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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 4944 | . . . . . 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 4217 | . . . 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 6906 | . 2 β’ (π β (πβ(βͺ π΅ β© π΄)) = (πβπ΄)) |
| 9 | totprobd.1 | . . . . . 6 β’ (π β π β Prob) | |
| 10 | domprobmeas 34071 | . . . . . 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 33881 | . . . . 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 582 | . . . 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 33869 | . . . 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 2729 | . . . 4 β’ (π β (π β dom π β¦ (πβ(π β© π΄))) = (π β dom π β¦ (πβ(π β© π΄)))) | |
| 20 | simpr 483 | . . . . . 6 β’ ((π β§ π = βͺ π΅) β π = βͺ π΅) | |
| 21 | 20 | ineq1d 4213 | . . . . 5 β’ ((π β§ π = βͺ π΅) β (π β© π΄) = (βͺ π΅ β© π΄)) |
| 22 | 21 | fveq2d 6906 | . . . 4 β’ ((π β§ π = βͺ π΅) β (πβ(π β© π΄)) = (πβ(βͺ π΅ β© π΄))) |
| 23 | domprobsiga 34072 | . . . . . 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 33777 | . . . . 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 33795 | . . . . . 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 34074 | . . . . 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 582 | . . . 4 β’ (π β (πβ(βͺ π΅ β© π΄)) β (0[,]1)) |
| 31 | 19, 22, 26, 30 | fvmptd 7017 | . . 3 β’ (π β ((π β dom π β¦ (πβ(π β© π΄)))ββͺ π΅) = (πβ(βͺ π΅ β© π΄))) |
| 32 | eqidd 2729 | . . . . 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 483 | . . . . . . 7 β’ (((π β§ π β π΅) β§ π = π) β π = π) | |
| 34 | 33 | ineq1d 4213 | . . . . . 6 β’ (((π β§ π β π΅) β§ π = π) β (π β© π΄) = (π β© π΄)) |
| 35 | 34 | fveq2d 6906 | . . . . 5 β’ (((π β§ π β π΅) β§ π = π) β (πβ(π β© π΄)) = (πβ(π β© π΄))) |
| 36 | simpr 483 | . . . . . 6 β’ ((π β§ π β π΅) β π β π΅) | |
| 37 | 14 | adantr 479 | . . . . . 6 β’ ((π β§ π β π΅) β π΅ β π« dom π) |
| 38 | elelpwi 4616 | . . . . . 6 β’ ((π β π΅ β§ π΅ β π« dom π) β π β dom π) | |
| 39 | 36, 37, 38 | syl2anc 582 | . . . . 5 β’ ((π β§ π β π΅) β π β dom π) |
| 40 | 9 | adantr 479 | . . . . . 6 β’ ((π β§ π β π΅) β π β Prob) |
| 41 | 24 | adantr 479 | . . . . . . 7 β’ ((π β§ π β π΅) β dom π β βͺ ran sigAlgebra) |
| 42 | 1 | adantr 479 | . . . . . . 7 β’ ((π β§ π β π΅) β π΄ β dom π) |
| 43 | inelsiga 33795 | . . . . . . 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 34074 | . . . . . 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 582 | . . . . 5 β’ ((π β§ π β π΅) β (πβ(π β© π΄)) β (0[,]1)) |
| 47 | 32, 35, 39, 46 | fvmptd 7017 | . . . 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 33698 | . . 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 2776 | . 2 β’ (π β (πβ(βͺ π΅ β© π΄)) = Ξ£*π β π΅(πβ(π β© π΄))) |
| 50 | 8, 49 | eqtr3d 2770 | 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 394 = wceq 1533 β wcel 2098 β© cin 3948 β wss 3949 π« cpw 4606 βͺ cuni 4912 Disj wdisj 5117 class class class wbr 5152 β¦ cmpt 5235 dom cdm 5682 ran crn 5683 βcfv 6553 (class class class)co 7426 Οcom 7878 βΌ cdom 8970 0cc0 11148 1c1 11149 [,]cicc 13369 Ξ£*cesum 33687 sigAlgebracsiga 33768 measurescmeas 33855 Probcprb 34068 |
| 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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-ac2 10496 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 ax-mulf 11228 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-fi 9444 df-sup 9475 df-inf 9476 df-oi 9543 df-dju 9934 df-card 9972 df-acn 9975 df-ac 10149 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13134 df-xadd 13135 df-xmul 13136 df-ioo 13370 df-ioc 13371 df-ico 13372 df-icc 13373 df-fz 13527 df-fzo 13670 df-fl 13799 df-mod 13877 df-seq 14009 df-exp 14069 df-fac 14275 df-bc 14304 df-hash 14332 df-shft 15056 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-limsup 15457 df-clim 15474 df-rlim 15475 df-sum 15675 df-ef 16053 df-sin 16055 df-cos 16056 df-pi 16058 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-rest 17413 df-topn 17414 df-0g 17432 df-gsum 17433 df-topgen 17434 df-pt 17435 df-prds 17438 df-ordt 17492 df-xrs 17493 df-qtop 17498 df-imas 17499 df-xps 17501 df-mre 17575 df-mrc 17576 df-acs 17578 df-ps 18567 df-tsr 18568 df-plusf 18608 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19038 df-subg 19092 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-cring 20190 df-subrng 20497 df-subrg 20522 df-abv 20711 df-lmod 20759 df-scaf 20760 df-sra 21072 df-rgmod 21073 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22824 df-topon 22841 df-topsp 22863 df-bases 22877 df-cld 22951 df-ntr 22952 df-cls 22953 df-nei 23030 df-lp 23068 df-perf 23069 df-cn 23159 df-cnp 23160 df-haus 23247 df-tx 23494 df-hmeo 23687 df-fil 23778 df-fm 23870 df-flim 23871 df-flf 23872 df-tmd 24004 df-tgp 24005 df-tsms 24059 df-trg 24092 df-xms 24254 df-ms 24255 df-tms 24256 df-nm 24519 df-ngp 24520 df-nrg 24522 df-nlm 24523 df-ii 24825 df-cncf 24826 df-limc 25823 df-dv 25824 df-log 26518 df-esum 33688 df-siga 33769 df-meas 33856 df-prob 34069 |
| This theorem is referenced by: totprob 34088 |
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