ovnlerp - Mathbox for Glauco Siliprandi

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Theorem ovnlerp 46577

Description: The Lebesgue outer measure of a subset of multidimensional real numbers can always be approximated by the total outer measure of a cover of half-open (multidimensional) intervals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
ovnlerp.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnlerp.n0	⊢ (𝜑 → 𝑋 ≠ ∅)
ovnlerp.a	⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑_m 𝑋))
ovnlerp.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
ovnlerp.m	⊢ 𝑀 = {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}

**Assertion**
Ref	Expression
ovnlerp	⊢ (𝜑 → ∃𝑧 ∈ 𝑀 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸))

Distinct variable groups: 𝐴,𝑖,𝑧 𝑧,𝐸 𝑖,𝑋,𝑗,𝑘,𝑧 𝜑,𝑖,𝑗,𝑘,𝑧 Allowed substitution hints: 𝐴(𝑗,𝑘) 𝐸(𝑖,𝑗,𝑘) 𝑀(𝑧,𝑖,𝑗,𝑘)

Proof of Theorem ovnlerp Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1914	. . . 4 ⊢ Ⅎ𝑥𝜑
2		ovnlerp.m	. . . . . 6 ⊢ 𝑀 = {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}
3		ssrab2 4080	. . . . . 6 ⊢ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⊆ ℝ^*
4	2, 3	eqsstri 4030	. . . . 5 ⊢ 𝑀 ⊆ ℝ^*
5	4	a1i 11	. . . 4 ⊢ (𝜑 → 𝑀 ⊆ ℝ^*)
6		ovnlerp.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin)
7		ovnlerp.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑_m 𝑋))
8	6, 7, 2	ovnpnfelsup 46574	. . . . 5 ⊢ (𝜑 → +∞ ∈ 𝑀)
9	8	ne0d 4342	. . . 4 ⊢ (𝜑 → 𝑀 ≠ ∅)
10		0red 11264	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
11	6, 7, 2	ovnsupge0 46572	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (0[,]+∞))
12		0xr 11308	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ∈ ℝ^*)
14		pnfxr 11315	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
15	14	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → +∞ ∈ ℝ^*)
16		ssel2 3978	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ (0[,]+∞))
17		iccgelb 13443	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
18	13, 15, 16, 17	syl3anc 1373	. . . . . . 7 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ≤ 𝑦)
19	18	ralrimiva 3146	. . . . . 6 ⊢ (𝑀 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)
20	11, 19	syl 17	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)
21		breq1 5146	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
22	21	ralbidv 3178	. . . . . 6 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦))
23	22	rspcev 3622	. . . . 5 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦)
24	10, 20, 23	syl2anc 584	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦)
25		ovnlerp.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
26	1, 5, 9, 24, 25	infrpge 45362	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ 𝑀 𝑤 ≤ (inf(𝑀, ℝ^*, < ) +_𝑒 𝐸))
27		nfv 1914	. . . 4 ⊢ Ⅎ𝑤𝜑
28		simp3 1139	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀 ∧ 𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸)) → 𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸))
29		ovnlerp.n0	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ ∅)
30	6, 29, 7, 2	ovnn0val 46566	. . . . . . . . 9 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = inf(𝑀, ℝ^, < ))
31	30	eqcomd 2743	. . . . . . . 8 ⊢ (𝜑 → inf(𝑀, ℝ^, < ) = ((voln‘𝑋)‘𝐴))
32	31	oveq1d 7446	. . . . . . 7 ⊢ (𝜑 → (inf(𝑀, ℝ^, < ) +_𝑒 𝐸) = (((voln‘𝑋)‘𝐴) +_𝑒 𝐸))
33	32	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀 ∧ 𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸)) → (inf(𝑀, ℝ^, < ) +_𝑒 𝐸) = (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸))
34	28, 33	breqtrd 5169	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀 ∧ 𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸)) → 𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸))
35	34	3exp 1120	. . . 4 ⊢ (𝜑 → (𝑤 ∈ 𝑀 → (𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸) → 𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸))))
36	27, 35	reximdai 3261	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ 𝑀 𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸) → ∃𝑤 ∈ 𝑀 𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸)))
37	26, 36	mpd 15	. 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑀 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸))
38		nfcv 2905	. . 3 ⊢ Ⅎ𝑤𝑀
39		nfrab1 3457	. . . 4 ⊢ Ⅎ𝑧{𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}
40	2, 39	nfcxfr 2903	. . 3 ⊢ Ⅎ𝑧𝑀
41		nfv 1914	. . 3 ⊢ Ⅎ𝑧 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸)
42		nfv 1914	. . 3 ⊢ Ⅎ𝑤 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸)
43		breq1 5146	. . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸) ↔ 𝑧 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸)))
44	38, 40, 41, 42, 43	cbvrexfw 3305	. 2 ⊢ (∃𝑤 ∈ 𝑀 𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸) ↔ ∃𝑧 ∈ 𝑀 𝑧 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸))
45	37, 44	sylib 218	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑀 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 ⊆ wss 3951 ∅c0 4333 ∪ ciun 4991 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ↑_m cmap 8866 Xcixp 8937 Fincfn 8985 infcinf 9481 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 ℕcn 12266 ℝ⁺crp 13034 +_𝑒 cxad 13152 [,)cico 13389 [,]cicc 13390 ∏cprod 15939 volcvol 25498 Σ^{^}csumge0 46377 voln*covoln 46551

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-prod 15940 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cmp 23395 df-ovol 25499 df-vol 25500 df-sumge0 46378 df-ovoln 46552

This theorem is referenced by: ovncvrrp 46579

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