Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omsval Structured version   Visualization version   GIF version

Theorem omsval 34295
Description: Value of the function mapping a content function to the corresponding outer measure. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
Assertion
Ref Expression
omsval (𝑅 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )))
Distinct variable group:   𝑥,𝑎,𝑦,𝑧,𝑅

Proof of Theorem omsval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 df-oms 34294 . 2 toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)), (0[,]+∞), < )))
2 dmeq 5914 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
32unieqd 4920 . . . 4 (𝑟 = 𝑅 dom 𝑟 = dom 𝑅)
43pweqd 4617 . . 3 (𝑟 = 𝑅 → 𝒫 dom 𝑟 = 𝒫 dom 𝑅)
52pweqd 4617 . . . . . . 7 (𝑟 = 𝑅 → 𝒫 dom 𝑟 = 𝒫 dom 𝑅)
6 rabeq 3451 . . . . . . 7 (𝒫 dom 𝑟 = 𝒫 dom 𝑅 → {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)})
75, 6syl 17 . . . . . 6 (𝑟 = 𝑅 → {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)})
8 simpl 482 . . . . . . . 8 ((𝑟 = 𝑅𝑦𝑥) → 𝑟 = 𝑅)
98fveq1d 6908 . . . . . . 7 ((𝑟 = 𝑅𝑦𝑥) → (𝑟𝑦) = (𝑅𝑦))
109esumeq2dv 34039 . . . . . 6 (𝑟 = 𝑅 → Σ*𝑦𝑥(𝑟𝑦) = Σ*𝑦𝑥(𝑅𝑦))
117, 10mpteq12dv 5233 . . . . 5 (𝑟 = 𝑅 → (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)) = (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)))
1211rneqd 5949 . . . 4 (𝑟 = 𝑅 → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)) = ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)))
1312infeq1d 9517 . . 3 (𝑟 = 𝑅 → inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)), (0[,]+∞), < ) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < ))
144, 13mpteq12dv 5233 . 2 (𝑟 = 𝑅 → (𝑎 ∈ 𝒫 dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)), (0[,]+∞), < )) = (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )))
15 id 22 . 2 (𝑅 ∈ V → 𝑅 ∈ V)
16 dmexg 7923 . . 3 (𝑅 ∈ V → dom 𝑅 ∈ V)
17 uniexg 7760 . . 3 (dom 𝑅 ∈ V → dom 𝑅 ∈ V)
18 pwexg 5378 . . 3 ( dom 𝑅 ∈ V → 𝒫 dom 𝑅 ∈ V)
19 mptexg 7241 . . 3 (𝒫 dom 𝑅 ∈ V → (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )) ∈ V)
2016, 17, 18, 194syl 19 . 2 (𝑅 ∈ V → (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )) ∈ V)
211, 14, 15, 20fvmptd3 7039 1 (𝑅 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686  cfv 6561  (class class class)co 7431  ωcom 7887  cdom 8983  infcinf 9481  0cc0 11155  +∞cpnf 11292   < clt 11295  [,]cicc 13390  Σ*cesum 34028  toOMeascoms 34293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-sup 9482  df-inf 9483  df-esum 34029  df-oms 34294
This theorem is referenced by:  omsfval  34296  omsf  34298
  Copyright terms: Public domain W3C validator