Nearby theorems
|
|
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
| 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 13480 | . . . 4 β’ (0[,]1) β β | |
| 2 | cndprob01 33963 | . . . . 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 1128 | . . . 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 3975 | . . 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 1200 | . . . . 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 1202 | . . . . 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 33941 | . . . . 5 β’ ((π β Prob β§ π΅ β dom π) β (πβπ΅) β (0[,]1)) | |
| 8 | 5, 6, 7 | syl2anc 583 | . . . 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 3975 | . . 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 1135 | . . 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 11997 | . 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 4196 | . . . . 5 β’ (π΄ β© π΅) = (π΅ β© π΄) | |
| 13 | 12 | fveq2i 6887 | . . . 4 β’ (πβ(π΄ β© π΅)) = (πβ(π΅ β© π΄)) |
| 14 | cndprobin 33962 | . . . . 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 1128 | . . . 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 1201 | . . . . 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 1134 | . . . . 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 33962 | . . . . 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 1370 | . . . 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 2792 | . . 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 7419 | . 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 2768 | 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 1084 = wceq 1533 β wcel 2098 β wne 2934 β© cin 3942 β¨cop 4629 dom cdm 5669 βcfv 6536 (class class class)co 7404 βcc 11107 0cc0 11109 1c1 11110 Β· cmul 11114 / cdiv 11872 [,]cicc 13330 Probcprb 33935 cprobccprob 33959 |
| 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 7721 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 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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 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 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15017 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-limsup 15418 df-clim 15435 df-rlim 15436 df-sum 15636 df-ef 16014 df-sin 16016 df-cos 16017 df-pi 16019 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-hom 17227 df-cco 17228 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-ordt 17453 df-xrs 17454 df-qtop 17459 df-imas 17460 df-xps 17462 df-mre 17536 df-mrc 17537 df-acs 17539 df-ps 18528 df-tsr 18529 df-plusf 18569 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-submnd 18711 df-grp 18863 df-minusg 18864 df-sbg 18865 df-mulg 18993 df-subg 19047 df-cntz 19230 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-subrng 20443 df-subrg 20468 df-abv 20657 df-lmod 20705 df-scaf 20706 df-sra 21018 df-rgmod 21019 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-fbas 21232 df-fg 21233 df-cnfld 21236 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-cld 22873 df-ntr 22874 df-cls 22875 df-nei 22952 df-lp 22990 df-perf 22991 df-cn 23081 df-cnp 23082 df-haus 23169 df-tx 23416 df-hmeo 23609 df-fil 23700 df-fm 23792 df-flim 23793 df-flf 23794 df-tmd 23926 df-tgp 23927 df-tsms 23981 df-trg 24014 df-xms 24176 df-ms 24177 df-tms 24178 df-nm 24441 df-ngp 24442 df-nrg 24444 df-nlm 24445 df-ii 24747 df-cncf 24748 df-limc 25745 df-dv 25746 df-log 26440 df-esum 33555 df-siga 33636 df-meas 33723 df-prob 33936 df-cndprob 33960 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |