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 44856
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 44852 . 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 7370 . . . 4 (𝑥 = 𝑋 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑋))
32pweqd 4582 . . 3 (𝑥 = 𝑋 → 𝒫 (ℝ ↑m 𝑥) = 𝒫 (ℝ ↑m 𝑋))
4 eqeq1 2741 . . . 4 (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
5 oveq2 7370 . . . . . . . 8 (𝑥 = 𝑋 → ((ℝ × ℝ) ↑m 𝑥) = ((ℝ × ℝ) ↑m 𝑋))
65oveq1d 7377 . . . . . . 7 (𝑥 = 𝑋 → (((ℝ × ℝ) ↑m 𝑥) ↑m ℕ) = (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
7 ixpeq1 8853 . . . . . . . . . 10 (𝑥 = 𝑋X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
87iuneq2d 4988 . . . . . . . . 9 (𝑥 = 𝑋 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
98sseq2d 3981 . . . . . . . 8 (𝑥 = 𝑋 → (𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
10 simpl 484 . . . . . . . . . . . 12 ((𝑥 = 𝑋𝑗 ∈ ℕ) → 𝑥 = 𝑋)
1110prodeq1d 15811 . . . . . . . . . . 11 ((𝑥 = 𝑋𝑗 ∈ ℕ) → ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
1211mpteq2dva 5210 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
1312fveq2d 6851 . . . . . . . . 9 (𝑥 = 𝑋 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
1413eqeq2d 2748 . . . . . . . 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 3323 . . . . . 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 3418 . . . . 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 9420 . . . 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 4517 . . 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 5201 . 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 7395 . . . . 5 (ℝ ↑m 𝑋) ∈ V
2322pwex 5340 . . . 4 𝒫 (ℝ ↑m 𝑋) ∈ V
2423mptex 7178 . . 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 6976 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 397   = wceq 1542  wcel 2107  wrex 3074  {crab 3410  Vcvv 3448  wss 3915  c0 4287  ifcif 4491  𝒫 cpw 4565   ciun 4959  cmpt 5193   × cxp 5636  ccom 5642  cfv 6501  (class class class)co 7362  m cmap 8772  Xcixp 8842  Fincfn 8890  infcinf 9384  cr 11057  0cc0 11058  *cxr 11195   < clt 11196  cn 12160  [,)cico 13273  cprod 15795  volcvol 24843  Σ^csumge0 44677  voln*covoln 44851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-ixp 8843  df-sup 9385  df-inf 9386  df-seq 13914  df-prod 15796  df-ovoln 44852
This theorem is referenced by:  ovnval2  44860  ovnf  44878
  Copyright terms: Public domain W3C validator