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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omessre | Structured version Visualization version GIF version |
Description: If the outer measure of a set is real, then the outer measure of any of its subset is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
omessre.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
omessre.x | ⊢ 𝑋 = ∪ dom 𝑂 |
omessre.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
omessre.re | ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) |
omessre.b | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
Ref | Expression |
---|---|
omessre | ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rge0ssre 13471 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 0xr 11297 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
4 | pnfxr 11304 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
6 | omessre.o | . . . 4 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
7 | omessre.x | . . . 4 ⊢ 𝑋 = ∪ dom 𝑂 | |
8 | omessre.b | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
9 | omessre.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
10 | 8, 9 | sstrd 3990 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) |
11 | 6, 7, 10 | omexrcl 45897 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
12 | 6, 7, 10 | omecl 45893 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,]+∞)) |
13 | iccgelb 13418 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝑂‘𝐵) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐵)) | |
14 | 3, 5, 12, 13 | syl3anc 1368 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐵)) |
15 | omessre.re | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) | |
16 | 15 | rexrd 11300 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) |
17 | 6, 7, 9, 8 | omessle 45888 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ≤ (𝑂‘𝐴)) |
18 | 15 | ltpnfd 13139 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) < +∞) |
19 | 11, 16, 5, 17, 18 | xrlelttrd 13177 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) < +∞) |
20 | 3, 5, 11, 14, 19 | elicod 13412 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,)+∞)) |
21 | 1, 20 | sselid 3978 | 1 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3947 ∪ cuni 4910 class class class wbr 5150 dom cdm 5680 ‘cfv 6551 (class class class)co 7424 ℝcr 11143 0cc0 11144 +∞cpnf 11281 ℝ*cxr 11283 ≤ cle 11285 [,)cico 13364 [,]cicc 13365 OutMeascome 45879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-addrcl 11205 ax-rnegex 11215 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-po 5592 df-so 5593 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-ico 13368 df-icc 13369 df-ome 45880 |
This theorem is referenced by: carageniuncllem1 45911 carageniuncllem2 45912 |
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