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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 4072 | . . . . . 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 4011 | . . . . 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 45829 | . . . . 5 ⊢ (𝜑 → +∞ ∈ 𝑀) |
9 | 8 | ne0d 4330 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ ∅) |
10 | 0red 11218 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℝ) | |
11 | 6, 7, 2 | ovnsupge0 45827 | . . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (0[,]+∞)) |
12 | 0xr 11262 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 0 ∈ ℝ*) |
14 | pnfxr 11269 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → +∞ ∈ ℝ*) |
16 | ssel2 3972 | . . . . . . . 8 ⊢ ((𝑀 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ (0[,]+∞)) | |
17 | iccgelb 13383 | . . . . . . . 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 3140 | . . . . . 6 ⊢ (𝑀 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦) |
20 | 11, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 0 ≤ 𝑦) |
21 | breq1 5144 | . . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦)) | |
22 | 21 | ralbidv 3171 | . . . . . 6 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦)) |
23 | 22 | rspcev 3606 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ 𝑀 0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦) |
24 | 10, 20, 23 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑀 𝑥 ≤ 𝑦) |
25 | ovnlerp.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
26 | 1, 5, 9, 24, 25 | infrpge 44615 | . . 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 45821 | . . . . . . . . 9 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = inf(𝑀, ℝ*, < )) |
31 | 30 | eqcomd 2732 | . . . . . . . 8 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) = ((voln*‘𝑋)‘𝐴)) |
32 | 31 | oveq1d 7419 | . . . . . . 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 5167 | . . . . 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 3252 | . . 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 2897 | . . 3 ⊢ Ⅎ𝑤𝑀 | |
39 | nfrab1 3445 | . . . 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 2895 | . . 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 5144 | . . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ↔ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) | |
44 | 38, 40, 41, 42, 43 | cbvrexfw 3296 | . 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 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ∃wrex 3064 {crab 3426 ⊆ wss 3943 ∅c0 4317 ∪ ciun 4990 class class class wbr 5141 ↦ cmpt 5224 × cxp 5667 ∘ ccom 5673 ‘cfv 6536 (class class class)co 7404 ↑m cmap 8819 Xcixp 8890 Fincfn 8938 infcinf 9435 ℝcr 11108 0cc0 11109 +∞cpnf 11246 ℝ*cxr 11248 < clt 11249 ≤ cle 11250 ℕcn 12213 ℝ+crp 12977 +𝑒 cxad 13093 [,)cico 13329 [,]cicc 13330 ∏cprod 15852 volcvol 25342 Σ^csumge0 45632 voln*covoln 45806 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-rlim 15436 df-sum 15636 df-prod 15853 df-rest 17374 df-topgen 17395 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-top 22746 df-topon 22763 df-bases 22799 df-cmp 23241 df-ovol 25343 df-vol 25344 df-sumge0 45633 df-ovoln 45807 |
This theorem is referenced by: ovncvrrp 45834 |
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