totprobd - Mathbox for Thierry Arnoux

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Theorem totprobd 31684

Description: Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
totprobd	⊢ (𝜑 → (𝑃‘𝐴) = Σ^*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))

Distinct variable groups: 𝐴,𝑏 𝐵,𝑏 𝑃,𝑏 𝜑,𝑏

Proof of Theorem totprobd Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		totprobd.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom 𝑃)
2		elssuni 4867	. . . . . 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 4007	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵)
6		sseqin2 4191	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ (∪ 𝐵 ∩ 𝐴) = 𝐴)
7	5, 6	sylib 220	. . 3 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐴) = 𝐴)
8	7	fveq2d 6673	. 2 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) = (𝑃‘𝐴))
9		totprobd.1	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Prob)
10		domprobmeas 31668	. . . . . 6 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (measures‘dom 𝑃))
12		measinb 31480	. . . . 5 ⊢ ((𝑃 ∈ (measures‘dom 𝑃) ∧ 𝐴 ∈ dom 𝑃) → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) ∈ (measures‘dom 𝑃))
13	11, 1, 12	syl2anc 586	. . . 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 31468	. . . 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 1370	. . 3 ⊢ (𝜑 → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘∪ 𝐵) = Σ^*𝑏 ∈ 𝐵((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏))
19		eqidd 2822	. . . 4 ⊢ (𝜑 → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) = (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))))
20		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → 𝑐 = ∪ 𝐵)
21	20	ineq1d 4187	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → (𝑐 ∩ 𝐴) = (∪ 𝐵 ∩ 𝐴))
22	21	fveq2d 6673	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → (𝑃‘(𝑐 ∩ 𝐴)) = (𝑃‘(∪ 𝐵 ∩ 𝐴)))
23		domprobsiga 31669	. . . . . 6 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
24	9, 23	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra)
25		sigaclcu 31376	. . . . 5 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ dom 𝑃)
26	24, 14, 15, 25	syl3anc 1367	. . . 4 ⊢ (𝜑 → ∪ 𝐵 ∈ dom 𝑃)
27		inelsiga 31394	. . . . . 6 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ ∪ 𝐵 ∈ dom 𝑃 ∧ 𝐴 ∈ dom 𝑃) → (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃)
28	24, 26, 1, 27	syl3anc 1367	. . . . 5 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃)
29		prob01 31671	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃) → (𝑃‘(∪ 𝐵 ∩ 𝐴)) ∈ (0[,]1))
30	9, 28, 29	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) ∈ (0[,]1))
31	19, 22, 26, 30	fvmptd 6774	. . 3 ⊢ (𝜑 → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘∪ 𝐵) = (𝑃‘(∪ 𝐵 ∩ 𝐴)))
32		eqidd 2822	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) = (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))))
33		simpr 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → 𝑐 = 𝑏)
34	33	ineq1d 4187	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → (𝑐 ∩ 𝐴) = (𝑏 ∩ 𝐴))
35	34	fveq2d 6673	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → (𝑃‘(𝑐 ∩ 𝐴)) = (𝑃‘(𝑏 ∩ 𝐴)))
36		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
37	14	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐵 ∈ 𝒫 dom 𝑃)
38		elelpwi 4550	. . . . . 6 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 dom 𝑃) → 𝑏 ∈ dom 𝑃)
39	36, 37, 38	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ dom 𝑃)
40	9	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑃 ∈ Prob)
41	24	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → dom 𝑃 ∈ ∪ ran sigAlgebra)
42	1	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐴 ∈ dom 𝑃)
43		inelsiga 31394	. . . . . . 7 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ 𝑏 ∈ dom 𝑃 ∧ 𝐴 ∈ dom 𝑃) → (𝑏 ∩ 𝐴) ∈ dom 𝑃)
44	41, 39, 42, 43	syl3anc 1367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ∩ 𝐴) ∈ dom 𝑃)
45		prob01 31671	. . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝑏 ∩ 𝐴) ∈ dom 𝑃) → (𝑃‘(𝑏 ∩ 𝐴)) ∈ (0[,]1))
46	40, 44, 45	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑃‘(𝑏 ∩ 𝐴)) ∈ (0[,]1))
47	32, 35, 39, 46	fvmptd 6774	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏) = (𝑃‘(𝑏 ∩ 𝐴)))
48	47	esumeq2dv 31297	. . 3 ⊢ (𝜑 → Σ^𝑏 ∈ 𝐵((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏) = Σ^𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))
49	18, 31, 48	3eqtr3d 2864	. 2 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) = Σ^*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))
50	8, 49	eqtr3d 2858	1 ⊢ (𝜑 → (𝑃‘𝐴) = Σ^*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 ∪ cuni 4837 Disj wdisj 5030 class class class wbr 5065 ↦ cmpt 5145 dom cdm 5554 ran crn 5555 ‘cfv 6354 (class class class)co 7155 ωcom 7579 ≼ cdom 8506 0cc0 10536 1c1 10537 [,]cicc 12740 Σ^*cesum 31286 sigAlgebracsiga 31367 measurescmeas 31454 Probcprb 31665

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-ac2 9884 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-disj 5031 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-dju 9329 df-card 9367 df-acn 9370 df-ac 9541 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-fac 13633 df-bc 13662 df-hash 13690 df-shft 14425 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-limsup 14827 df-clim 14844 df-rlim 14845 df-sum 15042 df-ef 15420 df-sin 15422 df-cos 15423 df-pi 15425 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-ordt 16773 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-ps 17809 df-tsr 17810 df-plusf 17850 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-cntz 18446 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-subrg 19532 df-abv 19587 df-lmod 19635 df-scaf 19636 df-sra 19943 df-rgmod 19944 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cn 21834 df-cnp 21835 df-haus 21922 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-tmd 22679 df-tgp 22680 df-tsms 22734 df-trg 22767 df-xms 22929 df-ms 22930 df-tms 22931 df-nm 23191 df-ngp 23192 df-nrg 23194 df-nlm 23195 df-ii 23484 df-cncf 23485 df-limc 24463 df-dv 24464 df-log 25139 df-esum 31287 df-siga 31368 df-meas 31455 df-prob 31666

This theorem is referenced by: totprob 31685

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