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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  ℝ*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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