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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 46500 | . 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < ))) |
5 | ovnn0val.2 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅) | |
6 | 5 | neneqd 2942 | . . 3 ⊢ (𝜑 → ¬ 𝑋 = ∅) |
7 | 6 | iffalsed 4541 | . 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < )) |
8 | 4, 7 | eqtrd 2774 | 1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = inf(𝑀, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 {crab 3432 ⊆ wss 3962 ∅c0 4338 ifcif 4530 ∪ ciun 4995 ↦ cmpt 5230 × cxp 5686 ∘ ccom 5692 ‘cfv 6562 (class class class)co 7430 ↑m cmap 8864 Xcixp 8935 Fincfn 8983 infcinf 9478 ℝcr 11151 0cc0 11152 ℝ*cxr 11291 < clt 11292 ℕcn 12263 [,)cico 13385 ∏cprod 15935 volcvol 25511 Σ^csumge0 46317 voln*covoln 46491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-mulcl 11214 ax-i2m1 11220 ax-pre-lttri 11226 ax-pre-lttrn 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-seq 14039 df-prod 15936 df-ovoln 46492 |
This theorem is referenced by: ovnlecvr 46513 ovnsslelem 46515 ovnlerp 46517 ovnhoilem2 46557 ovnlecvr2 46565 |
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