dstrvprob - Mathbox for Thierry Arnoux

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Theorem dstrvprob 33458

Description: The distribution of a random variable is a probability law. (TODO: could be shortened using dstrvval 33457). (Contributed by Thierry Arnoux, 10-Feb-2017.)

**Hypotheses**
Ref	Expression
dstrvprob.1	⊢ (𝜑 → 𝑃 ∈ Prob)
dstrvprob.2	⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
dstrvprob.3	⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))))

**Assertion**
Ref	Expression
dstrvprob	⊢ (𝜑 → 𝐷 ∈ Prob)

Distinct variable groups: 𝑃,𝑎 𝑋,𝑎 𝐷,𝑎 𝜑,𝑎

Proof of Theorem dstrvprob Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dstrvprob.3	. . . . . 6 ⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))))
2		dstrvprob.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ Prob)
3	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → 𝑃 ∈ Prob)
4		dstrvprob.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
5	4	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → 𝑋 ∈ (rRndVar‘𝑃))
6		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → 𝑎 ∈ 𝔅_ℝ)
7	3, 5, 6	orvcelel 33456	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → (𝑋∘_RV/𝑐 E 𝑎) ∈ dom 𝑃)
8		prob01 33400	. . . . . . . 8 ⊢ ((𝑃 ∈ Prob ∧ (𝑋∘_RV/𝑐 E 𝑎) ∈ dom 𝑃) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]1))
9	3, 7, 8	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]1))
10		elunitrn 13440	. . . . . . . . 9 ⊢ ((𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]1) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ ℝ)
11	10	rexrd 11260	. . . . . . . 8 ⊢ ((𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]1) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ ℝ^*)
12		elunitge0 32867	. . . . . . . 8 ⊢ ((𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]1) → 0 ≤ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))
13		elxrge0 13430	. . . . . . . 8 ⊢ ((𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]+∞) ↔ ((𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ ℝ^* ∧ 0 ≤ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))))
14	11, 12, 13	sylanbrc 583	. . . . . . 7 ⊢ ((𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]1) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]+∞))
15	9, 14	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]+∞))
16	1, 15	fmpt3d 7112	. . . . 5 ⊢ (𝜑 → 𝐷:𝔅_ℝ⟶(0[,]+∞))
17		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 = ∅) → 𝑎 = ∅)
18	17	oveq2d 7421	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 = ∅) → (𝑋∘_RV/𝑐 E 𝑎) = (𝑋∘_RV/𝑐 E ∅))
19	18	fveq2d 6892	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 = ∅) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = (𝑃‘(𝑋∘_RV/𝑐 E ∅)))
20		brsigarn 33170	. . . . . . . . 9 ⊢ 𝔅_ℝ ∈ (sigAlgebra‘ℝ)
21		elrnsiga 33112	. . . . . . . . 9 ⊢ (𝔅_ℝ ∈ (sigAlgebra‘ℝ) → 𝔅_ℝ ∈ ∪ ran sigAlgebra)
22		0elsiga 33100	. . . . . . . . 9 ⊢ (𝔅_ℝ ∈ ∪ ran sigAlgebra → ∅ ∈ 𝔅_ℝ)
23	20, 21, 22	mp2b 10	. . . . . . . 8 ⊢ ∅ ∈ 𝔅_ℝ
24	23	a1i 11	. . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝔅_ℝ)
25	2, 4, 24	orvcelel 33456	. . . . . . . 8 ⊢ (𝜑 → (𝑋∘_RV/𝑐 E ∅) ∈ dom 𝑃)
26	2, 25	probvalrnd 33411	. . . . . . 7 ⊢ (𝜑 → (𝑃‘(𝑋∘_RV/𝑐 E ∅)) ∈ ℝ)
27	1, 19, 24, 26	fvmptd 7002	. . . . . 6 ⊢ (𝜑 → (𝐷‘∅) = (𝑃‘(𝑋∘_RV/𝑐 E ∅)))
28	2, 4, 24	orvcelval 33455	. . . . . . 7 ⊢ (𝜑 → (𝑋∘_RV/𝑐 E ∅) = (^◡𝑋 “ ∅))
29	28	fveq2d 6892	. . . . . 6 ⊢ (𝜑 → (𝑃‘(𝑋∘_RV/𝑐 E ∅)) = (𝑃‘(^◡𝑋 “ ∅)))
30		ima0 6073	. . . . . . . 8 ⊢ (^◡𝑋 “ ∅) = ∅
31	30	fveq2i 6891	. . . . . . 7 ⊢ (𝑃‘(^◡𝑋 “ ∅)) = (𝑃‘∅)
32		probnul 33401	. . . . . . . 8 ⊢ (𝑃 ∈ Prob → (𝑃‘∅) = 0)
33	2, 32	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑃‘∅) = 0)
34	31, 33	eqtrid 2784	. . . . . 6 ⊢ (𝜑 → (𝑃‘(^◡𝑋 “ ∅)) = 0)
35	27, 29, 34	3eqtrd 2776	. . . . 5 ⊢ (𝜑 → (𝐷‘∅) = 0)
36	2, 4	rrvvf 33431	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ)
37	36	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝑋:∪ dom 𝑃⟶ℝ)
38	37	ffund 6718	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → Fun 𝑋)
39		unipreima 31856	. . . . . . . . . . 11 ⊢ (Fun 𝑋 → (^◡𝑋 “ ∪ 𝑥) = ∪ 𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎))
40	39	fveq2d 6892	. . . . . . . . . 10 ⊢ (Fun 𝑋 → (𝑃‘(^◡𝑋 “ ∪ 𝑥)) = (𝑃‘∪ 𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎)))
41	38, 40	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝑃‘(^◡𝑋 “ ∪ 𝑥)) = (𝑃‘∪ 𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎)))
42	2	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝑃 ∈ Prob)
43		domprobmeas 33397	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
44	42, 43	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝑃 ∈ (measures‘dom 𝑃))
45		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ)
46		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎 𝑥 ≼ ω
47		nfdisj1 5126	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎Disj 𝑎 ∈ 𝑥 𝑎
48	46, 47	nfan 1902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)
49	45, 48	nfan 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑎((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎))
50		simplll 773	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → 𝜑)
51		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → 𝑎 ∈ 𝑥)
52		simpllr 774	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → 𝑥 ∈ 𝒫 𝔅_ℝ)
53		elelpwi 4611	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) → 𝑎 ∈ 𝔅_ℝ)
54	51, 52, 53	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → 𝑎 ∈ 𝔅_ℝ)
55	2, 4	rrvfinvima 33437	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑎 ∈ 𝔅_ℝ (^◡𝑋 “ 𝑎) ∈ dom 𝑃)
56	55	r19.21bi 3248	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → (^◡𝑋 “ 𝑎) ∈ dom 𝑃)
57	50, 54, 56	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → (^◡𝑋 “ 𝑎) ∈ dom 𝑃)
58	57	ex 413	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝑎 ∈ 𝑥 → (^◡𝑋 “ 𝑎) ∈ dom 𝑃))
59	49, 58	ralrimi 3254	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → ∀𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎) ∈ dom 𝑃)
60		simprl 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝑥 ≼ ω)
61		simprr 771	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → Disj 𝑎 ∈ 𝑥 𝑎)
62		disjpreima 31802	. . . . . . . . . . 11 ⊢ ((Fun 𝑋 ∧ Disj 𝑎 ∈ 𝑥 𝑎) → Disj 𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎))
63	38, 61, 62	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → Disj 𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎))
64		measvuni 33200	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (measures‘dom 𝑃) ∧ ∀𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎) ∈ dom 𝑃 ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎))) → (𝑃‘∪ 𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎)) = Σ^*𝑎 ∈ 𝑥(𝑃‘(^◡𝑋 “ 𝑎)))
65	44, 59, 60, 63, 64	syl112anc 1374	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝑃‘∪ 𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎)) = Σ^*𝑎 ∈ 𝑥(𝑃‘(^◡𝑋 “ 𝑎)))
66	41, 65	eqtrd 2772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝑃‘(^◡𝑋 “ ∪ 𝑥)) = Σ^*𝑎 ∈ 𝑥(𝑃‘(^◡𝑋 “ 𝑎)))
67	4	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝑋 ∈ (rRndVar‘𝑃))
68	1	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝐷 = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))))
69	20, 21	mp1i 13	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝔅_ℝ ∈ ∪ ran sigAlgebra)
70		simplr 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝑥 ∈ 𝒫 𝔅_ℝ)
71		sigaclcu 33103	. . . . . . . . . 10 ⊢ ((𝔅_ℝ ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝔅_ℝ ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝔅_ℝ)
72	69, 70, 60, 71	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → ∪ 𝑥 ∈ 𝔅_ℝ)
73	42, 67, 68, 72	dstrvval 33457	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝐷‘∪ 𝑥) = (𝑃‘(^◡𝑋 “ ∪ 𝑥)))
74	1, 9	fvmpt2d 7008	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → (𝐷‘𝑎) = (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))
75	50, 54, 74	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → (𝐷‘𝑎) = (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))
76	42	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → 𝑃 ∈ Prob)
77	67	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → 𝑋 ∈ (rRndVar‘𝑃))
78	76, 77, 54	orvcelval 33455	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → (𝑋∘_RV/𝑐 E 𝑎) = (^◡𝑋 “ 𝑎))
79	78	fveq2d 6892	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = (𝑃‘(^◡𝑋 “ 𝑎)))
80	75, 79	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → (𝐷‘𝑎) = (𝑃‘(^◡𝑋 “ 𝑎)))
81	80	ex 413	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝑎 ∈ 𝑥 → (𝐷‘𝑎) = (𝑃‘(^◡𝑋 “ 𝑎))))
82	49, 81	ralrimi 3254	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → ∀𝑎 ∈ 𝑥 (𝐷‘𝑎) = (𝑃‘(^◡𝑋 “ 𝑎)))
83	49, 82	esumeq2d 33023	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → Σ^𝑎 ∈ 𝑥(𝐷‘𝑎) = Σ^𝑎 ∈ 𝑥(𝑃‘(^◡𝑋 “ 𝑎)))
84	66, 73, 83	3eqtr4d 2782	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝐷‘∪ 𝑥) = Σ^*𝑎 ∈ 𝑥(𝐷‘𝑎))
85	84	ex 413	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) → ((𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎) → (𝐷‘∪ 𝑥) = Σ^*𝑎 ∈ 𝑥(𝐷‘𝑎)))
86	85	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝔅_ℝ((𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎) → (𝐷‘∪ 𝑥) = Σ^*𝑎 ∈ 𝑥(𝐷‘𝑎)))
87		ismeas 33185	. . . . . 6 ⊢ (𝔅_ℝ ∈ ∪ ran sigAlgebra → (𝐷 ∈ (measures‘𝔅_ℝ) ↔ (𝐷:𝔅_ℝ⟶(0[,]+∞) ∧ (𝐷‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝔅_ℝ((𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎) → (𝐷‘∪ 𝑥) = Σ^*𝑎 ∈ 𝑥(𝐷‘𝑎)))))
88	20, 21, 87	mp2b 10	. . . . 5 ⊢ (𝐷 ∈ (measures‘𝔅_ℝ) ↔ (𝐷:𝔅_ℝ⟶(0[,]+∞) ∧ (𝐷‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝔅_ℝ((𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎) → (𝐷‘∪ 𝑥) = Σ^*𝑎 ∈ 𝑥(𝐷‘𝑎))))
89	16, 35, 86, 88	syl3anbrc 1343	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (measures‘𝔅_ℝ))
90	1	dmeqd 5903	. . . . . 6 ⊢ (𝜑 → dom 𝐷 = dom (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))))
91	15	ralrimiva 3146	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ 𝔅_ℝ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]+∞))
92		dmmptg 6238	. . . . . . 7 ⊢ (∀𝑎 ∈ 𝔅_ℝ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]+∞) → dom (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))) = 𝔅_ℝ)
93	91, 92	syl 17	. . . . . 6 ⊢ (𝜑 → dom (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))) = 𝔅_ℝ)
94	90, 93	eqtrd 2772	. . . . 5 ⊢ (𝜑 → dom 𝐷 = 𝔅_ℝ)
95	94	fveq2d 6892	. . . 4 ⊢ (𝜑 → (measures‘dom 𝐷) = (measures‘𝔅_ℝ))
96	89, 95	eleqtrrd 2836	. . 3 ⊢ (𝜑 → 𝐷 ∈ (measures‘dom 𝐷))
97		measbasedom 33188	. . 3 ⊢ (𝐷 ∈ ∪ ran measures ↔ 𝐷 ∈ (measures‘dom 𝐷))
98	96, 97	sylibr 233	. 2 ⊢ (𝜑 → 𝐷 ∈ ∪ ran measures)
99	94	unieqd 4921	. . . . 5 ⊢ (𝜑 → ∪ dom 𝐷 = ∪ 𝔅_ℝ)
100		unibrsiga 33172	. . . . 5 ⊢ ∪ 𝔅_ℝ = ℝ
101	99, 100	eqtrdi 2788	. . . 4 ⊢ (𝜑 → ∪ dom 𝐷 = ℝ)
102	101	fveq2d 6892	. . 3 ⊢ (𝜑 → (𝐷‘∪ dom 𝐷) = (𝐷‘ℝ))
103		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → 𝑎 = ℝ)
104	103	oveq2d 7421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑋∘_RV/𝑐 E 𝑎) = (𝑋∘_RV/𝑐 E ℝ))
105		baselsiga 33101	. . . . . . . . . 10 ⊢ (𝔅_ℝ ∈ (sigAlgebra‘ℝ) → ℝ ∈ 𝔅_ℝ)
106	20, 105	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ 𝔅_ℝ)
107	2, 4, 106	orvcelval 33455	. . . . . . . 8 ⊢ (𝜑 → (𝑋∘_RV/𝑐 E ℝ) = (^◡𝑋 “ ℝ))
108	107	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑋∘_RV/𝑐 E ℝ) = (^◡𝑋 “ ℝ))
109	104, 108	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑋∘_RV/𝑐 E 𝑎) = (^◡𝑋 “ ℝ))
110	109	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = (𝑃‘(^◡𝑋 “ ℝ)))
111		fimacnv 6736	. . . . . . . . 9 ⊢ (𝑋:∪ dom 𝑃⟶ℝ → (^◡𝑋 “ ℝ) = ∪ dom 𝑃)
112	36, 111	syl 17	. . . . . . . 8 ⊢ (𝜑 → (^◡𝑋 “ ℝ) = ∪ dom 𝑃)
113	112	fveq2d 6892	. . . . . . 7 ⊢ (𝜑 → (𝑃‘(^◡𝑋 “ ℝ)) = (𝑃‘∪ dom 𝑃))
114		probtot 33399	. . . . . . . 8 ⊢ (𝑃 ∈ Prob → (𝑃‘∪ dom 𝑃) = 1)
115	2, 114	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑃‘∪ dom 𝑃) = 1)
116	113, 115	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → (𝑃‘(^◡𝑋 “ ℝ)) = 1)
117	116	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑃‘(^◡𝑋 “ ℝ)) = 1)
118	110, 117	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = 1)
119		1red 11211	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
120	1, 118, 106, 119	fvmptd 7002	. . 3 ⊢ (𝜑 → (𝐷‘ℝ) = 1)
121	102, 120	eqtrd 2772	. 2 ⊢ (𝜑 → (𝐷‘∪ dom 𝐷) = 1)
122		elprob 33396	. 2 ⊢ (𝐷 ∈ Prob ↔ (𝐷 ∈ ∪ ran measures ∧ (𝐷‘∪ dom 𝐷) = 1))
123	98, 121, 122	sylanbrc 583	1 ⊢ (𝜑 → 𝐷 ∈ Prob)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∅c0 4321 𝒫 cpw 4601 ∪ cuni 4907 ∪ ciun 4996 Disj wdisj 5112 class class class wbr 5147 ↦ cmpt 5230 E cep 5578 ^◡ccnv 5674 dom cdm 5675 ran crn 5676 “ cima 5678 Fun wfun 6534 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ωcom 7851 ≼ cdom 8933 ℝcr 11105 0cc0 11106 1c1 11107 +∞cpnf 11241 ℝ^*cxr 11243 ≤ cle 11245 [,]cicc 13323 Σ^*cesum 33013 sigAlgebracsiga 33094 𝔅_ℝcbrsiga 33167 measurescmeas 33181 Probcprb 33394 rRndVarcrrv 33427 ∘_RV/𝑐corvc 33442

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-ac2 10454 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-ac 10107 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-ordt 17443 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-ps 18515 df-tsr 18516 df-plusf 18556 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-subrg 20353 df-abv 20417 df-lmod 20465 df-scaf 20466 df-sra 20777 df-rgmod 20778 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-tmd 23567 df-tgp 23568 df-tsms 23622 df-trg 23655 df-xms 23817 df-ms 23818 df-tms 23819 df-nm 24082 df-ngp 24083 df-nrg 24085 df-nlm 24086 df-ii 24384 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-esum 33014 df-siga 33095 df-sigagen 33125 df-brsiga 33168 df-meas 33182 df-mbfm 33236 df-prob 33395 df-rrv 33428 df-orvc 33443

This theorem is referenced by: (None)

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