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 11741 | . . . 4 ⊢ 0 ≤ 0 | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤ 0) |
3 | fveq2 6673 | . . . . . . 7 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅)) | |
4 | 3 | fveq1d 6675 | . . . . . 6 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘𝐴) = ((voln*‘∅)‘𝐴)) |
5 | 4 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = ((voln*‘∅)‘𝐴)) |
6 | ovnssle.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
7 | 6 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ 𝐵) |
8 | ovnssle.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑m 𝑋)) | |
9 | 8 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵 ⊆ (ℝ ↑m 𝑋)) |
10 | simpr 487 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅) | |
11 | 10 | oveq2d 7175 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑m 𝑋) = (ℝ ↑m ∅)) |
12 | 9, 11 | sseqtrd 4010 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵 ⊆ (ℝ ↑m ∅)) |
13 | 7, 12 | sstrd 3980 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ (ℝ ↑m ∅)) |
14 | 13 | ovn0val 42839 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐴) = 0) |
15 | 5, 14 | eqtrd 2859 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = 0) |
16 | 3 | fveq1d 6675 | . . . . . 6 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘𝐵) = ((voln*‘∅)‘𝐵)) |
17 | 16 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐵) = ((voln*‘∅)‘𝐵)) |
18 | 12 | ovn0val 42839 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐵) = 0) |
19 | 17, 18 | eqtrd 2859 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐵) = 0) |
20 | 15, 19 | breq12d 5082 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵) ↔ 0 ≤ 0)) |
21 | 2, 20 | mpbird 259 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) |
22 | ovnssle.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
23 | 22 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin) |
24 | neqne 3027 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
25 | 24 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
26 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴 ⊆ 𝐵) |
27 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵 ⊆ (ℝ ↑m 𝑋)) |
28 | eqid 2824 | . . 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 2824 | . . 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 42849 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) |
31 | 21, 30 | pm2.61dan 811 | 1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∃wrex 3142 {crab 3145 ⊆ wss 3939 ∅c0 4294 ∪ ciun 4922 class class class wbr 5069 ↦ cmpt 5149 × cxp 5556 ∘ ccom 5562 ‘cfv 6358 (class class class)co 7159 ↑m cmap 8409 Xcixp 8464 Fincfn 8512 ℝcr 10539 0cc0 10540 ℝ*cxr 10677 ≤ cle 10679 ℕcn 11641 [,)cico 12743 ∏cprod 15262 volcvol 24067 Σ^csumge0 42651 voln*covoln 42825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-seq 13373 df-prod 15263 df-ovoln 42826 |
This theorem is referenced by: ovnome 42862 hspmbllem3 42917 |
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