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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bayesth | Structured version Visualization version GIF version | ||
| Description: Bayes Theorem. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.) |
| Ref | Expression |
|---|---|
| bayesth | β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β ((cprobβπ)ββ¨π΄, π΅β©) = ((((cprobβπ)ββ¨π΅, π΄β©) Β· (πβπ΄)) / (πβπ΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unitsscn 13474 | . . . 4 β’ (0[,]1) β β | |
| 2 | cndprob01 33423 | . . . . 5 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 (0[,]1)) | |
| 3 | 2 | 3adant2 1132 | . . . 4 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β ((cprobβπ)ββ¨π΄, π΅β©) β (0[,]1)) |
| 4 | 1, 3 | sselid 3980 | . . 3 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β ((cprobβπ)ββ¨π΄, π΅β©) β β) |
| 5 | simp11 1204 | . . . . 5 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β π β Prob) | |
| 6 | simp13 1206 | . . . . 5 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β π΅ β dom π) | |
| 7 | prob01 33401 | . . . . 5 β’ ((π β Prob β§ π΅ β dom π) β (πβπ΅) β (0[,]1)) | |
| 8 | 5, 6, 7 | syl2anc 585 | . . . 4 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β (πβπ΅) β (0[,]1)) |
| 9 | 1, 8 | sselid 3980 | . . 3 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β (πβπ΅) β β) |
| 10 | simp3 1139 | . . 3 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β (πβπ΅) β 0) | |
| 11 | 4, 9, 10 | divcan4d 11993 | . 2 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β ((((cprobβπ)ββ¨π΄, π΅β©) Β· (πβπ΅)) / (πβπ΅)) = ((cprobβπ)ββ¨π΄, π΅β©)) |
| 12 | incom 4201 | . . . . 5 β’ (π΄ β© π΅) = (π΅ β© π΄) | |
| 13 | 12 | fveq2i 6892 | . . . 4 β’ (πβ(π΄ β© π΅)) = (πβ(π΅ β© π΄)) |
| 14 | cndprobin 33422 | . . . . 5 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΅))) | |
| 15 | 14 | 3adant2 1132 | . . . 4 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β (((cprobβπ)ββ¨π΄, π΅β©) Β· (πβπ΅)) = (πβ(π΄ β© π΅))) |
| 16 | simp12 1205 | . . . . 5 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β π΄ β dom π) | |
| 17 | simp2 1138 | . . . . 5 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β (πβπ΄) β 0) | |
| 18 | cndprobin 33422 | . . . . 5 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΄))) | |
| 19 | 5, 6, 16, 17, 18 | syl31anc 1374 | . . . 4 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β (((cprobβπ)ββ¨π΅, π΄β©) Β· (πβπ΄)) = (πβ(π΅ β© π΄))) |
| 20 | 13, 15, 19 | 3eqtr4a 2799 | . . 3 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β (((cprobβπ)ββ¨π΄, π΅β©) Β· (πβπ΅)) = (((cprobβπ)ββ¨π΅, π΄β©) Β· (πβπ΄))) |
| 21 | 20 | oveq1d 7421 | . 2 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β ((((cprobβπ)ββ¨π΄, π΅β©) Β· (πβπ΅)) / (πβπ΅)) = ((((cprobβπ)ββ¨π΅, π΄β©) Β· (πβπ΄)) / (πβπ΅))) |
| 22 | 11, 21 | eqtr3d 2775 | 1 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΄) β 0 β§ (πβπ΅) β 0) β ((cprobβπ)ββ¨π΄, π΅β©) = ((((cprobβπ)ββ¨π΅, π΄β©) Β· (πβπ΄)) / (πβπ΅))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 β© cin 3947 β¨cop 4634 dom cdm 5676 βcfv 6541 (class class class)co 7406 βcc 11105 0cc0 11107 1c1 11108 Β· cmul 11112 / cdiv 11868 [,]cicc 13324 Probcprb 33395 cprobccprob 33419 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-ac2 10455 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-dju 9893 df-card 9931 df-acn 9934 df-ac 10108 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-rest 17365 df-topn 17366 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-prds 17390 df-ordt 17444 df-xrs 17445 df-qtop 17450 df-imas 17451 df-xps 17453 df-mre 17527 df-mrc 17528 df-acs 17530 df-ps 18516 df-tsr 18517 df-plusf 18557 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mhm 18668 df-submnd 18669 df-grp 18819 df-minusg 18820 df-sbg 18821 df-mulg 18946 df-subg 18998 df-cntz 19176 df-cmn 19645 df-abl 19646 df-mgp 19983 df-ur 20000 df-ring 20052 df-cring 20053 df-subrg 20354 df-abv 20418 df-lmod 20466 df-scaf 20467 df-sra 20778 df-rgmod 20779 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-fbas 20934 df-fg 20935 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cld 22515 df-ntr 22516 df-cls 22517 df-nei 22594 df-lp 22632 df-perf 22633 df-cn 22723 df-cnp 22724 df-haus 22811 df-tx 23058 df-hmeo 23251 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-tmd 23568 df-tgp 23569 df-tsms 23623 df-trg 23656 df-xms 23818 df-ms 23819 df-tms 23820 df-nm 24083 df-ngp 24084 df-nrg 24086 df-nlm 24087 df-ii 24385 df-cncf 24386 df-limc 25375 df-dv 25376 df-log 26057 df-esum 33015 df-siga 33096 df-meas 33183 df-prob 33396 df-cndprob 33420 |
| This theorem is referenced by: (None) |
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