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 12832 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 0xr 10676 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
4 | pnfxr 10683 | . . . 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 3974 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) |
11 | 6, 7, 10 | omexrcl 42666 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
12 | 6, 7, 10 | omecl 42662 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,]+∞)) |
13 | iccgelb 12781 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝑂‘𝐵) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐵)) | |
14 | 3, 5, 12, 13 | syl3anc 1363 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐵)) |
15 | omessre.re | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) | |
16 | 15 | rexrd 10679 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) |
17 | 6, 7, 9, 8 | omessle 42657 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ≤ (𝑂‘𝐴)) |
18 | 15 | ltpnfd 12504 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) < +∞) |
19 | 11, 16, 5, 17, 18 | xrlelttrd 12541 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) < +∞) |
20 | 3, 5, 11, 14, 19 | elicod 12775 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,)+∞)) |
21 | 1, 20 | sseldi 3962 | 1 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ∪ cuni 4830 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 +∞cpnf 10660 ℝ*cxr 10662 ≤ cle 10664 [,)cico 12728 [,]cicc 12729 OutMeascome 42648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-addrcl 10586 ax-rnegex 10596 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-ico 12732 df-icc 12733 df-ome 42649 |
This theorem is referenced by: carageniuncllem1 42680 carageniuncllem2 42681 |
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