![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 13436 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 0xr 11262 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
4 | pnfxr 11269 | . . . 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 3987 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) |
11 | 6, 7, 10 | omexrcl 45776 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
12 | 6, 7, 10 | omecl 45772 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,]+∞)) |
13 | iccgelb 13383 | . . . 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 11265 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) |
17 | 6, 7, 9, 8 | omessle 45767 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ≤ (𝑂‘𝐴)) |
18 | 15 | ltpnfd 13104 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) < +∞) |
19 | 11, 16, 5, 17, 18 | xrlelttrd 13142 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) < +∞) |
20 | 3, 5, 11, 14, 19 | elicod 13377 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,)+∞)) |
21 | 1, 20 | sselid 3975 | 1 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 ∪ cuni 4902 class class class wbr 5141 dom cdm 5669 ‘cfv 6536 (class class class)co 7404 ℝcr 11108 0cc0 11109 +∞cpnf 11246 ℝ*cxr 11248 ≤ cle 11250 [,)cico 13329 [,]cicc 13330 OutMeascome 45758 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-addrcl 11170 ax-rnegex 11180 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-ico 13333 df-icc 13334 df-ome 45759 |
This theorem is referenced by: carageniuncllem1 45790 carageniuncllem2 45791 |
Copyright terms: Public domain | W3C validator |