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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 13380 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 0xr 11209 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
4 | pnfxr 11216 | . . . 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 3959 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) |
11 | 6, 7, 10 | omexrcl 44822 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
12 | 6, 7, 10 | omecl 44818 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,]+∞)) |
13 | iccgelb 13327 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝑂‘𝐵) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐵)) | |
14 | 3, 5, 12, 13 | syl3anc 1372 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐵)) |
15 | omessre.re | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) | |
16 | 15 | rexrd 11212 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) |
17 | 6, 7, 9, 8 | omessle 44813 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ≤ (𝑂‘𝐴)) |
18 | 15 | ltpnfd 13049 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) < +∞) |
19 | 11, 16, 5, 17, 18 | xrlelttrd 13086 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) < +∞) |
20 | 3, 5, 11, 14, 19 | elicod 13321 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,)+∞)) |
21 | 1, 20 | sselid 3947 | 1 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ⊆ wss 3915 ∪ cuni 4870 class class class wbr 5110 dom cdm 5638 ‘cfv 6501 (class class class)co 7362 ℝcr 11057 0cc0 11058 +∞cpnf 11193 ℝ*cxr 11195 ≤ cle 11197 [,)cico 13273 [,]cicc 13274 OutMeascome 44804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-addrcl 11119 ax-rnegex 11129 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-ico 13277 df-icc 13278 df-ome 44805 |
This theorem is referenced by: carageniuncllem1 44836 carageniuncllem2 44837 |
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