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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 45312 | . . 3 β’ (π β dom (volnβπ) = (CaraGenβ(voln*βπ))) |
| 3 | 2 | eleq2d 2819 | . 2 β’ (π β (πΈ β dom (volnβπ) β πΈ β (CaraGenβ(voln*βπ)))) |
| 4 | 1 | ovnome 45279 | . . 3 β’ (π β (voln*βπ) β OutMeas) |
| 5 | eqid 2732 | . . 3 β’ (CaraGenβ(voln*βπ)) = (CaraGenβ(voln*βπ)) | |
| 6 | 4, 5 | caragenel 45201 | . 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 4609 | . . . . . . 7 β’ (πΈ β π« βͺ dom (voln*βπ) β πΈ β βͺ dom (voln*βπ)) | |
| 8 | 7 | adantl 482 | . . . . . 6 β’ ((π β§ πΈ β π« βͺ dom (voln*βπ)) β πΈ β βͺ dom (voln*βπ)) |
| 9 | 1 | unidmovn 45319 | . . . . . . 7 β’ (π β βͺ dom (voln*βπ) = (β βm π)) |
| 10 | 9 | adantr 481 | . . . . . 6 β’ ((π β§ πΈ β π« βͺ dom (voln*βπ)) β βͺ dom (voln*βπ) = (β βm π)) |
| 11 | 8, 10 | sseqtrd 4022 | . . . . 5 β’ ((π β§ πΈ β π« βͺ dom (voln*βπ)) β πΈ β (β βm π)) |
| 12 | 11 | ex 413 | . . . 4 β’ (π β (πΈ β π« βͺ dom (voln*βπ) β πΈ β (β βm π))) |
| 13 | simpr 485 | . . . . . . 7 β’ ((π β§ πΈ β (β βm π)) β πΈ β (β βm π)) | |
| 14 | 9 | eqcomd 2738 | . . . . . . . 8 β’ (π β (β βm π) = βͺ dom (voln*βπ)) |
| 15 | 14 | adantr 481 | . . . . . . 7 β’ ((π β§ πΈ β (β βm π)) β (β βm π) = βͺ dom (voln*βπ)) |
| 16 | 13, 15 | sseqtrd 4022 | . . . . . 6 β’ ((π β§ πΈ β (β βm π)) β πΈ β βͺ dom (voln*βπ)) |
| 17 | ovex 7441 | . . . . . . . . 9 β’ (β βm π) β V | |
| 18 | 17 | ssex 5321 | . . . . . . . 8 β’ (πΈ β (β βm π) β πΈ β V) |
| 19 | elpwg 4605 | . . . . . . . 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 482 | . . . . . 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 256 | . . . . 5 β’ ((π β§ πΈ β (β βm π)) β πΈ β π« βͺ dom (voln*βπ)) |
| 23 | 22 | ex 413 | . . . 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 4619 | . . . 4 β’ (π β π« βͺ dom (voln*βπ) = π« (β βm π)) |
| 26 | raleq 3322 | . . . 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 631 | . 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 304 | 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 396 = wceq 1541 β wcel 2106 βwral 3061 Vcvv 3474 β cdif 3945 β© cin 3947 β wss 3948 π« cpw 4602 βͺ cuni 4908 dom cdm 5676 βcfv 6543 (class class class)co 7408 βm cmap 8819 Fincfn 8938 βcr 11108 +π cxad 13089 CaraGenccaragen 45197 voln*covoln 45242 volncvoln 45244 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cc 10429 ax-ac2 10457 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-acn 9936 df-ac 10110 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-seq 13966 df-exp 14027 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-clim 15431 df-rlim 15432 df-sum 15632 df-prod 15849 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-rest 17367 df-0g 17386 df-topgen 17388 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-minusg 18822 df-subg 19002 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-dvr 20214 df-drng 20358 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-cnfld 20944 df-top 22395 df-topon 22412 df-bases 22448 df-cmp 22890 df-ovol 24980 df-vol 24981 df-sumge0 45069 df-ome 45196 df-caragen 45198 df-ovoln 45243 df-voln 45245 |
| This theorem is referenced by: vonvolmbl 45367 |
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