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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 4941 | . . . . . 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 4023 | . . . 4 β’ (π β π΄ β βͺ π΅) |
| 6 | sseqin2 4215 | . . . 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 6895 | . 2 β’ (π β (πβ(βͺ π΅ β© π΄)) = (πβπ΄)) |
| 9 | totprobd.1 | . . . . . 6 β’ (π β π β Prob) | |
| 10 | domprobmeas 33404 | . . . . . 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 33214 | . . . . 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 584 | . . . 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 33202 | . . . 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 1374 | . . 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 2733 | . . . 4 β’ (π β (π β dom π β¦ (πβ(π β© π΄))) = (π β dom π β¦ (πβ(π β© π΄)))) | |
| 20 | simpr 485 | . . . . . 6 β’ ((π β§ π = βͺ π΅) β π = βͺ π΅) | |
| 21 | 20 | ineq1d 4211 | . . . . 5 β’ ((π β§ π = βͺ π΅) β (π β© π΄) = (βͺ π΅ β© π΄)) |
| 22 | 21 | fveq2d 6895 | . . . 4 β’ ((π β§ π = βͺ π΅) β (πβ(π β© π΄)) = (πβ(βͺ π΅ β© π΄))) |
| 23 | domprobsiga 33405 | . . . . . 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 33110 | . . . . 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 1371 | . . . 4 β’ (π β βͺ π΅ β dom π) |
| 27 | inelsiga 33128 | . . . . . 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 1371 | . . . . 5 β’ (π β (βͺ π΅ β© π΄) β dom π) |
| 29 | prob01 33407 | . . . . 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 584 | . . . 4 β’ (π β (πβ(βͺ π΅ β© π΄)) β (0[,]1)) |
| 31 | 19, 22, 26, 30 | fvmptd 7005 | . . 3 β’ (π β ((π β dom π β¦ (πβ(π β© π΄)))ββͺ π΅) = (πβ(βͺ π΅ β© π΄))) |
| 32 | eqidd 2733 | . . . . 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 485 | . . . . . . 7 β’ (((π β§ π β π΅) β§ π = π) β π = π) | |
| 34 | 33 | ineq1d 4211 | . . . . . 6 β’ (((π β§ π β π΅) β§ π = π) β (π β© π΄) = (π β© π΄)) |
| 35 | 34 | fveq2d 6895 | . . . . 5 β’ (((π β§ π β π΅) β§ π = π) β (πβ(π β© π΄)) = (πβ(π β© π΄))) |
| 36 | simpr 485 | . . . . . 6 β’ ((π β§ π β π΅) β π β π΅) | |
| 37 | 14 | adantr 481 | . . . . . 6 β’ ((π β§ π β π΅) β π΅ β π« dom π) |
| 38 | elelpwi 4612 | . . . . . 6 β’ ((π β π΅ β§ π΅ β π« dom π) β π β dom π) | |
| 39 | 36, 37, 38 | syl2anc 584 | . . . . 5 β’ ((π β§ π β π΅) β π β dom π) |
| 40 | 9 | adantr 481 | . . . . . 6 β’ ((π β§ π β π΅) β π β Prob) |
| 41 | 24 | adantr 481 | . . . . . . 7 β’ ((π β§ π β π΅) β dom π β βͺ ran sigAlgebra) |
| 42 | 1 | adantr 481 | . . . . . . 7 β’ ((π β§ π β π΅) β π΄ β dom π) |
| 43 | inelsiga 33128 | . . . . . . 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 1371 | . . . . . 6 β’ ((π β§ π β π΅) β (π β© π΄) β dom π) |
| 45 | prob01 33407 | . . . . . 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 584 | . . . . 5 β’ ((π β§ π β π΅) β (πβ(π β© π΄)) β (0[,]1)) |
| 47 | 32, 35, 39, 46 | fvmptd 7005 | . . . 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 33031 | . . 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 2780 | . 2 β’ (π β (πβ(βͺ π΅ β© π΄)) = Ξ£*π β π΅(πβ(π β© π΄))) |
| 50 | 8, 49 | eqtr3d 2774 | 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 396 = wceq 1541 β wcel 2106 β© cin 3947 β wss 3948 π« cpw 4602 βͺ cuni 4908 Disj wdisj 5113 class class class wbr 5148 β¦ cmpt 5231 dom cdm 5676 ran crn 5677 βcfv 6543 (class class class)co 7408 Οcom 7854 βΌ cdom 8936 0cc0 11109 1c1 11110 [,]cicc 13326 Ξ£*cesum 33020 sigAlgebracsiga 33101 measurescmeas 33188 Probcprb 33401 |
| 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-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-iin 5000 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-supp 8146 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-fsupp 9361 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-ioc 13328 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-mod 13834 df-seq 13966 df-exp 14027 df-fac 14233 df-bc 14262 df-hash 14290 df-shft 15013 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-limsup 15414 df-clim 15431 df-rlim 15432 df-sum 15632 df-ef 16010 df-sin 16012 df-cos 16013 df-pi 16015 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-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-ordt 17446 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-ps 18518 df-tsr 18519 df-plusf 18559 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-subrg 20316 df-abv 20424 df-lmod 20472 df-scaf 20473 df-sra 20784 df-rgmod 20785 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-tx 23065 df-hmeo 23258 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-tmd 23575 df-tgp 23576 df-tsms 23630 df-trg 23663 df-xms 23825 df-ms 23826 df-tms 23827 df-nm 24090 df-ngp 24091 df-nrg 24093 df-nlm 24094 df-ii 24392 df-cncf 24393 df-limc 25382 df-dv 25383 df-log 26064 df-esum 33021 df-siga 33102 df-meas 33189 df-prob 33402 |
| This theorem is referenced by: totprob 33421 |
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