totprobd - Mathbox for Thierry Arnoux

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Theorem totprobd 35104

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 4898	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑃 → 𝐴 ⊆ ∪ dom 𝑃)
3	1, 2	syl 18	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ∪ dom 𝑃)
4		totprobd.4	. . . . 5 ⊢ (𝜑 → ∪ 𝐵 = ∪ dom 𝑃)
5	3, 4	sseqtrrd 3945	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵)
6		sseqin2 4120	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ (∪ 𝐵 ∩ 𝐴) = 𝐴)
7	5, 6	sylib 221	. . 3 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐴) = 𝐴)
8	7	fveq2d 6889	. 2 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) = (𝑃‘𝐴))
9		totprobd.1	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Prob)
10		domprobmeas 35088	. . . . . 6 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
11	9, 10	syl 18	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (measures‘dom 𝑃))
12		measinb 34900	. . . . 5 ⊢ ((𝑃 ∈ (measures‘dom 𝑃) ∧ 𝐴 ∈ dom 𝑃) → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) ∈ (measures‘dom 𝑃))
13	11, 1, 12	syl2anc 596	. . . 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 34888	. . . 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 1401	. . 3 ⊢ (𝜑 → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘∪ 𝐵) = Σ^*𝑏 ∈ 𝐵((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏))
19		eqidd 2761	. . . 4 ⊢ (𝜑 → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) = (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))))
20		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → 𝑐 = ∪ 𝐵)
21	20	ineq1d 4116	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → (𝑐 ∩ 𝐴) = (∪ 𝐵 ∩ 𝐴))
22	21	fveq2d 6889	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = ∪ 𝐵) → (𝑃‘(𝑐 ∩ 𝐴)) = (𝑃‘(∪ 𝐵 ∩ 𝐴)))
23		domprobsiga 35089	. . . . . 6 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
24	9, 23	syl 18	. . . . 5 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra)
25		sigaclcu 34795	. . . . 5 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ dom 𝑃)
26	24, 14, 15, 25	syl3anc 1398	. . . 4 ⊢ (𝜑 → ∪ 𝐵 ∈ dom 𝑃)
27		inelsiga 34814	. . . . . 6 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ ∪ 𝐵 ∈ dom 𝑃 ∧ 𝐴 ∈ dom 𝑃) → (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃)
28	24, 26, 1, 27	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃)
29		prob01 35091	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (∪ 𝐵 ∩ 𝐴) ∈ dom 𝑃) → (𝑃‘(∪ 𝐵 ∩ 𝐴)) ∈ (0[,]1))
30	9, 28, 29	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) ∈ (0[,]1))
31	19, 22, 26, 30	fvmptd 7001	. . 3 ⊢ (𝜑 → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘∪ 𝐵) = (𝑃‘(∪ 𝐵 ∩ 𝐴)))
32		eqidd 2761	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))) = (𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴))))
33		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → 𝑐 = 𝑏)
34	33	ineq1d 4116	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → (𝑐 ∩ 𝐴) = (𝑏 ∩ 𝐴))
35	34	fveq2d 6889	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 = 𝑏) → (𝑃‘(𝑐 ∩ 𝐴)) = (𝑃‘(𝑏 ∩ 𝐴)))
36		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
37	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐵 ∈ 𝒫 dom 𝑃)
38		elelpwi 4566	. . . . . 6 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 dom 𝑃) → 𝑏 ∈ dom 𝑃)
39	36, 37, 38	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ dom 𝑃)
40	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑃 ∈ Prob)
41	24	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → dom 𝑃 ∈ ∪ ran sigAlgebra)
42	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐴 ∈ dom 𝑃)
43		inelsiga 34814	. . . . . . 7 ⊢ ((dom 𝑃 ∈ ∪ ran sigAlgebra ∧ 𝑏 ∈ dom 𝑃 ∧ 𝐴 ∈ dom 𝑃) → (𝑏 ∩ 𝐴) ∈ dom 𝑃)
44	41, 39, 42, 43	syl3anc 1398	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ∩ 𝐴) ∈ dom 𝑃)
45		prob01 35091	. . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝑏 ∩ 𝐴) ∈ dom 𝑃) → (𝑃‘(𝑏 ∩ 𝐴)) ∈ (0[,]1))
46	40, 44, 45	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑃‘(𝑏 ∩ 𝐴)) ∈ (0[,]1))
47	32, 35, 39, 46	fvmptd 7001	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏) = (𝑃‘(𝑏 ∩ 𝐴)))
48	47	esumeq2dv 34716	. . 3 ⊢ (𝜑 → Σ^𝑏 ∈ 𝐵((𝑐 ∈ dom 𝑃 ↦ (𝑃‘(𝑐 ∩ 𝐴)))‘𝑏) = Σ^𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))
49	18, 31, 48	3eqtr3d 2803	. 2 ⊢ (𝜑 → (𝑃‘(∪ 𝐵 ∩ 𝐴)) = Σ^*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))
50	8, 49	eqtr3d 2797	1 ⊢ (𝜑 → (𝑃‘𝐴) = Σ^*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ⊆ wss 3895 ∩ cin 4089 𝒫 cpw 4556 ∪ cuni 4866 Disj wdisj 5069 class class class wbr 5102 ↦ cmpt 5185 dom cdm 5650 ran crn 5651 ‘cfv 6538 (class class class)co 7420 ωcom 7877 ≼ cdom 8974 0cc0 11203 1c1 11204 [,]cicc 13482 Σ^*cesum 34705 sigAlgebracsiga 34786 measurescmeas 34874 Probcprb 35085

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9645 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281 ax-addf 11282 ax-mulf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-er 8720 df-map 8852 df-pm 8853 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9507 df-dju 9985 df-card 10023 df-acn 10026 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-q 13079 df-rp 13124 df-xneg 13244 df-xadd 13245 df-xmul 13246 df-ioo 13483 df-ioc 13484 df-ico 13485 df-icc 13486 df-fz 13643 df-fzo 13792 df-fl 13935 df-mod 14013 df-seq 14148 df-exp 14208 df-fac 14421 df-bc 14450 df-hash 14478 df-shft 15225 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-limsup 15643 df-clim 15660 df-rlim 15661 df-sum 15859 df-ef 16238 df-sin 16240 df-cos 16241 df-pi 16243 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-starv 17448 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-unif 17456 df-hom 17457 df-cco 17458 df-rest 17598 df-topn 17599 df-0g 17617 df-gsum 17618 df-topgen 17619 df-pt 17620 df-prds 17623 df-ordt 17678 df-xrs 17679 df-qtop 17684 df-imas 17685 df-xps 17687 df-mre 17761 df-mrc 17762 df-acs 17764 df-ps 18745 df-tsr 18746 df-plusf 18820 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-mhm 18983 df-submnd 18984 df-grp 19152 df-minusg 19153 df-sbg 19154 df-mulg 19283 df-subg 19338 df-cntz 19536 df-cmn 20001 df-abl 20002 df-mgp 20366 df-rng 20380 df-ur 20413 df-ring 20466 df-cring 20467 df-subrng 20803 df-subrg 20827 df-abv 21071 df-lmod 21142 df-scaf 21143 df-sra 21453 df-rgmod 21454 df-psmet 21675 df-xmet 21676 df-met 21677 df-bl 21678 df-mopn 21679 df-fbas 21680 df-fg 21681 df-cnfld 21684 df-top 23217 df-topon 23234 df-topsp 23256 df-bases 23269 df-cld 23342 df-ntr 23343 df-cls 23344 df-nei 23421 df-lp 23459 df-perf 23460 df-cn 23550 df-cnp 23551 df-haus 23638 df-tx 23886 df-hmeo 24079 df-fil 24170 df-fm 24262 df-flim 24263 df-flf 24264 df-tmd 24396 df-tgp 24397 df-tsms 24451 df-trg 24484 df-xms 24644 df-ms 24645 df-tms 24646 df-nm 24906 df-ngp 24907 df-nrg 24909 df-nlm 24910 df-ii 25203 df-cncf 25204 df-limc 26191 df-dv 26192 df-log 26889 df-esum 34706 df-siga 34787 df-meas 34875 df-prob 35086

This theorem is used by: totprob 35105

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