 8 (𝑘𝑍 → ((𝑗𝑍𝐵) ↾ (ℤ𝑘)) = (𝑗 ∈ (ℤ𝑘) ↦ 𝐵))
98rneqd 5803 . . . . . . 7 (𝑘𝑍 → ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)) = ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵))
109supeq1d 8904 . . . . . 6 (𝑘𝑍 → sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < ))
1110mpteq2ia 5150 . . . . 5 (𝑘𝑍 ↦ sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < ))
1211a1i 11 . . . 4 (𝜑 → (𝑘𝑍 ↦ sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < )) = (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < )))
1312rneqd 5803 . . 3 (𝜑 → ran (𝑘𝑍 ↦ sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < )) = ran (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < )))
1413infeq1d 8935 . 2 (𝜑 → inf(ran (𝑘𝑍 ↦ sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < )), ℝ*, < ))
156, 14eqtrd 2856 1 (𝜑 → (lim sup‘(𝑗𝑍𝐵)) = inf(ran (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < )), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wnf 1780  wcel 2110  cmpt 5139  ran crn 5551  cres 5552  cfv 6350  supcsup 8898  infcinf 8899  *cxr 10668   < clt 10669  cz 11975  cuz 12237  lim supclsp 14821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-sup 8900  df-inf 8901  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-ico 12738  df-fl 13156  df-limsup 14822
This theorem is referenced by:  smflimsuplem4  43090
  Copyright terms: Public domain W3C validator