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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cndprobprob | Structured version Visualization version GIF version | ||
| Description: The conditional probability defines a probability law. (Contributed by Thierry Arnoux, 23-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.) |
| Ref | Expression |
|---|---|
| cndprobprob | β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β (π β dom π β¦ ((cprobβπ)ββ¨π, π΅β©)) β Prob) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | domprobmeas 33708 | . . . 4 β’ (π β Prob β π β (measuresβdom π)) | |
| 2 | 1 | 3ad2ant1 1132 | . . 3 β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β π β (measuresβdom π)) |
| 3 | simp2 1136 | . . 3 β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β π΅ β dom π) | |
| 4 | prob01 33711 | . . . . . 6 β’ ((π β Prob β§ π΅ β dom π) β (πβπ΅) β (0[,]1)) | |
| 5 | 4 | 3adant3 1131 | . . . . 5 β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β (πβπ΅) β (0[,]1)) |
| 6 | elunitrn 13449 | . . . . 5 β’ ((πβπ΅) β (0[,]1) β (πβπ΅) β β) | |
| 7 | 5, 6 | syl 17 | . . . 4 β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β (πβπ΅) β β) |
| 8 | elunitge0 33178 | . . . . . 6 β’ ((πβπ΅) β (0[,]1) β 0 β€ (πβπ΅)) | |
| 9 | 5, 8 | syl 17 | . . . . 5 β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β 0 β€ (πβπ΅)) |
| 10 | simp3 1137 | . . . . 5 β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β (πβπ΅) β 0) | |
| 11 | 7, 9, 10 | ne0gt0d 11356 | . . . 4 β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β 0 < (πβπ΅)) |
| 12 | 7, 11 | elrpd 13018 | . . 3 β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β (πβπ΅) β β+) |
| 13 | probmeasb 33728 | . . 3 β’ ((π β (measuresβdom π) β§ π΅ β dom π β§ (πβπ΅) β β+) β (π β dom π β¦ ((πβ(π β© π΅)) / (πβπ΅))) β Prob) | |
| 14 | 2, 3, 12, 13 | syl3anc 1370 | . 2 β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β (π β dom π β¦ ((πβ(π β© π΅)) / (πβπ΅))) β Prob) |
| 15 | 3anan32 1096 | . . . . . 6 β’ ((π β Prob β§ π β dom π β§ π΅ β dom π) β ((π β Prob β§ π΅ β dom π) β§ π β dom π)) | |
| 16 | cndprobval 33731 | . . . . . 6 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΅))) | |
| 17 | 15, 16 | sylbir 234 | . . . . 5 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΅))) |
| 18 | 17 | mpteq2dva 5248 | . . . 4 β’ ((π β Prob β§ π΅ β dom π) β (π β dom π β¦ ((cprobβπ)ββ¨π, π΅β©)) = (π β dom π β¦ ((πβ(π β© π΅)) / (πβπ΅)))) |
| 19 | 18 | eleq1d 2817 | . . 3 β’ ((π β Prob β§ π΅ β dom π) β ((π β dom π β¦ ((cprobβπ)ββ¨π, π΅β©)) β Prob β (π β dom π β¦ ((πβ(π β© π΅)) / (πβπ΅))) β Prob)) |
| 20 | 19 | 3adant3 1131 | . 2 β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β ((π β dom π β¦ ((cprobβπ)ββ¨π, π΅β©)) β Prob β (π β dom π β¦ ((πβ(π β© π΅)) / (πβπ΅))) β Prob)) |
| 21 | 14, 20 | mpbird 257 | 1 β’ ((π β Prob β§ π΅ β dom π β§ (πβπ΅) β 0) β (π β dom π β¦ ((cprobβπ)ββ¨π, π΅β©)) β Prob) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2939 β© cin 3947 β¨cop 4634 class class class wbr 5148 β¦ cmpt 5231 dom cdm 5676 βcfv 6543 (class class class)co 7412 βcr 11113 0cc0 11114 1c1 11115 β€ cle 11254 / cdiv 11876 β+crp 12979 [,]cicc 13332 measurescmeas 33492 Probcprb 33705 cprobccprob 33729 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-ac2 10462 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-acn 9941 df-ac 10115 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-ef 16016 df-sin 16018 df-cos 16019 df-pi 16021 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-ordt 17452 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-ps 18524 df-tsr 18525 df-plusf 18565 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-subrng 20435 df-subrg 20460 df-abv 20569 df-lmod 20617 df-scaf 20618 df-sra 20931 df-rgmod 20932 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-tmd 23797 df-tgp 23798 df-tsms 23852 df-trg 23885 df-xms 24047 df-ms 24048 df-tms 24049 df-nm 24312 df-ngp 24313 df-nrg 24315 df-nlm 24316 df-ii 24618 df-cncf 24619 df-limc 25616 df-dv 25617 df-log 26302 df-xdiv 32352 df-esum 33325 df-siga 33406 df-meas 33493 df-prob 33706 df-cndprob 33730 |
| This theorem is referenced by: (None) |
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