Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 6672 | . . . . . . 7 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅)) | |
4 | 3 | fveq1d 6674 | . . . . . 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 7174 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑m 𝑋) = (ℝ ↑m ∅)) |
12 | 9, 11 | sseqtrd 4009 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵 ⊆ (ℝ ↑m ∅)) |
13 | 7, 12 | sstrd 3979 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ (ℝ ↑m ∅)) |
14 | 13 | ovn0val 42839 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐴) = 0) |
15 | 5, 14 | eqtrd 2858 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = 0) |
16 | 3 | fveq1d 6674 | . . . . . 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 2858 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐵) = 0) |
20 | 15, 19 | breq12d 5081 | . . 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 3026 | . . . 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 2823 | . . 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 2823 | . . 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 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 {crab 3144 ⊆ wss 3938 ∅c0 4293 ∪ ciun 4921 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ∘ ccom 5561 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 Xcixp 8463 Fincfn 8511 ℝcr 10538 0cc0 10539 ℝ*cxr 10676 ≤ cle 10678 ℕcn 11640 [,)cico 12743 ∏cprod 15261 volcvol 24066 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 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-seq 13373 df-prod 15262 df-ovoln 42826 |
This theorem is referenced by: ovnome 42862 hspmbllem3 42917 |
Copyright terms: Public domain | W3C validator |