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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 4737 | . . . . . 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 | sseqtr4d 3891 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵) |
6 | sseqin2 4073 | . . . 4 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ (∪ 𝐵 ∩ 𝐴) = 𝐴) | |
7 | 5, 6 | sylib 210 | . . 3 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐴) = 𝐴) |
8 | 7 | fveq2d 6500 | . 2 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) = (𝑃‘𝐴)) |
9 | totprobd.1 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
10 | domprobmeas 31346 | . . . . . 6 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃)) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ (measures‘dom 𝑃)) |
12 | measinb 31157 | . . . . 5 ⊢ ((𝑃 ∈ (measures‘dom 𝑃) ∧ 𝐴 ∈ dom 𝑃) → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) ∈ (measures‘dom 𝑃)) | |
13 | 11, 1, 12 | syl2anc 576 | . . . 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 31145 | . . . 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 1355 | . . 3 ⊢ (𝜑 → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘∪ 𝐵) = Σ*𝑏 ∈ 𝐵((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏)) |
19 | eqidd 2772 | . . . 4 ⊢ (𝜑 → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) = (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))) | |
20 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → 𝑐 = ∪ 𝐵) | |
21 | 20 | ineq1d 4069 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → (𝑐 ∩ 𝐴) = (∪ 𝐵 ∩ 𝐴)) |
22 | 21 | fveq2d 6500 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → (𝑃‘(𝑐 ∩ 𝐴)) = (𝑃‘(∪ 𝐵 ∩ 𝐴))) |
23 | domprobsiga 31347 | . . . . . 6 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
24 | 9, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
25 | sigaclcu 31053 | . . . . 5 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ dom 𝑃) | |
26 | 24, 14, 15, 25 | syl3anc 1352 | . . . 4 ⊢ (𝜑 → ∪ 𝐵 ∈ dom 𝑃) |
27 | inelsiga 31071 | . . . . . 6 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ ∪ 𝐵 ∈ dom 𝑃 ∧ 𝐴 ∈ dom 𝑃) → (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃) | |
28 | 24, 26, 1, 27 | syl3anc 1352 | . . . . 5 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃) |
29 | prob01 31349 | . . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃) → (𝑃‘(∪ 𝐵 ∩ 𝐴)) ∈ (0[,]1)) | |
30 | 9, 28, 29 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) ∈ (0[,]1)) |
31 | 19, 22, 26, 30 | fvmptd 6599 | . . 3 ⊢ (𝜑 → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘∪ 𝐵) = (𝑃‘(∪ 𝐵 ∩ 𝐴))) |
32 | eqidd 2772 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) = (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))) | |
33 | simpr 477 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → 𝑐 = 𝑏) | |
34 | 33 | ineq1d 4069 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → (𝑐 ∩ 𝐴) = (𝑏 ∩ 𝐴)) |
35 | 34 | fveq2d 6500 | . . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → (𝑃‘(𝑐 ∩ 𝐴)) = (𝑃‘(𝑏 ∩ 𝐴))) |
36 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵) | |
37 | 14 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐵 ∈ 𝒫 dom 𝑃) |
38 | elelpwi 4429 | . . . . . 6 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 dom 𝑃) → 𝑏 ∈ dom 𝑃) | |
39 | 36, 37, 38 | syl2anc 576 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ dom 𝑃) |
40 | 9 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑃 ∈ Prob) |
41 | 24 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → dom 𝑃 ∈ ∪ ran sigAlgebra) |
42 | 1 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐴 ∈ dom 𝑃) |
43 | inelsiga 31071 | . . . . . . 7 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ 𝑏 ∈ dom 𝑃 ∧ 𝐴 ∈ dom 𝑃) → (𝑏 ∩ 𝐴) ∈ dom 𝑃) | |
44 | 41, 39, 42, 43 | syl3anc 1352 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ∩ 𝐴) ∈ dom 𝑃) |
45 | prob01 31349 | . . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝑏 ∩ 𝐴) ∈ dom 𝑃) → (𝑃‘(𝑏 ∩ 𝐴)) ∈ (0[,]1)) | |
46 | 40, 44, 45 | syl2anc 576 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑃‘(𝑏 ∩ 𝐴)) ∈ (0[,]1)) |
47 | 32, 35, 39, 46 | fvmptd 6599 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏) = (𝑃‘(𝑏 ∩ 𝐴))) |
48 | 47 | esumeq2dv 30973 | . . 3 ⊢ (𝜑 → Σ*𝑏 ∈ 𝐵((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏) = Σ*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴))) |
49 | 18, 31, 48 | 3eqtr3d 2815 | . 2 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) = Σ*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴))) |
50 | 8, 49 | eqtr3d 2809 | 1 ⊢ (𝜑 → (𝑃‘𝐴) = Σ*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∩ cin 3821 ⊆ wss 3822 𝒫 cpw 4416 ∪ cuni 4708 Disj wdisj 4893 class class class wbr 4925 ↦ cmpt 5004 dom cdm 5403 ran crn 5404 ‘cfv 6185 (class class class)co 6974 ωcom 7394 ≼ cdom 8302 0cc0 10333 1c1 10334 [,]cicc 12555 Σ*cesum 30962 sigAlgebracsiga 31043 measurescmeas 31131 Probcprb 31343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-ac2 9681 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-disj 4894 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-fi 8668 df-sup 8699 df-inf 8700 df-oi 8767 df-dju 9122 df-card 9160 df-acn 9163 df-ac 9334 df-cda 9386 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ioo 12556 df-ioc 12557 df-ico 12558 df-icc 12559 df-fz 12707 df-fzo 12848 df-fl 12975 df-mod 13051 df-seq 13183 df-exp 13243 df-fac 13447 df-bc 13476 df-hash 13504 df-shft 14285 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-limsup 14687 df-clim 14704 df-rlim 14705 df-sum 14902 df-ef 15279 df-sin 15281 df-cos 15282 df-pi 15284 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-pt 16572 df-prds 16575 df-ordt 16628 df-xrs 16629 df-qtop 16634 df-imas 16635 df-xps 16637 df-mre 16727 df-mrc 16728 df-acs 16730 df-ps 17680 df-tsr 17681 df-plusf 17721 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-mulg 18024 df-subg 18072 df-cntz 18230 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-cring 19035 df-subrg 19268 df-abv 19322 df-lmod 19370 df-scaf 19371 df-sra 19678 df-rgmod 19679 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-fbas 20259 df-fg 20260 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cld 21346 df-ntr 21347 df-cls 21348 df-nei 21425 df-lp 21463 df-perf 21464 df-cn 21554 df-cnp 21555 df-haus 21642 df-tx 21889 df-hmeo 22082 df-fil 22173 df-fm 22265 df-flim 22266 df-flf 22267 df-tmd 22399 df-tgp 22400 df-tsms 22453 df-trg 22486 df-xms 22648 df-ms 22649 df-tms 22650 df-nm 22910 df-ngp 22911 df-nrg 22913 df-nlm 22914 df-ii 23203 df-cncf 23204 df-limc 24182 df-dv 24183 df-log 24856 df-esum 30963 df-siga 31044 df-meas 31132 df-prob 31344 |
This theorem is referenced by: totprob 31363 |
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