omessre - Mathbox for Glauco Siliprandi

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Theorem omessre 45898

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
omessre	⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ)

**Proof of Theorem omessre**
Step	Hyp	Ref	Expression
1		rge0ssre 13465	. 2 ⊢ (0[,)+∞) ⊆ ℝ
2		0xr 11291	. . . 4 ⊢ 0 ∈ ℝ^*
3	2	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ ℝ^*)
4		pnfxr 11298	. . . 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 45895	. . 3 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ^*)
12	6, 7, 10	omecl 45891	. . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,]+∞))
13		iccgelb 13412	. . . 4 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘𝐵) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐵))
14	3, 5, 12, 13	syl3anc 1369	. . 3 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐵))
15		omessre.re	. . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ)
16	15	rexrd 11294	. . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ^*)
17	6, 7, 9, 8	omessle 45886	. . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ≤ (𝑂‘𝐴))
18	15	ltpnfd 13133	. . . 4 ⊢ (𝜑 → (𝑂‘𝐴) < +∞)
19	11, 16, 5, 17, 18	xrlelttrd 13171	. . 3 ⊢ (𝜑 → (𝑂‘𝐵) < +∞)
20	3, 5, 11, 14, 19	elicod 13406	. 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 1534 ∈ wcel 2099 ⊆ wss 3947 ∪ cuni 4908 class class class wbr 5148 dom cdm 5678 ‘cfv 6548 (class class class)co 7420 ℝcr 11137 0cc0 11138 +∞cpnf 11275 ℝ^*cxr 11277 ≤ cle 11279 [,)cico 13358 [,]cicc 13359 OutMeascome 45877

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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-addrcl 11199 ax-rnegex 11209 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-ico 13362 df-icc 13363 df-ome 45878

This theorem is referenced by: carageniuncllem1 45909 carageniuncllem2 45910

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