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

Theorem limsupvaluzmpt 40470
 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 39959 . . 3 (𝜑 → (𝑗𝑍𝐵):𝑍⟶ℝ*)
61, 2, 5limsupvaluz 40461 . 2 (𝜑 → (lim sup‘(𝑗𝑍𝐵)) = inf(ran (𝑘𝑍 ↦ sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ))
72uzssd3 40169 . . . . . . . . 9 (𝑘𝑍 → (ℤ𝑘) ⊆ 𝑍)
87resmptd 5610 . . . . . . . 8 (𝑘𝑍 → ((𝑗𝑍𝐵) ↾ (ℤ𝑘)) = (𝑗 ∈ (ℤ𝑘) ↦ 𝐵))
98rneqd 5508 . . . . . . 7 (𝑘𝑍 → ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)) = ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵))
109supeq1d 8519 . . . . . 6 (𝑘𝑍 → sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < ))
1110mpteq2ia 4892 . . . . 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 5508 . . 3 (𝜑 → ran (𝑘𝑍 ↦ sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < )) = ran (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < )))
1413infeq1d 8550 . 2 (𝜑 → inf(ran (𝑘𝑍 ↦ sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < )), ℝ*, < ))
156, 14eqtrd 2794 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 383   = wceq 1632  Ⅎwnf 1857   ∈ wcel 2139   ↦ cmpt 4881  ran crn 5267   ↾ cres 5268  ‘cfv 6049  supcsup 8513  infcinf 8514  ℝ*cxr 10285   < clt 10286  ℤcz 11589  ℤ≥cuz 11899  lim supclsp 14420 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-sup 8515  df-inf 8516  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-z 11590  df-uz 11900  df-ico 12394  df-fl 12807  df-limsup 14421 This theorem is referenced by:  smflimsuplem4  41553
 Copyright terms: Public domain W3C validator