![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > measinb2 | Structured version Visualization version GIF version |
Description: Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
measinb2 | β’ ((π β (measuresβπ) β§ π΄ β π) β (π₯ β (π β© π« π΄) β¦ (πβ(π₯ β© π΄))) β (measuresβ(π β© π« π΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resmpt3 6029 | . . 3 β’ ((π₯ β π β¦ (πβ(π₯ β© π΄))) βΎ (π β© π« π΄)) = (π₯ β (π β© (π β© π« π΄)) β¦ (πβ(π₯ β© π΄))) | |
2 | inin 32225 | . . . 4 β’ (π β© (π β© π« π΄)) = (π β© π« π΄) | |
3 | eqid 2724 | . . . 4 β’ (πβ(π₯ β© π΄)) = (πβ(π₯ β© π΄)) | |
4 | 2, 3 | mpteq12i 5245 | . . 3 β’ (π₯ β (π β© (π β© π« π΄)) β¦ (πβ(π₯ β© π΄))) = (π₯ β (π β© π« π΄) β¦ (πβ(π₯ β© π΄))) |
5 | 1, 4 | eqtri 2752 | . 2 β’ ((π₯ β π β¦ (πβ(π₯ β© π΄))) βΎ (π β© π« π΄)) = (π₯ β (π β© π« π΄) β¦ (πβ(π₯ β© π΄))) |
6 | measinb 33711 | . . 3 β’ ((π β (measuresβπ) β§ π΄ β π) β (π₯ β π β¦ (πβ(π₯ β© π΄))) β (measuresβπ)) | |
7 | measbase 33687 | . . . 4 β’ (π β (measuresβπ) β π β βͺ ran sigAlgebra) | |
8 | sigainb 33626 | . . . . 5 β’ ((π β βͺ ran sigAlgebra β§ π΄ β π) β (π β© π« π΄) β (sigAlgebraβπ΄)) | |
9 | elrnsiga 33616 | . . . . 5 β’ ((π β© π« π΄) β (sigAlgebraβπ΄) β (π β© π« π΄) β βͺ ran sigAlgebra) | |
10 | 8, 9 | syl 17 | . . . 4 β’ ((π β βͺ ran sigAlgebra β§ π΄ β π) β (π β© π« π΄) β βͺ ran sigAlgebra) |
11 | 7, 10 | sylan 579 | . . 3 β’ ((π β (measuresβπ) β§ π΄ β π) β (π β© π« π΄) β βͺ ran sigAlgebra) |
12 | inss1 4221 | . . . 4 β’ (π β© π« π΄) β π | |
13 | 12 | a1i 11 | . . 3 β’ ((π β (measuresβπ) β§ π΄ β π) β (π β© π« π΄) β π) |
14 | measres 33712 | . . 3 β’ (((π₯ β π β¦ (πβ(π₯ β© π΄))) β (measuresβπ) β§ (π β© π« π΄) β βͺ ran sigAlgebra β§ (π β© π« π΄) β π) β ((π₯ β π β¦ (πβ(π₯ β© π΄))) βΎ (π β© π« π΄)) β (measuresβ(π β© π« π΄))) | |
15 | 6, 11, 13, 14 | syl3anc 1368 | . 2 β’ ((π β (measuresβπ) β§ π΄ β π) β ((π₯ β π β¦ (πβ(π₯ β© π΄))) βΎ (π β© π« π΄)) β (measuresβ(π β© π« π΄))) |
16 | 5, 15 | eqeltrrid 2830 | 1 β’ ((π β (measuresβπ) β§ π΄ β π) β (π₯ β (π β© π« π΄) β¦ (πβ(π₯ β© π΄))) β (measuresβ(π β© π« π΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β wcel 2098 β© cin 3940 β wss 3941 π« cpw 4595 βͺ cuni 4900 β¦ cmpt 5222 ran crn 5668 βΎ cres 5669 βcfv 6534 sigAlgebracsiga 33598 measurescmeas 33685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-ac2 10455 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-disj 5105 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-dju 9893 df-card 9931 df-acn 9934 df-ac 10108 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-q 12931 df-rp 12973 df-xneg 13090 df-xadd 13091 df-xmul 13092 df-ioo 13326 df-ioc 13327 df-ico 13328 df-icc 13329 df-fz 13483 df-fzo 13626 df-fl 13755 df-mod 13833 df-seq 13965 df-exp 14026 df-fac 14232 df-bc 14261 df-hash 14289 df-shft 15012 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15631 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-starv 17213 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-unif 17221 df-hom 17222 df-cco 17223 df-rest 17369 df-topn 17370 df-0g 17388 df-gsum 17389 df-topgen 17390 df-pt 17391 df-prds 17394 df-ordt 17448 df-xrs 17449 df-qtop 17454 df-imas 17455 df-xps 17457 df-mre 17531 df-mrc 17532 df-acs 17534 df-ps 18523 df-tsr 18524 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18988 df-subg 19042 df-cntz 19225 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-cring 20133 df-subrng 20438 df-subrg 20463 df-abv 20652 df-lmod 20700 df-scaf 20701 df-sra 21013 df-rgmod 21014 df-psmet 21222 df-xmet 21223 df-met 21224 df-bl 21225 df-mopn 21226 df-fbas 21227 df-fg 21228 df-cnfld 21231 df-top 22720 df-topon 22737 df-topsp 22759 df-bases 22773 df-cld 22847 df-ntr 22848 df-cls 22849 df-nei 22926 df-lp 22964 df-perf 22965 df-cn 23055 df-cnp 23056 df-haus 23143 df-tx 23390 df-hmeo 23583 df-fil 23674 df-fm 23766 df-flim 23767 df-flf 23768 df-tmd 23900 df-tgp 23901 df-tsms 23955 df-trg 23988 df-xms 24150 df-ms 24151 df-tms 24152 df-nm 24415 df-ngp 24416 df-nrg 24418 df-nlm 24419 df-ii 24721 df-cncf 24722 df-limc 25719 df-dv 25720 df-log 26409 df-esum 33518 df-siga 33599 df-meas 33686 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |