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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omess0 | Structured version Visualization version GIF version |
Description: If the outer measure of a set is 0, then the outer measure of its subsets is 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
omess0.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
omess0.x | ⊢ 𝑋 = ∪ dom 𝑂 |
omess0.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
omess0.z | ⊢ (𝜑 → (𝑂‘𝐴) = 0) |
omess0.s | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
Ref | Expression |
---|---|
omess0 | ⊢ (𝜑 → (𝑂‘𝐵) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omess0.o | . . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | omess0.x | . . 3 ⊢ 𝑋 = ∪ dom 𝑂 | |
3 | omess0.s | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
4 | omess0.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
5 | 3, 4 | sstrd 3952 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) |
6 | 1, 2, 5 | omexrcl 44680 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
7 | 0xr 11198 | . . 3 ⊢ 0 ∈ ℝ* | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ ℝ*) |
9 | 1, 2, 4, 3 | omessle 44671 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) ≤ (𝑂‘𝐴)) |
10 | omess0.z | . . 3 ⊢ (𝜑 → (𝑂‘𝐴) = 0) | |
11 | 9, 10 | breqtrd 5129 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ≤ 0) |
12 | 1, 2, 5 | omege0 44706 | . 2 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐵)) |
13 | 6, 8, 11, 12 | xrletrid 13066 | 1 ⊢ (𝜑 → (𝑂‘𝐵) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 ∪ cuni 4863 dom cdm 5631 ‘cfv 6493 0cc0 11047 ℝ*cxr 11184 ≤ cle 11186 OutMeascome 44662 |
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-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-addrcl 11108 ax-rnegex 11118 ax-cnre 11120 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-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-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-1st 7917 df-2nd 7918 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-icc 13263 df-ome 44663 |
This theorem is referenced by: caragencmpl 44708 voncmpl 44794 |
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