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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 4019 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) |
6 | 1, 2, 5 | omexrcl 46428 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
7 | 0xr 11337 | . . 3 ⊢ 0 ∈ ℝ* | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ ℝ*) |
9 | 1, 2, 4, 3 | omessle 46419 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) ≤ (𝑂‘𝐴)) |
10 | omess0.z | . . 3 ⊢ (𝜑 → (𝑂‘𝐴) = 0) | |
11 | 9, 10 | breqtrd 5192 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ≤ 0) |
12 | 1, 2, 5 | omege0 46454 | . 2 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐵)) |
13 | 6, 8, 11, 12 | xrletrid 13217 | 1 ⊢ (𝜑 → (𝑂‘𝐵) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 ∪ cuni 4931 dom cdm 5700 ‘cfv 6573 0cc0 11184 ℝ*cxr 11323 ≤ cle 11325 OutMeascome 46410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-addrcl 11245 ax-rnegex 11255 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-icc 13414 df-ome 46411 |
This theorem is referenced by: caragencmpl 46456 voncmpl 46542 |
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