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Theorem omsfval 33591
Description: Value of the outer measure evaluated for a given set 𝐴. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
Assertion
Ref Expression
omsfval ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ ((toOMeasβ€˜π‘…)β€˜π΄) = inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < ))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝑅   π‘₯,𝐴,𝑦,𝑧   π‘₯,𝑄,𝑦,𝑧   π‘₯,𝑉,𝑦,𝑧
Proof of Theorem omsfval
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 simp2 1135 . . . 4 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ 𝑅:π‘„βŸΆ(0[,]+∞))
2 simp1 1134 . . . 4 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ 𝑄 ∈ 𝑉)
31, 2fexd 7230 . . 3 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ 𝑅 ∈ V)
4 omsval 33590 . . 3 (𝑅 ∈ V β†’ (toOMeasβ€˜π‘…) = (π‘Ž ∈ 𝒫 βˆͺ dom 𝑅 ↦ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < )))
53, 4syl 17 . 2 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ (toOMeasβ€˜π‘…) = (π‘Ž ∈ 𝒫 βˆͺ dom 𝑅 ↦ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < )))
6 simpr 483 . . . . . . . 8 (((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) ∧ π‘Ž = 𝐴) β†’ π‘Ž = 𝐴)
76sseq1d 4012 . . . . . . 7 (((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) ∧ π‘Ž = 𝐴) β†’ (π‘Ž βŠ† βˆͺ 𝑧 ↔ 𝐴 βŠ† βˆͺ 𝑧))
87anbi1d 628 . . . . . 6 (((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) ∧ π‘Ž = 𝐴) β†’ ((π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰) ↔ (𝐴 βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)))
98rabbidv 3438 . . . . 5 (((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) ∧ π‘Ž = 𝐴) β†’ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)})
109mpteq1d 5242 . . . 4 (((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) ∧ π‘Ž = 𝐴) β†’ (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)) = (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)))
1110rneqd 5936 . . 3 (((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) ∧ π‘Ž = 𝐴) β†’ ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)) = ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)))
1211infeq1d 9474 . 2 (((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) ∧ π‘Ž = 𝐴) β†’ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < ) = inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < ))
13 simp3 1136 . . . 4 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ 𝐴 βŠ† βˆͺ 𝑄)
14 fdm 6725 . . . . . 6 (𝑅:π‘„βŸΆ(0[,]+∞) β†’ dom 𝑅 = 𝑄)
15143ad2ant2 1132 . . . . 5 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ dom 𝑅 = 𝑄)
1615unieqd 4921 . . . 4 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ βˆͺ dom 𝑅 = βˆͺ 𝑄)
1713, 16sseqtrrd 4022 . . 3 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ 𝐴 βŠ† βˆͺ dom 𝑅)
182uniexd 7734 . . . . 5 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ βˆͺ 𝑄 ∈ V)
19 ssexg 5322 . . . . 5 ((𝐴 βŠ† βˆͺ 𝑄 ∧ βˆͺ 𝑄 ∈ V) β†’ 𝐴 ∈ V)
2013, 18, 19syl2anc 582 . . . 4 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ 𝐴 ∈ V)
21 elpwg 4604 . . . 4 (𝐴 ∈ V β†’ (𝐴 ∈ 𝒫 βˆͺ dom 𝑅 ↔ 𝐴 βŠ† βˆͺ dom 𝑅))
2220, 21syl 17 . . 3 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ (𝐴 ∈ 𝒫 βˆͺ dom 𝑅 ↔ 𝐴 βŠ† βˆͺ dom 𝑅))
2317, 22mpbird 256 . 2 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ 𝐴 ∈ 𝒫 βˆͺ dom 𝑅)
24 xrltso 13124 . . . 4 < Or ℝ*
25 iccssxr 13411 . . . . 5 (0[,]+∞) βŠ† ℝ*
26 soss 5607 . . . . 5 ((0[,]+∞) βŠ† ℝ* β†’ ( < Or ℝ* β†’ < Or (0[,]+∞)))
2725, 26ax-mp 5 . . . 4 ( < Or ℝ* β†’ < Or (0[,]+∞))
2824, 27mp1i 13 . . 3 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ < Or (0[,]+∞))
2928infexd 9480 . 2 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < ) ∈ V)
305, 12, 23, 29fvmptd 7004 1 ((𝑄 ∈ 𝑉 ∧ 𝑅:π‘„βŸΆ(0[,]+∞) ∧ 𝐴 βŠ† βˆͺ 𝑄) β†’ ((toOMeasβ€˜π‘…)β€˜π΄) = inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  {crab 3430  Vcvv 3472   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907   class class class wbr 5147   ↦ cmpt 5230   Or wor 5586  dom cdm 5675  ran crn 5676  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  Ο‰com 7857   β‰Ό cdom 8939  infcinf 9438  0cc0 11112  +∞cpnf 11249  β„*cxr 11251   < clt 11252  [,]cicc 13331  Ξ£*cesum 33323  toOMeascoms 33588
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-pre-lttri 11186  ax-pre-lttrn 11187
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-icc 13335  df-esum 33324  df-oms 33589
This theorem is referenced by:  omsf  33593  oms0  33594  omsmon  33595  omssubaddlem  33596  omssubadd  33597
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