totprobd - Mathbox for Thierry Arnoux

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Theorem totprobd 34391

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 4961	. . . . . 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 4050	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵)
6		sseqin2 4244	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ (∪ 𝐵 ∩ 𝐴) = 𝐴)
7	5, 6	sylib 218	. . 3 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐴) = 𝐴)
8	7	fveq2d 6924	. 2 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) = (𝑃‘𝐴))
9		totprobd.1	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Prob)
10		domprobmeas 34375	. . . . . 6 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (measures‘dom 𝑃))
12		measinb 34185	. . . . 5 ⊢ ((𝑃 ∈ (measures‘dom 𝑃) ∧ 𝐴 ∈ dom 𝑃) → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) ∈ (measures‘dom 𝑃))
13	11, 1, 12	syl2anc 583	. . . 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 34173	. . . 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 2741	. . . 4 ⊢ (𝜑 → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) = (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))))
20		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → 𝑐 = ∪ 𝐵)
21	20	ineq1d 4240	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → (𝑐 ∩ 𝐴) = (∪ 𝐵 ∩ 𝐴))
22	21	fveq2d 6924	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → (𝑃‘(𝑐 ∩ 𝐴)) = (𝑃‘(∪ 𝐵 ∩ 𝐴)))
23		domprobsiga 34376	. . . . . 6 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
24	9, 23	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra)
25		sigaclcu 34081	. . . . 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 34099	. . . . . 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 34378	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃) → (𝑃‘(∪ 𝐵 ∩ 𝐴)) ∈ (0[,]1))
30	9, 28, 29	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) ∈ (0[,]1))
31	19, 22, 26, 30	fvmptd 7036	. . 3 ⊢ (𝜑 → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘∪ 𝐵) = (𝑃‘(∪ 𝐵 ∩ 𝐴)))
32		eqidd 2741	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) = (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))))
33		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → 𝑐 = 𝑏)
34	33	ineq1d 4240	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → (𝑐 ∩ 𝐴) = (𝑏 ∩ 𝐴))
35	34	fveq2d 6924	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → (𝑃‘(𝑐 ∩ 𝐴)) = (𝑃‘(𝑏 ∩ 𝐴)))
36		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
37	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐵 ∈ 𝒫 dom 𝑃)
38		elelpwi 4632	. . . . . 6 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 dom 𝑃) → 𝑏 ∈ dom 𝑃)
39	36, 37, 38	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ dom 𝑃)
40	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑃 ∈ Prob)
41	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → dom 𝑃 ∈ ∪ ran sigAlgebra)
42	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐴 ∈ dom 𝑃)
43		inelsiga 34099	. . . . . . 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 34378	. . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝑏 ∩ 𝐴) ∈ dom 𝑃) → (𝑃‘(𝑏 ∩ 𝐴)) ∈ (0[,]1))
46	40, 44, 45	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑃‘(𝑏 ∩ 𝐴)) ∈ (0[,]1))
47	32, 35, 39, 46	fvmptd 7036	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏) = (𝑃‘(𝑏 ∩ 𝐴)))
48	47	esumeq2dv 34002	. . 3 ⊢ (𝜑 → Σ^𝑏 ∈ 𝐵((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏) = Σ^𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))
49	18, 31, 48	3eqtr3d 2788	. 2 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) = Σ^*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))
50	8, 49	eqtr3d 2782	1 ⊢ (𝜑 → (𝑃‘𝐴) = Σ^*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∩ cin 3975 ⊆ wss 3976 𝒫 cpw 4622 ∪ cuni 4931 Disj wdisj 5133 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ran crn 5701 ‘cfv 6573 (class class class)co 7448 ωcom 7903 ≼ cdom 9001 0cc0 11184 1c1 11185 [,]cicc 13410 Σ^*cesum 33991 sigAlgebracsiga 34072 measurescmeas 34159 Probcprb 34372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-ac2 10532 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-ac 10185 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-ordt 17561 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-ps 18636 df-tsr 18637 df-plusf 18677 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-subrng 20572 df-subrg 20597 df-abv 20832 df-lmod 20882 df-scaf 20883 df-sra 21195 df-rgmod 21196 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-tmd 24101 df-tgp 24102 df-tsms 24156 df-trg 24189 df-xms 24351 df-ms 24352 df-tms 24353 df-nm 24616 df-ngp 24617 df-nrg 24619 df-nlm 24620 df-ii 24922 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-esum 33992 df-siga 34073 df-meas 34160 df-prob 34373

This theorem is referenced by: totprob 34392

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