ovnlerp - Mathbox for Glauco Siliprandi

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Theorem ovnlerp 43775

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 1922	. . . 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 3993	. . . . . 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 3935	. . . . 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 43772	. . . . 5 ⊢ (𝜑 → +∞ ∈ 𝑀)
9	8	ne0d 4250	. . . 4 ⊢ (𝜑 → 𝑀 ≠ ∅)
10		0red 10836	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
11	6, 7, 2	ovnsupge0 43770	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (0[,]+∞))
12		0xr 10880	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ∈ ℝ^*)
14		pnfxr 10887	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
15	14	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → +∞ ∈ ℝ^*)
16		ssel2 3895	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ (0[,]+∞))
17		iccgelb 12991	. . . . . . . 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 3105	. . . . . 6 ⊢ (𝑀 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)
20	11, 19	syl 17	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)
21		breq1 5056	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
22	21	ralbidv 3118	. . . . . 6 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦))
23	22	rspcev 3537	. . . . 5 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦)
24	10, 20, 23	syl2anc 587	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦)
25		ovnlerp.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
26	1, 5, 9, 24, 25	infrpge 42563	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ 𝑀 𝑤 ≤ (inf(𝑀, ℝ^*, < ) +_𝑒 𝐸))
27		nfv 1922	. . . 4 ⊢ Ⅎ𝑤𝜑
28		simp3 1140	. . . . . 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 43764	. . . . . . . . 9 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = inf(𝑀, ℝ^, < ))
31	30	eqcomd 2743	. . . . . . . 8 ⊢ (𝜑 → inf(𝑀, ℝ^, < ) = ((voln‘𝑋)‘𝐴))
32	31	oveq1d 7228	. . . . . . 7 ⊢ (𝜑 → (inf(𝑀, ℝ^, < ) +_𝑒 𝐸) = (((voln‘𝑋)‘𝐴) +_𝑒 𝐸))
33	32	3ad2ant1 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀 ∧ 𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸)) → (inf(𝑀, ℝ^, < ) +_𝑒 𝐸) = (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸))
34	28, 33	breqtrd 5079	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀 ∧ 𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸)) → 𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸))
35	34	3exp 1121	. . . 4 ⊢ (𝜑 → (𝑤 ∈ 𝑀 → (𝑤 ≤ (inf(𝑀, ℝ^, < ) +_𝑒 𝐸) → 𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸))))
36	27, 35	reximdai 3230	. . 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 2904	. . 3 ⊢ Ⅎ𝑤𝑀
39		nfrab1 3296	. . . 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 2902	. . 3 ⊢ Ⅎ𝑧𝑀
41		nfv 1922	. . 3 ⊢ Ⅎ𝑧 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸)
42		nfv 1922	. . 3 ⊢ Ⅎ𝑤 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸)
43		breq1 5056	. . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸) ↔ 𝑧 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸)))
44	38, 40, 41, 42, 43	cbvrexfw 3346	. 2 ⊢ (∃𝑤 ∈ 𝑀 𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸) ↔ ∃𝑧 ∈ 𝑀 𝑧 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸))
45	37, 44	sylib 221	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑀 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 ∃wrex 3062 {crab 3065 ⊆ wss 3866 ∅c0 4237 ∪ ciun 4904 class class class wbr 5053 ↦ cmpt 5135 × cxp 5549 ∘ ccom 5555 ‘cfv 6380 (class class class)co 7213 ↑_m cmap 8508 Xcixp 8578 Fincfn 8626 infcinf 9057 ℝcr 10728 0cc0 10729 +∞cpnf 10864 ℝ^*cxr 10866 < clt 10867 ≤ cle 10868 ℕcn 11830 ℝ⁺crp 12586 +_𝑒 cxad 12702 [,)cico 12937 [,]cicc 12938 ∏cprod 15467 volcvol 24360 Σ^{^}csumge0 43575 voln*covoln 43749

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 df-sum 15250 df-prod 15468 df-rest 16927 df-topgen 16948 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-top 21791 df-topon 21808 df-bases 21843 df-cmp 22284 df-ovol 24361 df-vol 24362 df-sumge0 43576 df-ovoln 43750

This theorem is referenced by: ovncvrrp 43777

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