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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cndprob01 | Structured version Visualization version GIF version | ||
| Description: The conditional probability has values in [0, 1]. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.) |
| Ref | Expression |
|---|---|
| cndprob01 | 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)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cndprobval 33887 | . . 3 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 | adantr 480 | . 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©) = ((πβ(π΄ β© π΅)) / (πβπ΅))) |
| 3 | simpl1 1188 | . . . . 5 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β π β Prob) | |
| 4 | domprobmeas 33864 | . . . . 5 β’ (π β Prob β π β (measuresβdom π)) | |
| 5 | 3, 4 | syl 17 | . . . 4 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β π β (measuresβdom π)) |
| 6 | domprobsiga 33865 | . . . . . 6 β’ (π β Prob β dom π β βͺ ran sigAlgebra) | |
| 7 | 3, 6 | syl 17 | . . . . 5 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β dom π β βͺ ran sigAlgebra) |
| 8 | simpl2 1189 | . . . . 5 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β π΄ β dom π) | |
| 9 | simpl3 1190 | . . . . 5 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β π΅ β dom π) | |
| 10 | inelsiga 33588 | . . . . 5 β’ ((dom π β βͺ ran sigAlgebra β§ π΄ β dom π β§ π΅ β dom π) β (π΄ β© π΅) β dom π) | |
| 11 | 7, 8, 9, 10 | syl3anc 1368 | . . . 4 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β (π΄ β© π΅) β dom π) |
| 12 | inss2 4221 | . . . . 5 β’ (π΄ β© π΅) β π΅ | |
| 13 | 12 | a1i 11 | . . . 4 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β (π΄ β© π΅) β π΅) |
| 14 | 5, 11, 9, 13 | measssd 33668 | . . 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€ (πβπ΅)) |
| 15 | prob01 33867 | . . . . 5 β’ ((π β Prob β§ (π΄ β© π΅) β dom π) β (πβ(π΄ β© π΅)) β (0[,]1)) | |
| 16 | 3, 11, 15 | syl2anc 583 | . . . 4 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β (πβ(π΄ β© π΅)) β (0[,]1)) |
| 17 | prob01 33867 | . . . . 5 β’ ((π β Prob β§ π΅ β dom π) β (πβπ΅) β (0[,]1)) | |
| 18 | 3, 9, 17 | 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[,]1)) |
| 19 | simpr 484 | . . . 4 β’ (((π β Prob β§ π΄ β dom π β§ π΅ β dom π) β§ (πβπ΅) β 0) β (πβπ΅) β 0) | |
| 20 | unitdivcld 33336 | . . . 4 β’ (((πβ(π΄ β© π΅)) β (0[,]1) β§ (πβπ΅) β (0[,]1) β§ (πβπ΅) β 0) β ((πβ(π΄ β© π΅)) β€ (πβπ΅) β ((πβ(π΄ β© π΅)) / (πβπ΅)) β (0[,]1))) | |
| 21 | 16, 18, 19, 20 | syl3anc 1368 | . . 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€ (πβπ΅) β ((πβ(π΄ β© π΅)) / (πβπ΅)) β (0[,]1))) |
| 22 | 14, 21 | mpbid 231 | . 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 (0[,]1)) |
| 23 | 2, 22 | eqeltrd 2825 | 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 (0[,]1)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2932 β© cin 3939 β wss 3940 β¨cop 4626 βͺ cuni 4899 class class class wbr 5138 dom cdm 5666 ran crn 5667 βcfv 6533 (class class class)co 7401 0cc0 11105 1c1 11106 β€ cle 11245 / cdiv 11867 [,]cicc 13323 sigAlgebracsiga 33561 measurescmeas 33648 Probcprb 33861 cprobccprob 33885 |
| 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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-ac2 10453 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 ax-mulf 11185 |
| 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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-disj 5104 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-dju 9891 df-card 9929 df-acn 9932 df-ac 10106 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-ordt 17443 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-ps 18518 df-tsr 18519 df-plusf 18559 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-mhm 18700 df-submnd 18701 df-grp 18853 df-minusg 18854 df-sbg 18855 df-mulg 18983 df-subg 19035 df-cntz 19218 df-cmn 19687 df-abl 19688 df-mgp 20025 df-rng 20043 df-ur 20072 df-ring 20125 df-cring 20126 df-subrng 20431 df-subrg 20456 df-abv 20645 df-lmod 20693 df-scaf 20694 df-sra 21006 df-rgmod 21007 df-psmet 21215 df-xmet 21216 df-met 21217 df-bl 21218 df-mopn 21219 df-fbas 21220 df-fg 21221 df-cnfld 21224 df-top 22706 df-topon 22723 df-topsp 22745 df-bases 22759 df-cld 22833 df-ntr 22834 df-cls 22835 df-nei 22912 df-lp 22950 df-perf 22951 df-cn 23041 df-cnp 23042 df-haus 23129 df-tx 23376 df-hmeo 23569 df-fil 23660 df-fm 23752 df-flim 23753 df-flf 23754 df-tmd 23886 df-tgp 23887 df-tsms 23941 df-trg 23974 df-xms 24136 df-ms 24137 df-tms 24138 df-nm 24401 df-ngp 24402 df-nrg 24404 df-nlm 24405 df-ii 24707 df-cncf 24708 df-limc 25705 df-dv 25706 df-log 26395 df-esum 33481 df-siga 33562 df-meas 33649 df-prob 33862 df-cndprob 33886 |
| This theorem is referenced by: bayesth 33893 |
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