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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 1909 | . . . 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 4077 | . . . . . 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 4016 | . . . . 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 45976 | . . . . 5 ⊢ (𝜑 → +∞ ∈ 𝑀) |
9 | 8 | ne0d 4339 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ ∅) |
10 | 0red 11255 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℝ) | |
11 | 6, 7, 2 | ovnsupge0 45974 | . . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (0[,]+∞)) |
12 | 0xr 11299 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ∈ ℝ*) |
14 | pnfxr 11306 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → +∞ ∈ ℝ*) |
16 | ssel2 3977 | . . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ (0[,]+∞)) | |
17 | iccgelb 13420 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦) | |
18 | 13, 15, 16, 17 | syl3anc 1368 | . . . . . . 7 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ≤ 𝑦) |
19 | 18 | ralrimiva 3143 | . . . . . 6 ⊢ (𝑀 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦) |
20 | 11, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦) |
21 | breq1 5155 | . . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦)) | |
22 | 21 | ralbidv 3175 | . . . . . 6 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)) |
23 | 22 | rspcev 3611 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦) |
24 | 10, 20, 23 | syl2anc 582 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦) |
25 | ovnlerp.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
26 | 1, 5, 9, 24, 25 | infrpge 44762 | . . 3 ⊢ (𝜑 → ∃𝑤 ∈ 𝑀 𝑤 ≤ (inf(𝑀, ℝ*, < ) +𝑒 𝐸)) |
27 | nfv 1909 | . . . 4 ⊢ Ⅎ𝑤𝜑 | |
28 | simp3 1135 | . . . . . 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 45968 | . . . . . . . . 9 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = inf(𝑀, ℝ*, < )) |
31 | 30 | eqcomd 2734 | . . . . . . . 8 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) = ((voln*‘𝑋)‘𝐴)) |
32 | 31 | oveq1d 7441 | . . . . . . 7 ⊢ (𝜑 → (inf(𝑀, ℝ*, < ) +𝑒 𝐸) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
33 | 32 | 3ad2ant1 1130 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀 ∧ 𝑤 ≤ (inf(𝑀, ℝ*, < ) +𝑒 𝐸)) → (inf(𝑀, ℝ*, < ) +𝑒 𝐸) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
34 | 28, 33 | breqtrd 5178 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀 ∧ 𝑤 ≤ (inf(𝑀, ℝ*, < ) +𝑒 𝐸)) → 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
35 | 34 | 3exp 1116 | . . . 4 ⊢ (𝜑 → (𝑤 ∈ 𝑀 → (𝑤 ≤ (inf(𝑀, ℝ*, < ) +𝑒 𝐸) → 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))) |
36 | 27, 35 | reximdai 3256 | . . 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 3450 | . . . 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 1909 | . . 3 ⊢ Ⅎ𝑧 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) | |
42 | nfv 1909 | . . 3 ⊢ Ⅎ𝑤 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) | |
43 | breq1 5155 | . . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ↔ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) | |
44 | 38, 40, 41, 42, 43 | cbvrexfw 3300 | . 2 ⊢ (∃𝑤 ∈ 𝑀 𝑤 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ↔ ∃𝑧 ∈ 𝑀 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
45 | 37, 44 | sylib 217 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑀 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 {crab 3430 ⊆ wss 3949 ∅c0 4326 ∪ ciun 5000 class class class wbr 5152 ↦ cmpt 5235 × cxp 5680 ∘ ccom 5686 ‘cfv 6553 (class class class)co 7426 ↑m cmap 8851 Xcixp 8922 Fincfn 8970 infcinf 9472 ℝcr 11145 0cc0 11146 +∞cpnf 11283 ℝ*cxr 11285 < clt 11286 ≤ cle 11287 ℕcn 12250 ℝ+crp 13014 +𝑒 cxad 13130 [,)cico 13366 [,]cicc 13367 ∏cprod 15889 volcvol 25412 Σ^csumge0 45779 voln*covoln 45953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-rlim 15473 df-sum 15673 df-prod 15890 df-rest 17411 df-topgen 17432 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-top 22816 df-topon 22833 df-bases 22869 df-cmp 23311 df-ovol 25413 df-vol 25414 df-sumge0 45780 df-ovoln 45954 |
This theorem is referenced by: ovncvrrp 45981 |
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