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 11110 | . . . 4 ⊢ 0 ≤ 0 | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤ 0) |
3 | fveq2 6191 | . . . . . . 7 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅)) | |
4 | 3 | fveq1d 6193 | . . . . . 6 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘𝐴) = ((voln*‘∅)‘𝐴)) |
5 | 4 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = ((voln*‘∅)‘𝐴)) |
6 | ovnssle.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
7 | 6 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ 𝐵) |
8 | ovnssle.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑𝑚 𝑋)) | |
9 | 8 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵 ⊆ (ℝ ↑𝑚 𝑋)) |
10 | simpr 477 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅) | |
11 | 10 | oveq2d 6666 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅)) |
12 | 9, 11 | sseqtrd 3641 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵 ⊆ (ℝ ↑𝑚 ∅)) |
13 | 7, 12 | sstrd 3613 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ (ℝ ↑𝑚 ∅)) |
14 | 13 | ovn0val 40764 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐴) = 0) |
15 | 5, 14 | eqtrd 2656 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = 0) |
16 | 3 | fveq1d 6193 | . . . . . 6 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘𝐵) = ((voln*‘∅)‘𝐵)) |
17 | 16 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐵) = ((voln*‘∅)‘𝐵)) |
18 | 12 | ovn0val 40764 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐵) = 0) |
19 | 17, 18 | eqtrd 2656 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐵) = 0) |
20 | 15, 19 | breq12d 4666 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵) ↔ 0 ≤ 0)) |
21 | 2, 20 | mpbird 247 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) |
22 | ovnssle.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
23 | 22 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin) |
24 | neqne 2802 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
25 | 24 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
26 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴 ⊆ 𝐵) |
27 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵 ⊆ (ℝ ↑𝑚 𝑋)) |
28 | eqid 2622 | . . 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 2622 | . . 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 40774 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) |
31 | 21, 30 | pm2.61dan 832 | 1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 {crab 2916 ⊆ wss 3574 ∅c0 3915 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Xcixp 7908 Fincfn 7955 ℝcr 9935 0cc0 9936 ℝ*cxr 10073 ≤ cle 10075 ℕcn 11020 [,)cico 12177 ∏cprod 14635 volcvol 23232 Σ^csumge0 40579 voln*covoln 40750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-seq 12802 df-prod 14636 df-ovoln 40751 |
This theorem is referenced by: ovnome 40787 hspmbllem3 40842 |
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