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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovnn0val | Structured version Visualization version GIF version |
Description: The value of a (multidimensional) Lebesgue outer measure, defined on a nonzero-dimensional space of reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
ovnn0val.1 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
ovnn0val.2 | ⊢ (𝜑 → 𝑋 ≠ ∅) |
ovnn0val.3 | ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) |
ovnn0val.4 | ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} |
Ref | Expression |
---|---|
ovnn0val | ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = inf(𝑀, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovnn0val.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
2 | ovnn0val.3 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) | |
3 | ovnn0val.4 | . . 3 ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} | |
4 | 1, 2, 3 | ovnval2 44718 | . 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < ))) |
5 | ovnn0val.2 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅) | |
6 | 5 | neneqd 2946 | . . 3 ⊢ (𝜑 → ¬ 𝑋 = ∅) |
7 | 6 | iffalsed 4495 | . 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < )) |
8 | 4, 7 | eqtrd 2776 | 1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = inf(𝑀, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∃wrex 3071 {crab 3405 ⊆ wss 3908 ∅c0 4280 ifcif 4484 ∪ ciun 4952 ↦ cmpt 5186 × cxp 5629 ∘ ccom 5635 ‘cfv 6493 (class class class)co 7353 ↑m cmap 8761 Xcixp 8831 Fincfn 8879 infcinf 9373 ℝcr 11046 0cc0 11047 ℝ*cxr 11184 < clt 11185 ℕcn 12149 [,)cico 13258 ∏cprod 15780 volcvol 24811 Σ^csumge0 44535 voln*covoln 44709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-mulcl 11109 ax-i2m1 11115 ax-pre-lttri 11121 ax-pre-lttrn 11122 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-inf 9375 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-seq 13899 df-prod 15781 df-ovoln 44710 |
This theorem is referenced by: ovnlecvr 44731 ovnsslelem 44733 ovnlerp 44735 ovnhoilem2 44775 ovnlecvr2 44783 |
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