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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cntmeas | Structured version Visualization version GIF version | ||
| Description: The Counting measure is a measure on any sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
| Ref | Expression |
|---|---|
| cntmeas | β’ (π β βͺ ran sigAlgebra β (β― βΎ π) β (measuresβπ)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hashf2 33070 | . . . 4 β’ β―:VβΆ(0[,]+β) | |
| 2 | ssv 4005 | . . . 4 β’ π β V | |
| 3 | fssres 6754 | . . . 4 β’ ((β―:VβΆ(0[,]+β) β§ π β V) β (β― βΎ π):πβΆ(0[,]+β)) | |
| 4 | 1, 2, 3 | mp2an 690 | . . 3 β’ (β― βΎ π):πβΆ(0[,]+β) |
| 5 | 4 | a1i 11 | . 2 β’ (π β βͺ ran sigAlgebra β (β― βΎ π):πβΆ(0[,]+β)) |
| 6 | 0elsiga 33100 | . . . 4 β’ (π β βͺ ran sigAlgebra β β β π) | |
| 7 | fvres 6907 | . . . 4 β’ (β β π β ((β― βΎ π)ββ ) = (β―ββ )) | |
| 8 | 6, 7 | syl 17 | . . 3 β’ (π β βͺ ran sigAlgebra β ((β― βΎ π)ββ ) = (β―ββ )) |
| 9 | hash0 14323 | . . 3 β’ (β―ββ ) = 0 | |
| 10 | 8, 9 | eqtrdi 2788 | . 2 β’ (π β βͺ ran sigAlgebra β ((β― βΎ π)ββ ) = 0) |
| 11 | vex 3478 | . . . . . . 7 β’ π₯ β V | |
| 12 | hasheuni 33071 | . . . . . . 7 β’ ((π₯ β V β§ Disj π¦ β π₯ π¦) β (β―ββͺ π₯) = Ξ£*π¦ β π₯(β―βπ¦)) | |
| 13 | 11, 12 | mpan 688 | . . . . . 6 β’ (Disj π¦ β π₯ π¦ β (β―ββͺ π₯) = Ξ£*π¦ β π₯(β―βπ¦)) |
| 14 | 13 | ad2antll 727 | . . . . 5 β’ (((π β βͺ ran sigAlgebra β§ π₯ β π« π) β§ (π₯ βΌ Ο β§ Disj π¦ β π₯ π¦)) β (β―ββͺ π₯) = Ξ£*π¦ β π₯(β―βπ¦)) |
| 15 | isrnsigau 33113 | . . . . . . . . . . 11 β’ (π β βͺ ran sigAlgebra β (π β π« βͺ π β§ (βͺ π β π β§ βπ₯ β π (βͺ π β π₯) β π β§ βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π)))) | |
| 16 | 15 | simprd 496 | . . . . . . . . . 10 β’ (π β βͺ ran sigAlgebra β (βͺ π β π β§ βπ₯ β π (βͺ π β π₯) β π β§ βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π))) |
| 17 | 16 | simp3d 1144 | . . . . . . . . 9 β’ (π β βͺ ran sigAlgebra β βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π)) |
| 18 | fvres 6907 | . . . . . . . . . . 11 β’ (βͺ π₯ β π β ((β― βΎ π)ββͺ π₯) = (β―ββͺ π₯)) | |
| 19 | 18 | imim2i 16 | . . . . . . . . . 10 β’ ((π₯ βΌ Ο β βͺ π₯ β π) β (π₯ βΌ Ο β ((β― βΎ π)ββͺ π₯) = (β―ββͺ π₯))) |
| 20 | 19 | ralimi 3083 | . . . . . . . . 9 β’ (βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π) β βπ₯ β π« π(π₯ βΌ Ο β ((β― βΎ π)ββͺ π₯) = (β―ββͺ π₯))) |
| 21 | 17, 20 | syl 17 | . . . . . . . 8 β’ (π β βͺ ran sigAlgebra β βπ₯ β π« π(π₯ βΌ Ο β ((β― βΎ π)ββͺ π₯) = (β―ββͺ π₯))) |
| 22 | 21 | r19.21bi 3248 | . . . . . . 7 β’ ((π β βͺ ran sigAlgebra β§ π₯ β π« π) β (π₯ βΌ Ο β ((β― βΎ π)ββͺ π₯) = (β―ββͺ π₯))) |
| 23 | 22 | imp 407 | . . . . . 6 β’ (((π β βͺ ran sigAlgebra β§ π₯ β π« π) β§ π₯ βΌ Ο) β ((β― βΎ π)ββͺ π₯) = (β―ββͺ π₯)) |
| 24 | 23 | adantrr 715 | . . . . 5 β’ (((π β βͺ ran sigAlgebra β§ π₯ β π« π) β§ (π₯ βΌ Ο β§ Disj π¦ β π₯ π¦)) β ((β― βΎ π)ββͺ π₯) = (β―ββͺ π₯)) |
| 25 | elpwi 4608 | . . . . . . . . . 10 β’ (π₯ β π« π β π₯ β π) | |
| 26 | 25 | sseld 3980 | . . . . . . . . 9 β’ (π₯ β π« π β (π¦ β π₯ β π¦ β π)) |
| 27 | fvres 6907 | . . . . . . . . 9 β’ (π¦ β π β ((β― βΎ π)βπ¦) = (β―βπ¦)) | |
| 28 | 26, 27 | syl6 35 | . . . . . . . 8 β’ (π₯ β π« π β (π¦ β π₯ β ((β― βΎ π)βπ¦) = (β―βπ¦))) |
| 29 | 28 | imp 407 | . . . . . . 7 β’ ((π₯ β π« π β§ π¦ β π₯) β ((β― βΎ π)βπ¦) = (β―βπ¦)) |
| 30 | 29 | esumeq2dv 33024 | . . . . . 6 β’ (π₯ β π« π β Ξ£*π¦ β π₯((β― βΎ π)βπ¦) = Ξ£*π¦ β π₯(β―βπ¦)) |
| 31 | 30 | ad2antlr 725 | . . . . 5 β’ (((π β βͺ ran sigAlgebra β§ π₯ β π« π) β§ (π₯ βΌ Ο β§ Disj π¦ β π₯ π¦)) β Ξ£*π¦ β π₯((β― βΎ π)βπ¦) = Ξ£*π¦ β π₯(β―βπ¦)) |
| 32 | 14, 24, 31 | 3eqtr4d 2782 | . . . 4 β’ (((π β βͺ ran sigAlgebra β§ π₯ β π« π) β§ (π₯ βΌ Ο β§ Disj π¦ β π₯ π¦)) β ((β― βΎ π)ββͺ π₯) = Ξ£*π¦ β π₯((β― βΎ π)βπ¦)) |
| 33 | 32 | ex 413 | . . 3 β’ ((π β βͺ ran sigAlgebra β§ π₯ β π« π) β ((π₯ βΌ Ο β§ Disj π¦ β π₯ π¦) β ((β― βΎ π)ββͺ π₯) = Ξ£*π¦ β π₯((β― βΎ π)βπ¦))) |
| 34 | 33 | ralrimiva 3146 | . 2 β’ (π β βͺ ran sigAlgebra β βπ₯ β π« π((π₯ βΌ Ο β§ Disj π¦ β π₯ π¦) β ((β― βΎ π)ββͺ π₯) = Ξ£*π¦ β π₯((β― βΎ π)βπ¦))) |
| 35 | ismeas 33185 | . 2 β’ (π β βͺ ran sigAlgebra β ((β― βΎ π) β (measuresβπ) β ((β― βΎ π):πβΆ(0[,]+β) β§ ((β― βΎ π)ββ ) = 0 β§ βπ₯ β π« π((π₯ βΌ Ο β§ Disj π¦ β π₯ π¦) β ((β― βΎ π)ββͺ π₯) = Ξ£*π¦ β π₯((β― βΎ π)βπ¦))))) | |
| 36 | 5, 10, 34, 35 | mpbir3and 1342 | 1 β’ (π β βͺ ran sigAlgebra β (β― βΎ π) β (measuresβπ)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3061 Vcvv 3474 β cdif 3944 β wss 3947 β c0 4321 π« cpw 4601 βͺ cuni 4907 Disj wdisj 5112 class class class wbr 5147 ran crn 5676 βΎ cres 5677 βΆwf 6536 βcfv 6540 (class class class)co 7405 Οcom 7851 βΌ cdom 8933 0cc0 11106 +βcpnf 11241 [,]cicc 13323 β―chash 14286 Ξ£*cesum 33013 sigAlgebracsiga 33094 measurescmeas 33181 |
| 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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
| 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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-ordt 17443 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-ps 18515 df-tsr 18516 df-plusf 18556 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-subrg 20353 df-abv 20417 df-lmod 20465 df-scaf 20466 df-sra 20777 df-rgmod 20778 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-tmd 23567 df-tgp 23568 df-tsms 23622 df-trg 23655 df-xms 23817 df-ms 23818 df-tms 23819 df-nm 24082 df-ngp 24083 df-nrg 24085 df-nlm 24086 df-ii 24384 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-esum 33014 df-siga 33095 df-meas 33182 |
| This theorem is referenced by: pwcntmeas 33213 |
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