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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovnssle | Structured version Visualization version GIF version |
Description: The (multidimensional) Lebesgue outer measure of a subset is less than the L.o.m. of the whole set. This is step (iii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
ovnssle.1 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
ovnssle.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
ovnssle.3 | ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑m 𝑋)) |
Ref | Expression |
---|---|
ovnssle | ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0le0 11726 | . . . 4 ⊢ 0 ≤ 0 | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤ 0) |
3 | fveq2 6645 | . . . . . . 7 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅)) | |
4 | 3 | fveq1d 6647 | . . . . . 6 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘𝐴) = ((voln*‘∅)‘𝐴)) |
5 | 4 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = ((voln*‘∅)‘𝐴)) |
6 | ovnssle.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
7 | 6 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ 𝐵) |
8 | ovnssle.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑m 𝑋)) | |
9 | 8 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵 ⊆ (ℝ ↑m 𝑋)) |
10 | simpr 488 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅) | |
11 | 10 | oveq2d 7151 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑m 𝑋) = (ℝ ↑m ∅)) |
12 | 9, 11 | sseqtrd 3955 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵 ⊆ (ℝ ↑m ∅)) |
13 | 7, 12 | sstrd 3925 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ (ℝ ↑m ∅)) |
14 | 13 | ovn0val 43189 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐴) = 0) |
15 | 5, 14 | eqtrd 2833 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = 0) |
16 | 3 | fveq1d 6647 | . . . . . 6 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘𝐵) = ((voln*‘∅)‘𝐵)) |
17 | 16 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐵) = ((voln*‘∅)‘𝐵)) |
18 | 12 | ovn0val 43189 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐵) = 0) |
19 | 17, 18 | eqtrd 2833 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐵) = 0) |
20 | 15, 19 | breq12d 5043 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵) ↔ 0 ≤ 0)) |
21 | 2, 20 | mpbird 260 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) |
22 | ovnssle.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
23 | 22 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin) |
24 | neqne 2995 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
25 | 24 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
26 | 6 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴 ⊆ 𝐵) |
27 | 8 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵 ⊆ (ℝ ↑m 𝑋)) |
28 | eqid 2798 | . . 3 ⊢ {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)))))} = {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)))))} | |
29 | eqid 2798 | . . 3 ⊢ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐵 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐵 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} | |
30 | 23, 25, 26, 27, 28, 29 | ovnsslelem 43199 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) |
31 | 21, 30 | pm2.61dan 812 | 1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 {crab 3110 ⊆ wss 3881 ∅c0 4243 ∪ ciun 4881 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 Xcixp 8444 Fincfn 8492 ℝcr 10525 0cc0 10526 ℝ*cxr 10663 ≤ cle 10665 ℕcn 11625 [,)cico 12728 ∏cprod 15251 volcvol 24067 Σ^csumge0 43001 voln*covoln 43175 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-seq 13365 df-prod 15252 df-ovoln 43176 |
This theorem is referenced by: ovnome 43212 hspmbllem3 43267 |
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