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Theorem limsupvaluzmpt 40693
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 40190 . . 3 (𝜑 → (𝑗𝑍𝐵):𝑍⟶ℝ*)
61, 2, 5limsupvaluz 40684 . 2 (𝜑 → (lim sup‘(𝑗𝑍𝐵)) = inf(ran (𝑘𝑍 ↦ sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ))
72uzssd3 40396 . . . . . . . . 9 (𝑘𝑍 → (ℤ𝑘) ⊆ 𝑍)
87resmptd 5664 . . . . . . . 8 (𝑘𝑍 → ((𝑗𝑍𝐵) ↾ (ℤ𝑘)) = (𝑗 ∈ (ℤ𝑘) ↦ 𝐵))
98rneqd 5556 . . . . . . 7 (𝑘𝑍 → ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)) = ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵))
109supeq1d 8594 . . . . . 6 (𝑘𝑍 → sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < ))
1110mpteq2ia 4933 . . . . 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 5556 . . 3 (𝜑 → ran (𝑘𝑍 ↦ sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < )) = ran (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < )))
1413infeq1d 8625 . 2 (𝜑 → inf(ran (𝑘𝑍 ↦ sup(ran ((𝑗𝑍𝐵) ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < )), ℝ*, < ))
156, 14eqtrd 2833 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 385   = wceq 1653  wnf 1879  wcel 2157  cmpt 4922  ran crn 5313  cres 5314  cfv 6101  supcsup 8588  infcinf 8589  *cxr 10362   < clt 10363  cz 11666  cuz 11930  lim supclsp 14542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-sup 8590  df-inf 8591  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-n0 11581  df-z 11667  df-uz 11931  df-ico 12430  df-fl 12848  df-limsup 14543
This theorem is referenced by:  smflimsuplem4  41775
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