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Theorem ovnval 40049
Description: Value of the Lebesgue outer measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypothesis
Ref Expression
ovnval.1 (𝜑𝑋 ∈ Fin)
Assertion
Ref Expression
ovnval (𝜑 → (voln*‘𝑋) = (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))))
Distinct variable group:   𝑖,𝑋,𝑗,𝑘,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑖,𝑗,𝑘)

Proof of Theorem ovnval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-ovoln 40045 . . 3 voln* = (𝑥 ∈ Fin ↦ (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))))
21a1i 11 . 2 (𝜑 → voln* = (𝑥 ∈ Fin ↦ (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < )))))
3 oveq2 6613 . . . . 5 (𝑥 = 𝑋 → (ℝ ↑𝑚 𝑥) = (ℝ ↑𝑚 𝑋))
43pweqd 4140 . . . 4 (𝑥 = 𝑋 → 𝒫 (ℝ ↑𝑚 𝑥) = 𝒫 (ℝ ↑𝑚 𝑋))
5 eqeq1 2630 . . . . 5 (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
6 oveq2 6613 . . . . . . . . 9 (𝑥 = 𝑋 → ((ℝ × ℝ) ↑𝑚 𝑥) = ((ℝ × ℝ) ↑𝑚 𝑋))
76oveq1d 6620 . . . . . . . 8 (𝑥 = 𝑋 → (((ℝ × ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ) = (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
8 ixpeq1 7864 . . . . . . . . . . 11 (𝑥 = 𝑋X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
98iuneq2d 4518 . . . . . . . . . 10 (𝑥 = 𝑋 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
109sseq2d 3617 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
11 simpl 473 . . . . . . . . . . . . 13 ((𝑥 = 𝑋𝑗 ∈ ℕ) → 𝑥 = 𝑋)
1211prodeq1d 14571 . . . . . . . . . . . 12 ((𝑥 = 𝑋𝑗 ∈ ℕ) → ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
1312mpteq2dva 4709 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
1413fveq2d 6154 . . . . . . . . . 10 (𝑥 = 𝑋 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
1514eqeq2d 2636 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
1610, 15anbi12d 746 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ (𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
177, 16rexeqbidv 3147 . . . . . . 7 (𝑥 = 𝑋 → (∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
1817rabbidv 3182 . . . . . 6 (𝑥 = 𝑋 → {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
1918infeq1d 8328 . . . . 5 (𝑥 = 𝑋 → inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ) = inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))
205, 19ifbieq2d 4088 . . . 4 (𝑥 = 𝑋 → if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < )) = if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < )))
214, 20mpteq12dv 4698 . . 3 (𝑥 = 𝑋 → (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))) = (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))))
2221adantl 482 . 2 ((𝜑𝑥 = 𝑋) → (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))) = (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))))
23 ovnval.1 . 2 (𝜑𝑋 ∈ Fin)
24 ovex 6633 . . . . 5 (ℝ ↑𝑚 𝑋) ∈ V
2524pwex 4813 . . . 4 𝒫 (ℝ ↑𝑚 𝑋) ∈ V
2625mptex 6441 . . 3 (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))) ∈ V
2726a1i 11 . 2 (𝜑 → (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))) ∈ V)
282, 22, 23, 27fvmptd 6246 1 (𝜑 → (voln*‘𝑋) = (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wrex 2913  {crab 2916  Vcvv 3191  wss 3560  c0 3896  ifcif 4063  𝒫 cpw 4135   ciun 4490  cmpt 4678   × cxp 5077  ccom 5083  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  Xcixp 7853  Fincfn 7900  infcinf 8292  cr 9880  0cc0 9881  *cxr 10018   < clt 10019  cn 10965  [,)cico 12116  cprod 14555  volcvol 23134  Σ^csumge0 39873  voln*covoln 40044
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-ixp 7854  df-sup 8293  df-inf 8294  df-seq 12739  df-prod 14556  df-ovoln 40045
This theorem is referenced by:  ovnval2  40053  ovnf  40071
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