Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovnlerp | Structured version Visualization version GIF version |
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.) |
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)))))} |
Ref | Expression |
---|---|
ovnlerp | ⊢ (𝜑 → ∃𝑧 ∈ 𝑀 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1911 | . . . 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 4000 | . . . . 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 42840 | . . . . 5 ⊢ (𝜑 → +∞ ∈ 𝑀) |
9 | 8 | ne0d 4300 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ ∅) |
10 | 0red 10643 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℝ) | |
11 | 6, 7, 2 | ovnsupge0 42838 | . . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (0[,]+∞)) |
12 | 0xr 10687 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ∈ ℝ*) |
14 | pnfxr 10694 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → +∞ ∈ ℝ*) |
16 | ssel2 3961 | . . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ (0[,]+∞)) | |
17 | iccgelb 12792 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦) | |
18 | 13, 15, 16, 17 | syl3anc 1367 | . . . . . . 7 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ≤ 𝑦) |
19 | 18 | ralrimiva 3182 | . . . . . 6 ⊢ (𝑀 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦) |
20 | 11, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦) |
21 | breq1 5068 | . . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦)) | |
22 | 21 | ralbidv 3197 | . . . . . 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 586 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦) |
25 | ovnlerp.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
26 | 1, 5, 9, 24, 25 | infrpge 41617 | . . 3 ⊢ (𝜑 → ∃𝑤 ∈ 𝑀 𝑤 ≤ (inf(𝑀, ℝ*, < ) +𝑒 𝐸)) |
27 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑤𝜑 | |
28 | simp3 1134 | . . . . . 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 42832 | . . . . . . . . 9 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = inf(𝑀, ℝ*, < )) |
31 | 30 | eqcomd 2827 | . . . . . . . 8 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) = ((voln*‘𝑋)‘𝐴)) |
32 | 31 | oveq1d 7170 | . . . . . . 7 ⊢ (𝜑 → (inf(𝑀, ℝ*, < ) +𝑒 𝐸) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
33 | 32 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀 ∧ 𝑤 ≤ (inf(𝑀, ℝ*, < ) +𝑒 𝐸)) → (inf(𝑀, ℝ*, < ) +𝑒 𝐸) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
34 | 28, 33 | breqtrd 5091 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀 ∧ 𝑤 ≤ (inf(𝑀, ℝ*, < ) +𝑒 𝐸)) → 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
35 | 34 | 3exp 1115 | . . . 4 ⊢ (𝜑 → (𝑤 ∈ 𝑀 → (𝑤 ≤ (inf(𝑀, ℝ*, < ) +𝑒 𝐸) → 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))) |
36 | 27, 35 | reximdai 3311 | . . 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 2977 | . . 3 ⊢ Ⅎ𝑤𝑀 | |
39 | nfrab1 3384 | . . . 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 2975 | . . 3 ⊢ Ⅎ𝑧𝑀 |
41 | nfv 1911 | . . 3 ⊢ Ⅎ𝑧 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) | |
42 | nfv 1911 | . . 3 ⊢ Ⅎ𝑤 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) | |
43 | breq1 5068 | . . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ↔ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) | |
44 | 38, 40, 41, 42, 43 | cbvrexfw 3438 | . 2 ⊢ (∃𝑤 ∈ 𝑀 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ↔ ∃𝑧 ∈ 𝑀 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
45 | 37, 44 | sylib 220 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑀 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 ⊆ wss 3935 ∅c0 4290 ∪ ciun 4918 class class class wbr 5065 ↦ cmpt 5145 × cxp 5552 ∘ ccom 5558 ‘cfv 6354 (class class class)co 7155 ↑m cmap 8405 Xcixp 8460 Fincfn 8508 infcinf 8904 ℝcr 10535 0cc0 10536 +∞cpnf 10671 ℝ*cxr 10673 < clt 10674 ≤ cle 10675 ℕcn 11637 ℝ+crp 12388 +𝑒 cxad 12504 [,)cico 12739 [,]cicc 12740 ∏cprod 15258 volcvol 24063 Σ^csumge0 42643 voln*covoln 42817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-rlim 14845 df-sum 15042 df-prod 15259 df-rest 16695 df-topgen 16716 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-top 21501 df-topon 21518 df-bases 21553 df-cmp 21994 df-ovol 24064 df-vol 24065 df-sumge0 42644 df-ovoln 42818 |
This theorem is referenced by: ovncvrrp 42845 |
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