ovnlerp - Mathbox for Glauco Siliprandi

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Theorem ovnlerp 46567

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 4046	. . . . . 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 3996	. . . . 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 46564	. . . . 5 ⊢ (𝜑 → +∞ ∈ 𝑀)
9	8	ne0d 4308	. . . 4 ⊢ (𝜑 → 𝑀 ≠ ∅)
10		0red 11184	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
11	6, 7, 2	ovnsupge0 46562	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (0[,]+∞))
12		0xr 11228	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ∈ ℝ^*)
14		pnfxr 11235	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
15	14	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → +∞ ∈ ℝ^*)
16		ssel2 3944	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ (0[,]+∞))
17		iccgelb 13370	. . . . . . . 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 3126	. . . . . 6 ⊢ (𝑀 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)
20	11, 19	syl 17	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)
21		breq1 5113	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
22	21	ralbidv 3157	. . . . . 6 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦))
23	22	rspcev 3591	. . . . 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 45354	. . 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 46556	. . . . . . . . 9 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = inf(𝑀, ℝ^, < ))
31	30	eqcomd 2736	. . . . . . . 8 ⊢ (𝜑 → inf(𝑀, ℝ^, < ) = ((voln‘𝑋)‘𝐴))
32	31	oveq1d 7405	. . . . . . 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 5136	. . . . 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 3240	. . 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 2892	. . 3 ⊢ Ⅎ𝑤𝑀
39		nfrab1 3429	. . . 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 2890	. . 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 5113	. . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸) ↔ 𝑧 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸)))
44	38, 40, 41, 42, 43	cbvrexfw 3281	. 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 2109 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 {crab 3408 ⊆ wss 3917 ∅c0 4299 ∪ ciun 4958 class class class wbr 5110 ↦ cmpt 5191 × cxp 5639 ∘ ccom 5645 ‘cfv 6514 (class class class)co 7390 ↑_m cmap 8802 Xcixp 8873 Fincfn 8921 infcinf 9399 ℝcr 11074 0cc0 11075 +∞cpnf 11212 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 ℕcn 12193 ℝ⁺crp 12958 +_𝑒 cxad 13077 [,)cico 13315 [,]cicc 13316 ∏cprod 15876 volcvol 25371 Σ^{^}csumge0 46367 voln*covoln 46541

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-prod 15877 df-rest 17392 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-top 22788 df-topon 22805 df-bases 22840 df-cmp 23281 df-ovol 25372 df-vol 25373 df-sumge0 46368 df-ovoln 46542

This theorem is referenced by: ovncvrrp 46569

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