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 12836 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 0xr 10680 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
4 | pnfxr 10687 | . . . 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 3975 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) |
11 | 6, 7, 10 | omexrcl 42780 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
12 | 6, 7, 10 | omecl 42776 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,]+∞)) |
13 | iccgelb 12785 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝑂‘𝐵) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐵)) | |
14 | 3, 5, 12, 13 | syl3anc 1366 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐵)) |
15 | omessre.re | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) | |
16 | 15 | rexrd 10683 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) |
17 | 6, 7, 9, 8 | omessle 42771 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ≤ (𝑂‘𝐴)) |
18 | 15 | ltpnfd 12508 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) < +∞) |
19 | 11, 16, 5, 17, 18 | xrlelttrd 12545 | . . 3 ⊢ (𝜑 → (𝑂‘𝐵) < +∞) |
20 | 3, 5, 11, 14, 19 | elicod 12779 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,)+∞)) |
21 | 1, 20 | sseldi 3963 | 1 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ⊆ wss 3934 ∪ cuni 4830 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 (class class class)co 7148 ℝcr 10528 0cc0 10529 +∞cpnf 10664 ℝ*cxr 10666 ≤ cle 10668 [,)cico 12732 [,]cicc 12733 OutMeascome 42762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-addrcl 10590 ax-rnegex 10600 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 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 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-ico 12736 df-icc 12737 df-ome 42763 |
This theorem is referenced by: carageniuncllem1 42794 carageniuncllem2 42795 |
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