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Theorem omsfval 30137
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 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → ((toOMeas‘𝑅)‘𝐴) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < ))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑥,𝑄,𝑦,𝑧   𝑥,𝑉,𝑦,𝑧

Proof of Theorem omsfval
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 simp2 1060 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝑅:𝑄⟶(0[,]+∞))
2 simp1 1059 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝑄𝑉)
3 fex 6444 . . . 4 ((𝑅:𝑄⟶(0[,]+∞) ∧ 𝑄𝑉) → 𝑅 ∈ V)
41, 2, 3syl2anc 692 . . 3 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝑅 ∈ V)
5 omsval 30136 . . 3 (𝑅 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )))
64, 5syl 17 . 2 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )))
7 simpr 477 . . . . . . . 8 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
87sseq1d 3611 . . . . . . 7 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → (𝑎 𝑧𝐴 𝑧))
98anbi1d 740 . . . . . 6 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → ((𝑎 𝑧𝑧 ≼ ω) ↔ (𝐴 𝑧𝑧 ≼ ω)))
109rabbidv 3177 . . . . 5 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)})
1110mpteq1d 4698 . . . 4 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)) = (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)))
1211rneqd 5313 . . 3 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → 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 8327 . 2 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → 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[,]+∞), < ))
14 simp3 1061 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝐴 𝑄)
15 fdm 6008 . . . . . 6 (𝑅:𝑄⟶(0[,]+∞) → dom 𝑅 = 𝑄)
16153ad2ant2 1081 . . . . 5 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → dom 𝑅 = 𝑄)
1716unieqd 4412 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → dom 𝑅 = 𝑄)
1814, 17sseqtr4d 3621 . . 3 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝐴 dom 𝑅)
19 elex 3198 . . . . . 6 (𝑄𝑉𝑄 ∈ V)
20 uniexb 6921 . . . . . . 7 (𝑄 ∈ V ↔ 𝑄 ∈ V)
2120biimpi 206 . . . . . 6 (𝑄 ∈ V → 𝑄 ∈ V)
222, 19, 213syl 18 . . . . 5 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝑄 ∈ V)
23 ssexg 4764 . . . . 5 ((𝐴 𝑄 𝑄 ∈ V) → 𝐴 ∈ V)
2414, 22, 23syl2anc 692 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝐴 ∈ V)
25 elpwg 4138 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 dom 𝑅𝐴 dom 𝑅))
2624, 25syl 17 . . 3 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → (𝐴 ∈ 𝒫 dom 𝑅𝐴 dom 𝑅))
2718, 26mpbird 247 . 2 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝐴 ∈ 𝒫 dom 𝑅)
28 xrltso 11918 . . . 4 < Or ℝ*
29 iccssxr 12198 . . . . 5 (0[,]+∞) ⊆ ℝ*
30 soss 5013 . . . . 5 ((0[,]+∞) ⊆ ℝ* → ( < Or ℝ* → < Or (0[,]+∞)))
3129, 30ax-mp 5 . . . 4 ( < Or ℝ* → < Or (0[,]+∞))
3228, 31mp1i 13 . . 3 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → < Or (0[,]+∞))
3332infexd 8333 . 2 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < ) ∈ V)
346, 13, 27, 33fvmptd 6245 1 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → ((toOMeas‘𝑅)‘𝐴) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  cmpt 4673   Or wor 4994  dom cdm 5074  ran crn 5075  wf 5843  cfv 5847  (class class class)co 6604  ωcom 7012  cdom 7897  infcinf 8291  0cc0 9880  +∞cpnf 10015  *cxr 10017   < clt 10018  [,]cicc 12120  Σ*cesum 29870  toOMeascoms 30134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-icc 12124  df-esum 29871  df-oms 30135
This theorem is referenced by:  omsf  30139  oms0  30140  omsmon  30141  omssubaddlem  30142  omssubadd  30143
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