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Theorem ovnval2b 46110
Description: Value of the Lebesgue outer measure of a subset 𝐴 of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
ovnval2b.1 (𝜑𝑋 ∈ Fin)
ovnval2b.2 (𝜑𝐴 ⊆ (ℝ ↑m 𝑋))
ovnval2b.3 𝐿 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
Assertion
Ref Expression
ovnval2b (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf((𝐿𝐴), ℝ*, < )))
Distinct variable groups:   𝐴,𝑎,𝑖,𝑧   𝑋,𝑎,𝑖,𝑗,𝑘,𝑧   𝜑,𝑎
Allowed substitution hints:   𝜑(𝑧,𝑖,𝑗,𝑘)   𝐴(𝑗,𝑘)   𝐿(𝑧,𝑖,𝑗,𝑘,𝑎)

Proof of Theorem ovnval2b
StepHypRef Expression
1 ovnval2b.1 . . 3 (𝜑𝑋 ∈ Fin)
2 ovnval2b.2 . . 3 (𝜑𝐴 ⊆ (ℝ ↑m 𝑋))
3 eqid 2725 . . 3 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}
41, 2, 3ovnval2 46103 . 2 (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < )))
5 biidd 261 . . 3 (𝜑 → (𝑋 = ∅ ↔ 𝑋 = ∅))
6 ovnval2b.3 . . . . . 6 𝐿 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
7 cleq1lem 14982 . . . . . . . 8 (𝑎 = 𝐴 → ((𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
87rexbidv 3168 . . . . . . 7 (𝑎 = 𝐴 → (∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
98rabbidv 3426 . . . . . 6 (𝑎 = 𝐴 → {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
10 ovexd 7458 . . . . . . . 8 (𝜑 → (ℝ ↑m 𝑋) ∈ V)
1110, 2ssexd 5328 . . . . . . 7 (𝜑𝐴 ∈ V)
1211, 2elpwd 4612 . . . . . 6 (𝜑𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
13 xrex 13018 . . . . . . . 8 * ∈ V
1413rabex 5338 . . . . . . 7 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ∈ V
1514a1i 11 . . . . . 6 (𝜑 → {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ∈ V)
166, 9, 12, 15fvmptd3 7031 . . . . 5 (𝜑 → (𝐿𝐴) = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
1716eqcomd 2731 . . . 4 (𝜑 → {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} = (𝐿𝐴))
1817infeq1d 9516 . . 3 (𝜑 → inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ) = inf((𝐿𝐴), ℝ*, < ))
195, 18ifbieq2d 4558 . 2 (𝜑 → if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < )) = if(𝑋 = ∅, 0, inf((𝐿𝐴), ℝ*, < )))
204, 19eqtrd 2765 1 (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf((𝐿𝐴), ℝ*, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wrex 3059  {crab 3418  Vcvv 3461  wss 3946  c0 4324  ifcif 4532  𝒫 cpw 4606   ciun 5000  cmpt 5235   × cxp 5679  ccom 5685  cfv 6553  (class class class)co 7423  m cmap 8854  Xcixp 8925  Fincfn 8973  infcinf 9480  cr 11153  0cc0 11154  *cxr 11293   < clt 11294  cn 12259  [,)cico 13375  cprod 15902  volcvol 25475  Σ^csumge0 45920  voln*covoln 46094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-cnex 11210  ax-resscn 11211  ax-1cn 11212  ax-icn 11213  ax-addcl 11214  ax-mulcl 11216  ax-i2m1 11222  ax-pre-lttri 11228  ax-pre-lttrn 11229
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-po 5593  df-so 5594  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7426  df-oprab 7427  df-mpo 7428  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-er 8733  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9481  df-inf 9482  df-pnf 11296  df-mnf 11297  df-xr 11298  df-ltxr 11299  df-seq 14017  df-prod 15903  df-ovoln 46095
This theorem is referenced by:  ovnsubaddlem1  46128
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