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Theorem omsfval 30665
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 1132 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝑅:𝑄⟶(0[,]+∞))
2 simp1 1131 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝑄𝑉)
3 fex 6653 . . . 4 ((𝑅:𝑄⟶(0[,]+∞) ∧ 𝑄𝑉) → 𝑅 ∈ V)
41, 2, 3syl2anc 696 . . 3 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝑅 ∈ V)
5 omsval 30664 . . 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 479 . . . . . . . 8 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
87sseq1d 3773 . . . . . . 7 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → (𝑎 𝑧𝐴 𝑧))
98anbi1d 743 . . . . . 6 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → ((𝑎 𝑧𝑧 ≼ ω) ↔ (𝐴 𝑧𝑧 ≼ ω)))
109rabbidv 3329 . . . . 5 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)})
1110mpteq1d 4890 . . . 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 5508 . . 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 8548 . 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 1133 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝐴 𝑄)
15 fdm 6212 . . . . . 6 (𝑅:𝑄⟶(0[,]+∞) → dom 𝑅 = 𝑄)
16153ad2ant2 1129 . . . . 5 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → dom 𝑅 = 𝑄)
1716unieqd 4598 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → dom 𝑅 = 𝑄)
1814, 17sseqtr4d 3783 . . 3 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝐴 dom 𝑅)
19 elex 3352 . . . . . 6 (𝑄𝑉𝑄 ∈ V)
20 uniexb 7138 . . . . . . 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 4956 . . . . 5 ((𝐴 𝑄 𝑄 ∈ V) → 𝐴 ∈ V)
2414, 22, 23syl2anc 696 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝐴 ∈ V)
25 elpwg 4310 . . . 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 12167 . . . 4 < Or ℝ*
29 iccssxr 12449 . . . . 5 (0[,]+∞) ⊆ ℝ*
30 soss 5205 . . . . 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 8554 . 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 6450 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 383  w3a 1072   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  wss 3715  𝒫 cpw 4302   cuni 4588   class class class wbr 4804  cmpt 4881   Or wor 5186  dom cdm 5266  ran crn 5267  wf 6045  cfv 6049  (class class class)co 6813  ωcom 7230  cdom 8119  infcinf 8512  0cc0 10128  +∞cpnf 10263  *cxr 10265   < clt 10266  [,]cicc 12371  Σ*cesum 30398  toOMeascoms 30662
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 7114  ax-cnex 10184  ax-resscn 10185  ax-pre-lttri 10202  ax-pre-lttrn 10203
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-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  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-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-sup 8513  df-inf 8514  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-icc 12375  df-esum 30399  df-oms 30663
This theorem is referenced by:  omsf  30667  oms0  30668  omsmon  30669  omssubaddlem  30670  omssubadd  30671
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