Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovnval Structured version   Visualization version   GIF version

Theorem ovnval 46497
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 46493 . 2 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*, < ))))
2 oveq2 7439 . . . 4 (𝑥 = 𝑋 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑋))
32pweqd 4622 . . 3 (𝑥 = 𝑋 → 𝒫 (ℝ ↑m 𝑥) = 𝒫 (ℝ ↑m 𝑋))
4 eqeq1 2739 . . . 4 (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
5 oveq2 7439 . . . . . . . 8 (𝑥 = 𝑋 → ((ℝ × ℝ) ↑m 𝑥) = ((ℝ × ℝ) ↑m 𝑋))
65oveq1d 7446 . . . . . . 7 (𝑥 = 𝑋 → (((ℝ × ℝ) ↑m 𝑥) ↑m ℕ) = (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
7 ixpeq1 8947 . . . . . . . . . 10 (𝑥 = 𝑋X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
87iuneq2d 5027 . . . . . . . . 9 (𝑥 = 𝑋 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
98sseq2d 4028 . . . . . . . 8 (𝑥 = 𝑋 → (𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
10 simpl 482 . . . . . . . . . . . 12 ((𝑥 = 𝑋𝑗 ∈ ℕ) → 𝑥 = 𝑋)
1110prodeq1d 15953 . . . . . . . . . . 11 ((𝑥 = 𝑋𝑗 ∈ ℕ) → ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
1211mpteq2dva 5248 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
1312fveq2d 6911 . . . . . . . . 9 (𝑥 = 𝑋 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
1413eqeq2d 2746 . . . . . . . 8 (𝑥 = 𝑋 → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
159, 14anbi12d 632 . . . . . . 7 (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)))))))
166, 15rexeqbidv 3345 . . . . . 6 (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)))))))
1716rabbidv 3441 . . . . 5 (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)))))})
1817infeq1d 9515 . . . 4 (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*, < ))
194, 18ifbieq2d 4557 . . 3 (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*, < )))
203, 19mpteq12dv 5239 . 2 (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*, < ))))
21 ovnval.1 . 2 (𝜑𝑋 ∈ Fin)
22 ovex 7464 . . . . 5 (ℝ ↑m 𝑋) ∈ V
2322pwex 5386 . . . 4 𝒫 (ℝ ↑m 𝑋) ∈ V
2423mptex 7243 . . 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
2524a1i 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)
261, 20, 21, 25fvmptd3 7039 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 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339  ifcif 4531  𝒫 cpw 4605   ciun 4996  cmpt 5231   × cxp 5687  ccom 5693  cfv 6563  (class class class)co 7431  m cmap 8865  Xcixp 8936  Fincfn 8984  infcinf 9479  cr 11152  0cc0 11153  *cxr 11292   < clt 11293  cn 12264  [,)cico 13386  cprod 15936  volcvol 25512  Σ^csumge0 46318  voln*covoln 46492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-ixp 8937  df-sup 9480  df-inf 9481  df-seq 14040  df-prod 15937  df-ovoln 46493
This theorem is referenced by:  ovnval2  46501  ovnf  46519
  Copyright terms: Public domain W3C validator