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Mirrors > Home > MPE Home > Th. List > Mathboxes > totprobd | Structured version Visualization version GIF version |
Description: Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
totprobd.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
totprobd.2 | ⊢ (𝜑 → 𝐴 ∈ dom 𝑃) |
totprobd.3 | ⊢ (𝜑 → 𝐵 ∈ 𝒫 dom 𝑃) |
totprobd.4 | ⊢ (𝜑 → ∪ 𝐵 = ∪ dom 𝑃) |
totprobd.5 | ⊢ (𝜑 → 𝐵 ≼ ω) |
totprobd.6 | ⊢ (𝜑 → Disj 𝑏 ∈ 𝐵 𝑏) |
Ref | Expression |
---|---|
totprobd | ⊢ (𝜑 → (𝑃‘𝐴) = Σ*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | totprobd.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom 𝑃) | |
2 | elssuni 4898 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝑃 → 𝐴 ⊆ ∪ dom 𝑃) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ∪ dom 𝑃) |
4 | totprobd.4 | . . . . 5 ⊢ (𝜑 → ∪ 𝐵 = ∪ dom 𝑃) | |
5 | 3, 4 | sseqtrrd 3985 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵) |
6 | sseqin2 4175 | . . . 4 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ (∪ 𝐵 ∩ 𝐴) = 𝐴) | |
7 | 5, 6 | sylib 217 | . . 3 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐴) = 𝐴) |
8 | 7 | fveq2d 6846 | . 2 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) = (𝑃‘𝐴)) |
9 | totprobd.1 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
10 | domprobmeas 32950 | . . . . . 6 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃)) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ (measures‘dom 𝑃)) |
12 | measinb 32760 | . . . . 5 ⊢ ((𝑃 ∈ (measures‘dom 𝑃) ∧ 𝐴 ∈ dom 𝑃) → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) ∈ (measures‘dom 𝑃)) | |
13 | 11, 1, 12 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) ∈ (measures‘dom 𝑃)) |
14 | totprobd.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝒫 dom 𝑃) | |
15 | totprobd.5 | . . . 4 ⊢ (𝜑 → 𝐵 ≼ ω) | |
16 | totprobd.6 | . . . 4 ⊢ (𝜑 → Disj 𝑏 ∈ 𝐵 𝑏) | |
17 | measvun 32748 | . . . 4 ⊢ (((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) ∈ (measures‘dom 𝑃) ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ (𝐵 ≼ ω ∧ Disj 𝑏 ∈ 𝐵 𝑏)) → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘∪ 𝐵) = Σ*𝑏 ∈ 𝐵((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏)) | |
18 | 13, 14, 15, 16, 17 | syl112anc 1374 | . . 3 ⊢ (𝜑 → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘∪ 𝐵) = Σ*𝑏 ∈ 𝐵((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏)) |
19 | eqidd 2737 | . . . 4 ⊢ (𝜑 → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) = (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))) | |
20 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → 𝑐 = ∪ 𝐵) | |
21 | 20 | ineq1d 4171 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → (𝑐 ∩ 𝐴) = (∪ 𝐵 ∩ 𝐴)) |
22 | 21 | fveq2d 6846 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → (𝑃‘(𝑐 ∩ 𝐴)) = (𝑃‘(∪ 𝐵 ∩ 𝐴))) |
23 | domprobsiga 32951 | . . . . . 6 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
24 | 9, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
25 | sigaclcu 32656 | . . . . 5 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ dom 𝑃) | |
26 | 24, 14, 15, 25 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → ∪ 𝐵 ∈ dom 𝑃) |
27 | inelsiga 32674 | . . . . . 6 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ ∪ 𝐵 ∈ dom 𝑃 ∧ 𝐴 ∈ dom 𝑃) → (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃) | |
28 | 24, 26, 1, 27 | syl3anc 1371 | . . . . 5 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃) |
29 | prob01 32953 | . . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃) → (𝑃‘(∪ 𝐵 ∩ 𝐴)) ∈ (0[,]1)) | |
30 | 9, 28, 29 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) ∈ (0[,]1)) |
31 | 19, 22, 26, 30 | fvmptd 6955 | . . 3 ⊢ (𝜑 → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘∪ 𝐵) = (𝑃‘(∪ 𝐵 ∩ 𝐴))) |
32 | eqidd 2737 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) = (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))) | |
33 | simpr 485 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → 𝑐 = 𝑏) | |
34 | 33 | ineq1d 4171 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → (𝑐 ∩ 𝐴) = (𝑏 ∩ 𝐴)) |
35 | 34 | fveq2d 6846 | . . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → (𝑃‘(𝑐 ∩ 𝐴)) = (𝑃‘(𝑏 ∩ 𝐴))) |
36 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵) | |
37 | 14 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐵 ∈ 𝒫 dom 𝑃) |
38 | elelpwi 4570 | . . . . . 6 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 dom 𝑃) → 𝑏 ∈ dom 𝑃) | |
39 | 36, 37, 38 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ dom 𝑃) |
40 | 9 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑃 ∈ Prob) |
41 | 24 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → dom 𝑃 ∈ ∪ ran sigAlgebra) |
42 | 1 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐴 ∈ dom 𝑃) |
43 | inelsiga 32674 | . . . . . . 7 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ 𝑏 ∈ dom 𝑃 ∧ 𝐴 ∈ dom 𝑃) → (𝑏 ∩ 𝐴) ∈ dom 𝑃) | |
44 | 41, 39, 42, 43 | syl3anc 1371 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ∩ 𝐴) ∈ dom 𝑃) |
45 | prob01 32953 | . . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝑏 ∩ 𝐴) ∈ dom 𝑃) → (𝑃‘(𝑏 ∩ 𝐴)) ∈ (0[,]1)) | |
46 | 40, 44, 45 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑃‘(𝑏 ∩ 𝐴)) ∈ (0[,]1)) |
47 | 32, 35, 39, 46 | fvmptd 6955 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏) = (𝑃‘(𝑏 ∩ 𝐴))) |
48 | 47 | esumeq2dv 32577 | . . 3 ⊢ (𝜑 → Σ*𝑏 ∈ 𝐵((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏) = Σ*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴))) |
49 | 18, 31, 48 | 3eqtr3d 2784 | . 2 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) = Σ*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴))) |
50 | 8, 49 | eqtr3d 2778 | 1 ⊢ (𝜑 → (𝑃‘𝐴) = Σ*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3909 ⊆ wss 3910 𝒫 cpw 4560 ∪ cuni 4865 Disj wdisj 5070 class class class wbr 5105 ↦ cmpt 5188 dom cdm 5633 ran crn 5634 ‘cfv 6496 (class class class)co 7356 ωcom 7801 ≼ cdom 8880 0cc0 11050 1c1 11051 [,]cicc 13266 Σ*cesum 32566 sigAlgebracsiga 32647 measurescmeas 32734 Probcprb 32947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-ac2 10398 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 ax-addf 11129 ax-mulf 11130 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-supp 8092 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8647 df-map 8766 df-pm 8767 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9305 df-fi 9346 df-sup 9377 df-inf 9378 df-oi 9445 df-dju 9836 df-card 9874 df-acn 9877 df-ac 10051 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-q 12873 df-rp 12915 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13267 df-ioc 13268 df-ico 13269 df-icc 13270 df-fz 13424 df-fzo 13567 df-fl 13696 df-mod 13774 df-seq 13906 df-exp 13967 df-fac 14173 df-bc 14202 df-hash 14230 df-shft 14951 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-limsup 15352 df-clim 15369 df-rlim 15370 df-sum 15570 df-ef 15949 df-sin 15951 df-cos 15952 df-pi 15954 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-starv 17147 df-sca 17148 df-vsca 17149 df-ip 17150 df-tset 17151 df-ple 17152 df-ds 17154 df-unif 17155 df-hom 17156 df-cco 17157 df-rest 17303 df-topn 17304 df-0g 17322 df-gsum 17323 df-topgen 17324 df-pt 17325 df-prds 17328 df-ordt 17382 df-xrs 17383 df-qtop 17388 df-imas 17389 df-xps 17391 df-mre 17465 df-mrc 17466 df-acs 17468 df-ps 18454 df-tsr 18455 df-plusf 18495 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-mhm 18600 df-submnd 18601 df-grp 18750 df-minusg 18751 df-sbg 18752 df-mulg 18871 df-subg 18923 df-cntz 19095 df-cmn 19562 df-abl 19563 df-mgp 19895 df-ur 19912 df-ring 19964 df-cring 19965 df-subrg 20218 df-abv 20274 df-lmod 20322 df-scaf 20323 df-sra 20631 df-rgmod 20632 df-psmet 20786 df-xmet 20787 df-met 20788 df-bl 20789 df-mopn 20790 df-fbas 20791 df-fg 20792 df-cnfld 20795 df-top 22241 df-topon 22258 df-topsp 22280 df-bases 22294 df-cld 22368 df-ntr 22369 df-cls 22370 df-nei 22447 df-lp 22485 df-perf 22486 df-cn 22576 df-cnp 22577 df-haus 22664 df-tx 22911 df-hmeo 23104 df-fil 23195 df-fm 23287 df-flim 23288 df-flf 23289 df-tmd 23421 df-tgp 23422 df-tsms 23476 df-trg 23509 df-xms 23671 df-ms 23672 df-tms 23673 df-nm 23936 df-ngp 23937 df-nrg 23939 df-nlm 23940 df-ii 24238 df-cncf 24239 df-limc 25228 df-dv 25229 df-log 25910 df-esum 32567 df-siga 32648 df-meas 32735 df-prob 32948 |
This theorem is referenced by: totprob 32967 |
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