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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 | ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑃‘𝐵) ≠ 0) → (((cprob‘𝑃)‘〈𝐴, 𝐵〉) · (𝑃‘𝐵)) = (𝑃‘(𝐴 ∩ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cndprobval 33363 | . . . 4 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((cprob‘𝑃)‘〈𝐴, 𝐵〉) = ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵))) | |
2 | 1 | oveq1d 7411 | . . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (((cprob‘𝑃)‘〈𝐴, 𝐵〉) · (𝑃‘𝐵)) = (((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)) · (𝑃‘𝐵))) |
3 | 2 | adantr 482 | . 2 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑃‘𝐵) ≠ 0) → (((cprob‘𝑃)‘〈𝐴, 𝐵〉) · (𝑃‘𝐵)) = (((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)) · (𝑃‘𝐵))) |
4 | unitsscn 13464 | . . . . 5 ⊢ (0[,]1) ⊆ ℂ | |
5 | simp1 1137 | . . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → 𝑃 ∈ Prob) | |
6 | domprobsiga 33341 | . . . . . . 7 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
7 | inelsiga 33064 | . . . . . . 7 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝐴 ∩ 𝐵) ∈ dom 𝑃) | |
8 | 6, 7 | syl3an1 1164 | . . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝐴 ∩ 𝐵) ∈ dom 𝑃) |
9 | prob01 33343 | . . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∩ 𝐵) ∈ dom 𝑃) → (𝑃‘(𝐴 ∩ 𝐵)) ∈ (0[,]1)) | |
10 | 5, 8, 9 | syl2anc 585 | . . . . 5 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘(𝐴 ∩ 𝐵)) ∈ (0[,]1)) |
11 | 4, 10 | sselid 3978 | . . . 4 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘(𝐴 ∩ 𝐵)) ∈ ℂ) |
12 | 11 | adantr 482 | . . 3 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑃‘𝐵) ≠ 0) → (𝑃‘(𝐴 ∩ 𝐵)) ∈ ℂ) |
13 | prob01 33343 | . . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘𝐵) ∈ (0[,]1)) | |
14 | 13 | 3adant2 1132 | . . . . 5 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘𝐵) ∈ (0[,]1)) |
15 | 4, 14 | sselid 3978 | . . . 4 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘𝐵) ∈ ℂ) |
16 | 15 | adantr 482 | . . 3 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑃‘𝐵) ≠ 0) → (𝑃‘𝐵) ∈ ℂ) |
17 | simpr 486 | . . 3 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑃‘𝐵) ≠ 0) → (𝑃‘𝐵) ≠ 0) | |
18 | 12, 16, 17 | divcan1d 11978 | . 2 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑃‘𝐵) ≠ 0) → (((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)) · (𝑃‘𝐵)) = (𝑃‘(𝐴 ∩ 𝐵))) |
19 | 3, 18 | eqtrd 2773 | 1 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑃‘𝐵) ≠ 0) → (((cprob‘𝑃)‘〈𝐴, 𝐵〉) · (𝑃‘𝐵)) = (𝑃‘(𝐴 ∩ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∩ cin 3945 〈cop 4630 ∪ cuni 4904 dom cdm 5672 ran crn 5673 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 0cc0 11097 1c1 11098 · cmul 11102 / cdiv 11858 [,]cicc 13314 sigAlgebracsiga 33037 Probcprb 33337 cprobccprob 33361 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-inf2 9623 ax-ac2 10445 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 ax-addf 11176 ax-mulf 11177 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-disj 5110 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 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 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-om 7843 df-1st 7962 df-2nd 7963 df-supp 8134 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8691 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9350 df-fi 9393 df-sup 9424 df-inf 9425 df-oi 9492 df-dju 9883 df-card 9921 df-acn 9924 df-ac 10098 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-q 12920 df-rp 12962 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13315 df-ioc 13316 df-ico 13317 df-icc 13318 df-fz 13472 df-fzo 13615 df-fl 13744 df-mod 13822 df-seq 13954 df-exp 14015 df-fac 14221 df-bc 14250 df-hash 14278 df-shft 15001 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-limsup 15402 df-clim 15419 df-rlim 15420 df-sum 15620 df-ef 15998 df-sin 16000 df-cos 16001 df-pi 16003 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-starv 17199 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-unif 17207 df-hom 17208 df-cco 17209 df-rest 17355 df-topn 17356 df-0g 17374 df-gsum 17375 df-topgen 17376 df-pt 17377 df-prds 17380 df-ordt 17434 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-ps 18506 df-tsr 18507 df-plusf 18547 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-mhm 18658 df-submnd 18659 df-grp 18809 df-minusg 18810 df-sbg 18811 df-mulg 18936 df-subg 18988 df-cntz 19166 df-cmn 19634 df-abl 19635 df-mgp 19971 df-ur 19988 df-ring 20040 df-cring 20041 df-subrg 20338 df-abv 20402 df-lmod 20450 df-scaf 20451 df-sra 20762 df-rgmod 20763 df-psmet 20910 df-xmet 20911 df-met 20912 df-bl 20913 df-mopn 20914 df-fbas 20915 df-fg 20916 df-cnfld 20919 df-top 22365 df-topon 22382 df-topsp 22404 df-bases 22418 df-cld 22492 df-ntr 22493 df-cls 22494 df-nei 22571 df-lp 22609 df-perf 22610 df-cn 22700 df-cnp 22701 df-haus 22788 df-tx 23035 df-hmeo 23228 df-fil 23319 df-fm 23411 df-flim 23412 df-flf 23413 df-tmd 23545 df-tgp 23546 df-tsms 23600 df-trg 23633 df-xms 23795 df-ms 23796 df-tms 23797 df-nm 24060 df-ngp 24061 df-nrg 24063 df-nlm 24064 df-ii 24362 df-cncf 24363 df-limc 25352 df-dv 25353 df-log 26034 df-esum 32957 df-siga 33038 df-meas 33125 df-prob 33338 df-cndprob 33362 |
This theorem is referenced by: bayesth 33369 |
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