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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 33612 | . . . 4 β’ β―:VβΆ(0[,]+β) | |
| 2 | ssv 4001 | . . . 4 β’ π β V | |
| 3 | fssres 6751 | . . . 4 β’ ((β―:VβΆ(0[,]+β) β§ π β V) β (β― βΎ π):πβΆ(0[,]+β)) | |
| 4 | 1, 2, 3 | mp2an 689 | . . 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 33642 | . . . 4 β’ (π β βͺ ran sigAlgebra β β β π) | |
| 7 | fvres 6904 | . . . 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 14332 | . . 3 β’ (β―ββ ) = 0 | |
| 10 | 8, 9 | eqtrdi 2782 | . 2 β’ (π β βͺ ran sigAlgebra β ((β― βΎ π)ββ ) = 0) |
| 11 | vex 3472 | . . . . . . 7 β’ π₯ β V | |
| 12 | hasheuni 33613 | . . . . . . 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 687 | . . . . . 6 β’ (Disj π¦ β π₯ π¦ β (β―ββͺ π₯) = Ξ£*π¦ β π₯(β―βπ¦)) |
| 14 | 13 | ad2antll 726 | . . . . 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 33655 | . . . . . . . . . . 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 495 | . . . . . . . . . 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 1141 | . . . . . . . . 9 β’ (π β βͺ ran sigAlgebra β βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π)) |
| 18 | fvres 6904 | . . . . . . . . . . 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 3077 | . . . . . . . . 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 3242 | . . . . . . 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 406 | . . . . . 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 714 | . . . . 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 4604 | . . . . . . . . . 10 β’ (π₯ β π« π β π₯ β π) | |
| 26 | 25 | sseld 3976 | . . . . . . . . 9 β’ (π₯ β π« π β (π¦ β π₯ β π¦ β π)) |
| 27 | fvres 6904 | . . . . . . . . 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 406 | . . . . . . 7 β’ ((π₯ β π« π β§ π¦ β π₯) β ((β― βΎ π)βπ¦) = (β―βπ¦)) |
| 30 | 29 | esumeq2dv 33566 | . . . . . 6 β’ (π₯ β π« π β Ξ£*π¦ β π₯((β― βΎ π)βπ¦) = Ξ£*π¦ β π₯(β―βπ¦)) |
| 31 | 30 | ad2antlr 724 | . . . . 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 2776 | . . . 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 412 | . . 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 3140 | . 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 33727 | . 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 1339 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 Vcvv 3468 β cdif 3940 β wss 3943 β c0 4317 π« cpw 4597 βͺ cuni 4902 Disj wdisj 5106 class class class wbr 5141 ran crn 5670 βΎ cres 5671 βΆwf 6533 βcfv 6537 (class class class)co 7405 Οcom 7852 βΌ cdom 8939 0cc0 11112 +βcpnf 11249 [,]cicc 13333 β―chash 14295 Ξ£*cesum 33555 sigAlgebracsiga 33636 measurescmeas 33723 |
| 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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
| 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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-ef 16017 df-sin 16019 df-cos 16020 df-pi 16022 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-ordt 17456 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-ps 18531 df-tsr 18532 df-plusf 18572 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-subrng 20446 df-subrg 20471 df-abv 20660 df-lmod 20708 df-scaf 20709 df-sra 21021 df-rgmod 21022 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-tmd 23931 df-tgp 23932 df-tsms 23986 df-trg 24019 df-xms 24181 df-ms 24182 df-tms 24183 df-nm 24446 df-ngp 24447 df-nrg 24449 df-nlm 24450 df-ii 24752 df-cncf 24753 df-limc 25750 df-dv 25751 df-log 26445 df-esum 33556 df-siga 33637 df-meas 33724 |
| This theorem is referenced by: pwcntmeas 33755 |
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