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| Mirrors > Home > MPE Home > Th. List > Mathboxes > isvonmbl | Structured version Visualization version GIF version | ||
| Description: The predicate "π΄ is measurable w.r.t. the n-dimensional Lebesgue measure". A set is measurable if it splits every other set π₯ in a "nice" way, that is, if the measure of the pieces π₯ β© π΄ and π₯ β π΄ sum up to the measure of π₯. Definition 114E of [Fremlin1] p. 25. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
| Ref | Expression |
|---|---|
| isvonmbl.1 | β’ (π β π β Fin) |
| Ref | Expression |
|---|---|
| isvonmbl | β’ (π β (πΈ β dom (volnβπ) β (πΈ β (β βm π) β§ βπ β π« (β βm π)(((voln*βπ)β(π β© πΈ)) +π ((voln*βπ)β(π β πΈ))) = ((voln*βπ)βπ)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isvonmbl.1 | . . . 4 β’ (π β π β Fin) | |
| 2 | 1 | dmvon 45322 | . . 3 β’ (π β dom (volnβπ) = (CaraGenβ(voln*βπ))) |
| 3 | 2 | eleq2d 2820 | . 2 β’ (π β (πΈ β dom (volnβπ) β πΈ β (CaraGenβ(voln*βπ)))) |
| 4 | 1 | ovnome 45289 | . . 3 β’ (π β (voln*βπ) β OutMeas) |
| 5 | eqid 2733 | . . 3 β’ (CaraGenβ(voln*βπ)) = (CaraGenβ(voln*βπ)) | |
| 6 | 4, 5 | caragenel 45211 | . 2 β’ (π β (πΈ β (CaraGenβ(voln*βπ)) β (πΈ β π« βͺ dom (voln*βπ) β§ βπ β π« βͺ dom (voln*βπ)(((voln*βπ)β(π β© πΈ)) +π ((voln*βπ)β(π β πΈ))) = ((voln*βπ)βπ)))) |
| 7 | elpwi 4610 | . . . . . . 7 β’ (πΈ β π« βͺ dom (voln*βπ) β πΈ β βͺ dom (voln*βπ)) | |
| 8 | 7 | adantl 483 | . . . . . 6 β’ ((π β§ πΈ β π« βͺ dom (voln*βπ)) β πΈ β βͺ dom (voln*βπ)) |
| 9 | 1 | unidmovn 45329 | . . . . . . 7 β’ (π β βͺ dom (voln*βπ) = (β βm π)) |
| 10 | 9 | adantr 482 | . . . . . 6 β’ ((π β§ πΈ β π« βͺ dom (voln*βπ)) β βͺ dom (voln*βπ) = (β βm π)) |
| 11 | 8, 10 | sseqtrd 4023 | . . . . 5 β’ ((π β§ πΈ β π« βͺ dom (voln*βπ)) β πΈ β (β βm π)) |
| 12 | 11 | ex 414 | . . . 4 β’ (π β (πΈ β π« βͺ dom (voln*βπ) β πΈ β (β βm π))) |
| 13 | simpr 486 | . . . . . . 7 β’ ((π β§ πΈ β (β βm π)) β πΈ β (β βm π)) | |
| 14 | 9 | eqcomd 2739 | . . . . . . . 8 β’ (π β (β βm π) = βͺ dom (voln*βπ)) |
| 15 | 14 | adantr 482 | . . . . . . 7 β’ ((π β§ πΈ β (β βm π)) β (β βm π) = βͺ dom (voln*βπ)) |
| 16 | 13, 15 | sseqtrd 4023 | . . . . . 6 β’ ((π β§ πΈ β (β βm π)) β πΈ β βͺ dom (voln*βπ)) |
| 17 | ovex 7442 | . . . . . . . . 9 β’ (β βm π) β V | |
| 18 | 17 | ssex 5322 | . . . . . . . 8 β’ (πΈ β (β βm π) β πΈ β V) |
| 19 | elpwg 4606 | . . . . . . . 8 β’ (πΈ β V β (πΈ β π« βͺ dom (voln*βπ) β πΈ β βͺ dom (voln*βπ))) | |
| 20 | 18, 19 | syl 17 | . . . . . . 7 β’ (πΈ β (β βm π) β (πΈ β π« βͺ dom (voln*βπ) β πΈ β βͺ dom (voln*βπ))) |
| 21 | 20 | adantl 483 | . . . . . 6 β’ ((π β§ πΈ β (β βm π)) β (πΈ β π« βͺ dom (voln*βπ) β πΈ β βͺ dom (voln*βπ))) |
| 22 | 16, 21 | mpbird 257 | . . . . 5 β’ ((π β§ πΈ β (β βm π)) β πΈ β π« βͺ dom (voln*βπ)) |
| 23 | 22 | ex 414 | . . . 4 β’ (π β (πΈ β (β βm π) β πΈ β π« βͺ dom (voln*βπ))) |
| 24 | 12, 23 | impbid 211 | . . 3 β’ (π β (πΈ β π« βͺ dom (voln*βπ) β πΈ β (β βm π))) |
| 25 | 9 | pweqd 4620 | . . . 4 β’ (π β π« βͺ dom (voln*βπ) = π« (β βm π)) |
| 26 | raleq 3323 | . . . 4 β’ (π« βͺ dom (voln*βπ) = π« (β βm π) β (βπ β π« βͺ dom (voln*βπ)(((voln*βπ)β(π β© πΈ)) +π ((voln*βπ)β(π β πΈ))) = ((voln*βπ)βπ) β βπ β π« (β βm π)(((voln*βπ)β(π β© πΈ)) +π ((voln*βπ)β(π β πΈ))) = ((voln*βπ)βπ))) | |
| 27 | 25, 26 | syl 17 | . . 3 β’ (π β (βπ β π« βͺ dom (voln*βπ)(((voln*βπ)β(π β© πΈ)) +π ((voln*βπ)β(π β πΈ))) = ((voln*βπ)βπ) β βπ β π« (β βm π)(((voln*βπ)β(π β© πΈ)) +π ((voln*βπ)β(π β πΈ))) = ((voln*βπ)βπ))) |
| 28 | 24, 27 | anbi12d 632 | . 2 β’ (π β ((πΈ β π« βͺ dom (voln*βπ) β§ βπ β π« βͺ dom (voln*βπ)(((voln*βπ)β(π β© πΈ)) +π ((voln*βπ)β(π β πΈ))) = ((voln*βπ)βπ)) β (πΈ β (β βm π) β§ βπ β π« (β βm π)(((voln*βπ)β(π β© πΈ)) +π ((voln*βπ)β(π β πΈ))) = ((voln*βπ)βπ)))) |
| 29 | 3, 6, 28 | 3bitrd 305 | 1 β’ (π β (πΈ β dom (volnβπ) β (πΈ β (β βm π) β§ βπ β π« (β βm π)(((voln*βπ)β(π β© πΈ)) +π ((voln*βπ)β(π β πΈ))) = ((voln*βπ)βπ)))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3062 Vcvv 3475 β cdif 3946 β© cin 3948 β wss 3949 π« cpw 4603 βͺ cuni 4909 dom cdm 5677 βcfv 6544 (class class class)co 7409 βm cmap 8820 Fincfn 8939 βcr 11109 +π cxad 13090 CaraGenccaragen 45207 voln*covoln 45252 volncvoln 45254 |
| 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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-ac2 10458 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-ac 10111 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-prod 15850 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-rest 17368 df-0g 17387 df-topgen 17389 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-subg 19003 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-drng 20359 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-cnfld 20945 df-top 22396 df-topon 22413 df-bases 22449 df-cmp 22891 df-ovol 24981 df-vol 24982 df-sumge0 45079 df-ome 45206 df-caragen 45208 df-ovoln 45253 df-voln 45255 |
| This theorem is referenced by: vonvolmbl 45377 |
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