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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 3897 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) |
6 | 1, 2, 5 | omexrcl 43663 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
7 | 0xr 10845 | . . 3 ⊢ 0 ∈ ℝ* | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ ℝ*) |
9 | 1, 2, 4, 3 | omessle 43654 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) ≤ (𝑂‘𝐴)) |
10 | omess0.z | . . 3 ⊢ (𝜑 → (𝑂‘𝐴) = 0) | |
11 | 9, 10 | breqtrd 5065 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ≤ 0) |
12 | 1, 2, 5 | omege0 43689 | . 2 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐵)) |
13 | 6, 8, 11, 12 | xrletrid 12710 | 1 ⊢ (𝜑 → (𝑂‘𝐵) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 ∪ cuni 4805 dom cdm 5536 ‘cfv 6358 0cc0 10694 ℝ*cxr 10831 ≤ cle 10833 OutMeascome 43645 |
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-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-addrcl 10755 ax-rnegex 10765 ax-cnre 10767 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-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-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-1st 7739 df-2nd 7740 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-icc 12907 df-ome 43646 |
This theorem is referenced by: caragencmpl 43691 voncmpl 43777 |
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