Nearby theorems
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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cndprobin | Structured version Visualization version GIF version | ||
| Description: An identity linking conditional probability and intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.) |
| Ref | Expression |
|---|---|
| cndprobin | β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β (((cprobβπ)ββ¨π΄, π΅β©) Β· (πβπ΅)) = (πβ(π΄ β© π΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cndprobval 34110 | . . . 4 β’ ((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β ((cprobβπ)ββ¨π΄, π΅β©) = ((πβ(π΄ β© π΅)) / (πβπ΅))) | |
| 2 | 1 | oveq1d 7431 | . . 3 β’ ((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β (((cprobβπ)ββ¨π΄, π΅β©) Β· (πβπ΅)) = (((πβ(π΄ β© π΅)) / (πβπ΅)) Β· (πβπ΅))) |
| 3 | 2 | adantr 479 | . 2 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β (((cprobβπ)ββ¨π΄, π΅β©) Β· (πβπ΅)) = (((πβ(π΄ β© π΅)) / (πβπ΅)) Β· (πβπ΅))) |
| 4 | unitsscn 13509 | . . . . 5 β’ (0[,]1) β β | |
| 5 | simp1 1133 | . . . . . 6 β’ ((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β π β Prob) | |
| 6 | domprobsiga 34088 | . . . . . . 7 β’ (π β Prob β dom π β βͺ ran sigAlgebra) | |
| 7 | inelsiga 33811 | . . . . . . 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) | |
| 8 | 6, 7 | syl3an1 1160 | . . . . . 6 β’ ((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β (π΄ β© π΅) β dom π) |
| 9 | prob01 34090 | . . . . . 6 β’ ((π β Prob β§ (π΄ β© π΅) β dom π) β (πβ(π΄ β© π΅)) β (0[,]1)) | |
| 10 | 5, 8, 9 | syl2anc 582 | . . . . 5 β’ ((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β (πβ(π΄ β© π΅)) β (0[,]1)) |
| 11 | 4, 10 | sselid 3970 | . . . 4 β’ ((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β (πβ(π΄ β© π΅)) β β) |
| 12 | 11 | adantr 479 | . . 3 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β (πβ(π΄ β© π΅)) β β) |
| 13 | prob01 34090 | . . . . . 6 β’ ((π β Prob β§ π΅ β dom π) β (πβπ΅) β (0[,]1)) | |
| 14 | 13 | 3adant2 1128 | . . . . 5 β’ ((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β (πβπ΅) β (0[,]1)) |
| 15 | 4, 14 | sselid 3970 | . . . 4 β’ ((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β (πβπ΅) β β) |
| 16 | 15 | adantr 479 | . . 3 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β (πβπ΅) β β) |
| 17 | simpr 483 | . . 3 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β (πβπ΅) β 0) | |
| 18 | 12, 16, 17 | divcan1d 12021 | . 2 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β (((πβ(π΄ β© π΅)) / (πβπ΅)) Β· (πβπ΅)) = (πβ(π΄ β© π΅))) |
| 19 | 3, 18 | eqtrd 2765 | 1 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β (((cprobβπ)ββ¨π΄, π΅β©) Β· (πβπ΅)) = (πβ(π΄ β© π΅))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 β© cin 3938 β¨cop 4630 βͺ cuni 4903 dom cdm 5672 ran crn 5673 βcfv 6543 (class class class)co 7416 βcc 11136 0cc0 11138 1c1 11139 Β· cmul 11143 / cdiv 11901 [,]cicc 13359 sigAlgebracsiga 33784 Probcprb 34084 cprobccprob 34108 |
| 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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-ac2 10486 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 |
| 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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-acn 9965 df-ac 10139 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-ef 16043 df-sin 16045 df-cos 16046 df-pi 16048 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-ordt 17482 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-ps 18557 df-tsr 18558 df-plusf 18598 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-subrng 20487 df-subrg 20512 df-abv 20701 df-lmod 20749 df-scaf 20750 df-sra 21062 df-rgmod 21063 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-tmd 23994 df-tgp 23995 df-tsms 24049 df-trg 24082 df-xms 24244 df-ms 24245 df-tms 24246 df-nm 24509 df-ngp 24510 df-nrg 24512 df-nlm 24513 df-ii 24815 df-cncf 24816 df-limc 25813 df-dv 25814 df-log 26508 df-esum 33704 df-siga 33785 df-meas 33872 df-prob 34085 df-cndprob 34109 |
| This theorem is referenced by: bayesth 34116 |
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