ovnlerp - Mathbox for Glauco Siliprandi

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Theorem ovnlerp 46949

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 1916	. . . 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 4034	. . . . . 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 3982	. . . . 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 46946	. . . . 5 ⊢ (𝜑 → +∞ ∈ 𝑀)
9	8	ne0d 4296	. . . 4 ⊢ (𝜑 → 𝑀 ≠ ∅)
10		0red 11149	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
11	6, 7, 2	ovnsupge0 46944	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (0[,]+∞))
12		0xr 11193	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ∈ ℝ^*)
14		pnfxr 11200	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
15	14	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → +∞ ∈ ℝ^*)
16		ssel2 3930	. . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ (0[,]+∞))
17		iccgelb 13332	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
18	13, 15, 16, 17	syl3anc 1374	. . . . . . 7 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ≤ 𝑦)
19	18	ralrimiva 3130	. . . . . 6 ⊢ (𝑀 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)
20	11, 19	syl 17	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)
21		breq1 5103	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
22	21	ralbidv 3161	. . . . . 6 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦))
23	22	rspcev 3578	. . . . 5 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦)
24	10, 20, 23	syl2anc 585	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦)
25		ovnlerp.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
26	1, 5, 9, 24, 25	infrpge 45739	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ 𝑀 𝑤 ≤ (inf(𝑀, ℝ^*, < ) +_𝑒 𝐸))
27		nfv 1916	. . . 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 46938	. . . . . . . . 9 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = inf(𝑀, ℝ^, < ))
31	30	eqcomd 2743	. . . . . . . 8 ⊢ (𝜑 → inf(𝑀, ℝ^, < ) = ((voln‘𝑋)‘𝐴))
32	31	oveq1d 7385	. . . . . . 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 5126	. . . . 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 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 2899	. . 3 ⊢ Ⅎ𝑤𝑀
39		nfrab1 3421	. . . 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 2897	. . 3 ⊢ Ⅎ𝑧𝑀
41		nfv 1916	. . 3 ⊢ Ⅎ𝑧 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸)
42		nfv 1916	. . 3 ⊢ Ⅎ𝑤 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝐸)
43		breq1 5103	. . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸) ↔ 𝑧 ≤ (((voln‘𝑋)‘𝐴) +_𝑒 𝐸)))
44	38, 40, 41, 42, 43	cbvrexfw 3279	. 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 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3401 ⊆ wss 3903 ∅c0 4287 ∪ ciun 4948 class class class wbr 5100 ↦ cmpt 5181 × cxp 5632 ∘ ccom 5638 ‘cfv 6502 (class class class)co 7370 ↑_m cmap 8777 Xcixp 8849 Fincfn 8897 infcinf 9358 ℝcr 11039 0cc0 11040 +∞cpnf 11177 ℝ^*cxr 11179 < clt 11180 ≤ cle 11181 ℕcn 12159 ℝ⁺crp 12919 +_𝑒 cxad 13038 [,)cico 13277 [,]cicc 13278 ∏cprod 15840 volcvol 25437 Σ^{^}csumge0 46749 voln*covoln 46923

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-inf2 9564 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-pm 8780 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fi 9328 df-sup 9359 df-inf 9360 df-oi 9429 df-dju 9827 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-n0 12416 df-z 12503 df-uz 12766 df-q 12876 df-rp 12920 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13279 df-ico 13281 df-icc 13282 df-fz 13438 df-fzo 13585 df-fl 13726 df-seq 13939 df-exp 13999 df-hash 14268 df-cj 15036 df-re 15037 df-im 15038 df-sqrt 15172 df-abs 15173 df-clim 15425 df-rlim 15426 df-sum 15624 df-prod 15841 df-rest 17356 df-topgen 17377 df-psmet 21318 df-xmet 21319 df-met 21320 df-bl 21321 df-mopn 21322 df-top 22855 df-topon 22872 df-bases 22907 df-cmp 23348 df-ovol 25438 df-vol 25439 df-sumge0 46750 df-ovoln 46924

This theorem is referenced by: ovncvrrp 46951

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