ovnlerp - Mathbox for Glauco Siliprandi

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Theorem ovnlerp 46591

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 4055	. . . . . 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 4005	. . . . 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 46588	. . . . 5 ⊢ (𝜑 → +∞ ∈ 𝑀)
9	8	ne0d 4317	. . . 4 ⊢ (𝜑 → 𝑀 ≠ ∅)
10		0red 11238	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
11	6, 7, 2	ovnsupge0 46586	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (0[,]+∞))
12		0xr 11282	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ∈ ℝ^*)
14		pnfxr 11289	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
15	14	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → +∞ ∈ ℝ^*)
16		ssel2 3953	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ (0[,]+∞))
17		iccgelb 13419	. . . . . . . 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 3132	. . . . . 6 ⊢ (𝑀 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)
20	11, 19	syl 17	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)
21		breq1 5122	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
22	21	ralbidv 3163	. . . . . 6 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦))
23	22	rspcev 3601	. . . . 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 45378	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ 𝑀 𝑤 ≤ (inf(𝑀, ℝ^*, < ) +_𝑒 𝐸))
27		nfv 1914	. . . 4 ⊢ Ⅎ𝑤𝜑
28		simp3 1138	. . . . . 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 46580	. . . . . . . . 9 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = inf(𝑀, ℝ^, < ))
31	30	eqcomd 2741	. . . . . . . 8 ⊢ (𝜑 → inf(𝑀, ℝ^, < ) = ((voln‘𝑋)‘𝐴))
32	31	oveq1d 7420	. . . . . . 7 ⊢ (𝜑 → (inf(𝑀, ℝ^, < ) +_𝑒 𝐸) = (((voln‘𝑋)‘𝐴) +_𝑒 𝐸))
33	32	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀 ∧ 𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸)) → (inf(𝑀, ℝ^, < ) +_𝑒 𝐸) = (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸))
34	28, 33	breqtrd 5145	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀 ∧ 𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸)) → 𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸))
35	34	3exp 1119	. . . 4 ⊢ (𝜑 → (𝑤 ∈ 𝑀 → (𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸) → 𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸))))
36	27, 35	reximdai 3244	. . 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 2898	. . 3 ⊢ Ⅎ𝑤𝑀
39		nfrab1 3436	. . . 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 2896	. . 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 5122	. . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸) ↔ 𝑧 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸)))
44	38, 40, 41, 42, 43	cbvrexfw 3285	. 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 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 {crab 3415 ⊆ wss 3926 ∅c0 4308 ∪ ciun 4967 class class class wbr 5119 ↦ cmpt 5201 × cxp 5652 ∘ ccom 5658 ‘cfv 6531 (class class class)co 7405 ↑_m cmap 8840 Xcixp 8911 Fincfn 8959 infcinf 9453 ℝcr 11128 0cc0 11129 +∞cpnf 11266 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 ℕcn 12240 ℝ⁺crp 13008 +_𝑒 cxad 13126 [,)cico 13364 [,]cicc 13365 ∏cprod 15919 volcvol 25416 Σ^{^}csumge0 46391 voln*covoln 46565

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-n0 12502 df-z 12589 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-seq 14020 df-exp 14080 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-clim 15504 df-rlim 15505 df-sum 15703 df-prod 15920 df-rest 17436 df-topgen 17457 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-top 22832 df-topon 22849 df-bases 22884 df-cmp 23325 df-ovol 25417 df-vol 25418 df-sumge0 46392 df-ovoln 46566

This theorem is referenced by: ovncvrrp 46593

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