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Theorem omsfval 30958
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 1128 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝑅:𝑄⟶(0[,]+∞))
2 simp1 1127 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝑄𝑉)
3 fex 6763 . . . 4 ((𝑅:𝑄⟶(0[,]+∞) ∧ 𝑄𝑉) → 𝑅 ∈ V)
41, 2, 3syl2anc 579 . . 3 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝑅 ∈ V)
5 omsval 30957 . . 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 3851 . . . . . . 7 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → (𝑎 𝑧𝐴 𝑧))
98anbi1d 623 . . . . . 6 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → ((𝑎 𝑧𝑧 ≼ ω) ↔ (𝐴 𝑧𝑧 ≼ ω)))
109rabbidv 3386 . . . . 5 (((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) ∧ 𝑎 = 𝐴) → {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)})
1110mpteq1d 4975 . . . 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 5600 . . 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 8673 . 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 1129 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝐴 𝑄)
15 fdm 6301 . . . . . 6 (𝑅:𝑄⟶(0[,]+∞) → dom 𝑅 = 𝑄)
16153ad2ant2 1125 . . . . 5 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → dom 𝑅 = 𝑄)
1716unieqd 4683 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → dom 𝑅 = 𝑄)
1814, 17sseqtr4d 3861 . . 3 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝐴 dom 𝑅)
19 elex 3414 . . . . . 6 (𝑄𝑉𝑄 ∈ V)
20 uniexb 7252 . . . . . . 7 (𝑄 ∈ V ↔ 𝑄 ∈ V)
2120biimpi 208 . . . . . 6 (𝑄 ∈ V → 𝑄 ∈ V)
222, 19, 213syl 18 . . . . 5 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝑄 ∈ V)
23 ssexg 5043 . . . . 5 ((𝐴 𝑄 𝑄 ∈ V) → 𝐴 ∈ V)
2414, 22, 23syl2anc 579 . . . 4 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝐴 ∈ V)
25 elpwg 4387 . . . 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 249 . 2 ((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → 𝐴 ∈ 𝒫 dom 𝑅)
28 xrltso 12288 . . . 4 < Or ℝ*
29 iccssxr 12572 . . . . 5 (0[,]+∞) ⊆ ℝ*
30 soss 5295 . . . . 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 8679 . 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 6550 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 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  {crab 3094  Vcvv 3398  wss 3792  𝒫 cpw 4379   cuni 4673   class class class wbr 4888  cmpt 4967   Or wor 5275  dom cdm 5357  ran crn 5358  wf 6133  cfv 6137  (class class class)co 6924  ωcom 7345  cdom 8241  infcinf 8637  0cc0 10274  +∞cpnf 10410  *cxr 10412   < clt 10413  [,]cicc 12494  Σ*cesum 30691  toOMeascoms 30955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-pre-lttri 10348  ax-pre-lttrn 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-po 5276  df-so 5277  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-sup 8638  df-inf 8639  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-icc 12498  df-esum 30692  df-oms 30956
This theorem is referenced by:  omsf  30960  oms0  30961  omsmon  30962  omssubaddlem  30963  omssubadd  30964
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