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 43701 | . 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < ))) |
5 | ovnn0val.2 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅) | |
6 | 5 | neneqd 2937 | . . 3 ⊢ (𝜑 → ¬ 𝑋 = ∅) |
7 | 6 | iffalsed 4436 | . 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < )) |
8 | 4, 7 | eqtrd 2771 | 1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = inf(𝑀, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∃wrex 3052 {crab 3055 ⊆ wss 3853 ∅c0 4223 ifcif 4425 ∪ ciun 4890 ↦ cmpt 5120 × cxp 5534 ∘ ccom 5540 ‘cfv 6358 (class class class)co 7191 ↑m cmap 8486 Xcixp 8556 Fincfn 8604 infcinf 9035 ℝcr 10693 0cc0 10694 ℝ*cxr 10831 < clt 10832 ℕcn 11795 [,)cico 12902 ∏cprod 15430 volcvol 24314 Σ^csumge0 43518 voln*covoln 43692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-mulcl 10756 ax-i2m1 10762 ax-pre-lttri 10768 ax-pre-lttrn 10769 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-seq 13540 df-prod 15431 df-ovoln 43693 |
This theorem is referenced by: ovnlecvr 43714 ovnsslelem 43716 ovnlerp 43718 ovnhoilem2 43758 ovnlecvr2 43766 |
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