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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > totprobd | Structured version Visualization version GIF version | ||
| Description: Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
| Ref | Expression |
|---|---|
| totprobd.1 | β’ (π β π β Prob) |
| totprobd.2 | β’ (π β π΄ β dom π) |
| totprobd.3 | β’ (π β π΅ β π« dom π) |
| totprobd.4 | β’ (π β βͺ π΅ = βͺ dom π) |
| totprobd.5 | β’ (π β π΅ βΌ Ο) |
| totprobd.6 | β’ (π β Disj π β π΅ π) |
| Ref | Expression |
|---|---|
| totprobd | β’ (π β (πβπ΄) = Ξ£*π β π΅(πβ(π β© π΄))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | totprobd.2 | . . . . . 6 β’ (π β π΄ β dom π) | |
| 2 | elssuni 4934 | . . . . . 6 β’ (π΄ β dom π β π΄ β βͺ dom π) | |
| 3 | 1, 2 | syl 17 | . . . . 5 β’ (π β π΄ β βͺ dom π) |
| 4 | totprobd.4 | . . . . 5 β’ (π β βͺ π΅ = βͺ dom π) | |
| 5 | 3, 4 | sseqtrrd 4018 | . . . 4 β’ (π β π΄ β βͺ π΅) |
| 6 | sseqin2 4210 | . . . 4 β’ (π΄ β βͺ π΅ β (βͺ π΅ β© π΄) = π΄) | |
| 7 | 5, 6 | sylib 217 | . . 3 β’ (π β (βͺ π΅ β© π΄) = π΄) |
| 8 | 7 | fveq2d 6889 | . 2 β’ (π β (πβ(βͺ π΅ β© π΄)) = (πβπ΄)) |
| 9 | totprobd.1 | . . . . . 6 β’ (π β π β Prob) | |
| 10 | domprobmeas 33939 | . . . . . 6 β’ (π β Prob β π β (measuresβdom π)) | |
| 11 | 9, 10 | syl 17 | . . . . 5 β’ (π β π β (measuresβdom π)) |
| 12 | measinb 33749 | . . . . 5 β’ ((π β (measuresβdom π) β§ π΄ β dom π) β (π β dom π β¦ (πβ(π β© π΄))) β (measuresβdom π)) | |
| 13 | 11, 1, 12 | syl2anc 583 | . . . 4 β’ (π β (π β dom π β¦ (πβ(π β© π΄))) β (measuresβdom π)) |
| 14 | totprobd.3 | . . . 4 β’ (π β π΅ β π« dom π) | |
| 15 | totprobd.5 | . . . 4 β’ (π β π΅ βΌ Ο) | |
| 16 | totprobd.6 | . . . 4 β’ (π β Disj π β π΅ π) | |
| 17 | measvun 33737 | . . . 4 β’ (((π β dom π β¦ (πβ(π β© π΄))) β (measuresβdom π) β§ π΅ β π« dom π β§ (π΅ βΌ Ο β§ Disj π β π΅ π)) β ((π β dom π β¦ (πβ(π β© π΄)))ββͺ π΅) = Ξ£*π β π΅((π β dom π β¦ (πβ(π β© π΄)))βπ)) | |
| 18 | 13, 14, 15, 16, 17 | syl112anc 1371 | . . 3 β’ (π β ((π β dom π β¦ (πβ(π β© π΄)))ββͺ π΅) = Ξ£*π β π΅((π β dom π β¦ (πβ(π β© π΄)))βπ)) |
| 19 | eqidd 2727 | . . . 4 β’ (π β (π β dom π β¦ (πβ(π β© π΄))) = (π β dom π β¦ (πβ(π β© π΄)))) | |
| 20 | simpr 484 | . . . . . 6 β’ ((π β§ π = βͺ π΅) β π = βͺ π΅) | |
| 21 | 20 | ineq1d 4206 | . . . . 5 β’ ((π β§ π = βͺ π΅) β (π β© π΄) = (βͺ π΅ β© π΄)) |
| 22 | 21 | fveq2d 6889 | . . . 4 β’ ((π β§ π = βͺ π΅) β (πβ(π β© π΄)) = (πβ(βͺ π΅ β© π΄))) |
| 23 | domprobsiga 33940 | . . . . . 6 β’ (π β Prob β dom π β βͺ ran sigAlgebra) | |
| 24 | 9, 23 | syl 17 | . . . . 5 β’ (π β dom π β βͺ ran sigAlgebra) |
| 25 | sigaclcu 33645 | . . . . 5 β’ ((dom π β βͺ ran sigAlgebra β§ π΅ β π« dom π β§ π΅ βΌ Ο) β βͺ π΅ β dom π) | |
| 26 | 24, 14, 15, 25 | syl3anc 1368 | . . . 4 β’ (π β βͺ π΅ β dom π) |
| 27 | inelsiga 33663 | . . . . . 6 β’ ((dom π β βͺ ran sigAlgebra β§ βͺ π΅ β dom π β§ π΄ β dom π) β (βͺ π΅ β© π΄) β dom π) | |
| 28 | 24, 26, 1, 27 | syl3anc 1368 | . . . . 5 β’ (π β (βͺ π΅ β© π΄) β dom π) |
| 29 | prob01 33942 | . . . . 5 β’ ((π β Prob β§ (βͺ π΅ β© π΄) β dom π) β (πβ(βͺ π΅ β© π΄)) β (0[,]1)) | |
| 30 | 9, 28, 29 | syl2anc 583 | . . . 4 β’ (π β (πβ(βͺ π΅ β© π΄)) β (0[,]1)) |
| 31 | 19, 22, 26, 30 | fvmptd 6999 | . . 3 β’ (π β ((π β dom π β¦ (πβ(π β© π΄)))ββͺ π΅) = (πβ(βͺ π΅ β© π΄))) |
| 32 | eqidd 2727 | . . . . 5 β’ ((π β§ π β π΅) β (π β dom π β¦ (πβ(π β© π΄))) = (π β dom π β¦ (πβ(π β© π΄)))) | |
| 33 | simpr 484 | . . . . . . 7 β’ (((π β§ π β π΅) β§ π = π) β π = π) | |
| 34 | 33 | ineq1d 4206 | . . . . . 6 β’ (((π β§ π β π΅) β§ π = π) β (π β© π΄) = (π β© π΄)) |
| 35 | 34 | fveq2d 6889 | . . . . 5 β’ (((π β§ π β π΅) β§ π = π) β (πβ(π β© π΄)) = (πβ(π β© π΄))) |
| 36 | simpr 484 | . . . . . 6 β’ ((π β§ π β π΅) β π β π΅) | |
| 37 | 14 | adantr 480 | . . . . . 6 β’ ((π β§ π β π΅) β π΅ β π« dom π) |
| 38 | elelpwi 4607 | . . . . . 6 β’ ((π β π΅ β§ π΅ β π« dom π) β π β dom π) | |
| 39 | 36, 37, 38 | syl2anc 583 | . . . . 5 β’ ((π β§ π β π΅) β π β dom π) |
| 40 | 9 | adantr 480 | . . . . . 6 β’ ((π β§ π β π΅) β π β Prob) |
| 41 | 24 | adantr 480 | . . . . . . 7 β’ ((π β§ π β π΅) β dom π β βͺ ran sigAlgebra) |
| 42 | 1 | adantr 480 | . . . . . . 7 β’ ((π β§ π β π΅) β π΄ β dom π) |
| 43 | inelsiga 33663 | . . . . . . 7 β’ ((dom π β βͺ ran sigAlgebra β§ π β dom π β§ π΄ β dom π) β (π β© π΄) β dom π) | |
| 44 | 41, 39, 42, 43 | syl3anc 1368 | . . . . . 6 β’ ((π β§ π β π΅) β (π β© π΄) β dom π) |
| 45 | prob01 33942 | . . . . . 6 β’ ((π β Prob β§ (π β© π΄) β dom π) β (πβ(π β© π΄)) β (0[,]1)) | |
| 46 | 40, 44, 45 | syl2anc 583 | . . . . 5 β’ ((π β§ π β π΅) β (πβ(π β© π΄)) β (0[,]1)) |
| 47 | 32, 35, 39, 46 | fvmptd 6999 | . . . 4 β’ ((π β§ π β π΅) β ((π β dom π β¦ (πβ(π β© π΄)))βπ) = (πβ(π β© π΄))) |
| 48 | 47 | esumeq2dv 33566 | . . 3 β’ (π β Ξ£*π β π΅((π β dom π β¦ (πβ(π β© π΄)))βπ) = Ξ£*π β π΅(πβ(π β© π΄))) |
| 49 | 18, 31, 48 | 3eqtr3d 2774 | . 2 β’ (π β (πβ(βͺ π΅ β© π΄)) = Ξ£*π β π΅(πβ(π β© π΄))) |
| 50 | 8, 49 | eqtr3d 2768 | 1 β’ (π β (πβπ΄) = Ξ£*π β π΅(πβ(π β© π΄))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β© cin 3942 β wss 3943 π« cpw 4597 βͺ cuni 4902 Disj wdisj 5106 class class class wbr 5141 β¦ cmpt 5224 dom cdm 5669 ran crn 5670 βcfv 6537 (class class class)co 7405 Οcom 7852 βΌ cdom 8939 0cc0 11112 1c1 11113 [,]cicc 13333 Ξ£*cesum 33555 sigAlgebracsiga 33636 measurescmeas 33723 Probcprb 33936 |
| 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-ac2 10460 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-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-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 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-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 df-prob 33937 |
| This theorem is referenced by: totprob 33956 |
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