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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 3979 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) |
6 | 1, 2, 5 | omexrcl 42796 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
7 | 0xr 10690 | . . 3 ⊢ 0 ∈ ℝ* | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ ℝ*) |
9 | 1, 2, 4, 3 | omessle 42787 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) ≤ (𝑂‘𝐴)) |
10 | omess0.z | . . 3 ⊢ (𝜑 → (𝑂‘𝐴) = 0) | |
11 | 9, 10 | breqtrd 5094 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ≤ 0) |
12 | 1, 2, 5 | omege0 42822 | . 2 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐵)) |
13 | 6, 8, 11, 12 | xrletrid 12551 | 1 ⊢ (𝜑 → (𝑂‘𝐵) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∪ cuni 4840 dom cdm 5557 ‘cfv 6357 0cc0 10539 ℝ*cxr 10676 ≤ cle 10678 OutMeascome 42778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-addrcl 10600 ax-rnegex 10610 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-icc 12748 df-ome 42779 |
This theorem is referenced by: caragencmpl 42824 voncmpl 42910 |
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