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Mirrors > Home > MPE Home > Th. List > Mathboxes > von0val | Structured version Visualization version GIF version |
Description: The Lebesgue measure (for the zero dimensional space of reals) of every measurable set is zero. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
von0val.1 | ⊢ (𝜑 → 𝐴 ∈ dom (voln‘∅)) |
Ref | Expression |
---|---|
von0val | ⊢ (𝜑 → ((voln‘∅)‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0fin 8954 | . . . 4 ⊢ ∅ ∈ Fin | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ Fin) |
3 | von0val.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom (voln‘∅)) | |
4 | 2, 3 | mblvon 44177 | . 2 ⊢ (𝜑 → ((voln‘∅)‘𝐴) = ((voln*‘∅)‘𝐴)) |
5 | 2, 3 | vonmblss2 44180 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m ∅)) |
6 | 5 | ovn0val 44088 | . 2 ⊢ (𝜑 → ((voln*‘∅)‘𝐴) = 0) |
7 | 4, 6 | eqtrd 2778 | 1 ⊢ (𝜑 → ((voln‘∅)‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∅c0 4256 dom cdm 5589 ‘cfv 6433 Fincfn 8733 0cc0 10871 voln*covoln 44074 volncvoln 44076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-ac2 10219 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-acn 9700 df-ac 9872 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-prod 15616 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-rest 17133 df-0g 17152 df-topgen 17154 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-subg 18752 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-top 22043 df-topon 22060 df-bases 22096 df-cmp 22538 df-ovol 24628 df-vol 24629 df-sumge0 43901 df-ome 44028 df-caragen 44030 df-ovoln 44075 df-voln 44077 |
This theorem is referenced by: vonhoire 44210 vonioo 44220 vonicc 44223 vonsn 44229 |
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