dstrvprob - Mathbox for Thierry Arnoux

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Theorem dstrvprob 33768

Description: The distribution of a random variable is a probability law. (TODO: could be shortened using dstrvval 33767). (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 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → 𝑃 ∈ Prob)
4		dstrvprob.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
5	4	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → 𝑋 ∈ (rRndVar‘𝑃))
6		simpr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → 𝑎 ∈ 𝔅_ℝ)
7	3, 5, 6	orvcelel 33766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → (𝑋∘_RV/𝑐 E 𝑎) ∈ dom 𝑃)
8		prob01 33710	. . . . . . . 8 ⊢ ((𝑃 ∈ Prob ∧ (𝑋∘_RV/𝑐 E 𝑎) ∈ dom 𝑃) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]1))
9	3, 7, 8	syl2anc 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]1))
10		elunitrn 13448	. . . . . . . . 9 ⊢ ((𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]1) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ ℝ)
11	10	rexrd 11268	. . . . . . . 8 ⊢ ((𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]1) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ ℝ^*)
12		elunitge0 33177	. . . . . . . 8 ⊢ ((𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]1) → 0 ≤ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))
13		elxrge0 13438	. . . . . . . 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 581	. . . . . . 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 7116	. . . . 5 ⊢ (𝜑 → 𝐷:𝔅_ℝ⟶(0[,]+∞))
17		simpr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 = ∅) → 𝑎 = ∅)
18	17	oveq2d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 = ∅) → (𝑋∘_RV/𝑐 E 𝑎) = (𝑋∘_RV/𝑐 E ∅))
19	18	fveq2d 6894	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 = ∅) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = (𝑃‘(𝑋∘_RV/𝑐 E ∅)))
20		brsigarn 33480	. . . . . . . . 9 ⊢ 𝔅_ℝ ∈ (sigAlgebra‘ℝ)
21		elrnsiga 33422	. . . . . . . . 9 ⊢ (𝔅_ℝ ∈ (sigAlgebra‘ℝ) → 𝔅_ℝ ∈ ∪ ran sigAlgebra)
22		0elsiga 33410	. . . . . . . . 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 33766	. . . . . . . 8 ⊢ (𝜑 → (𝑋∘_RV/𝑐 E ∅) ∈ dom 𝑃)
26	2, 25	probvalrnd 33721	. . . . . . 7 ⊢ (𝜑 → (𝑃‘(𝑋∘_RV/𝑐 E ∅)) ∈ ℝ)
27	1, 19, 24, 26	fvmptd 7004	. . . . . 6 ⊢ (𝜑 → (𝐷‘∅) = (𝑃‘(𝑋∘_RV/𝑐 E ∅)))
28	2, 4, 24	orvcelval 33765	. . . . . . 7 ⊢ (𝜑 → (𝑋∘_RV/𝑐 E ∅) = (^◡𝑋 “ ∅))
29	28	fveq2d 6894	. . . . . 6 ⊢ (𝜑 → (𝑃‘(𝑋∘_RV/𝑐 E ∅)) = (𝑃‘(^◡𝑋 “ ∅)))
30		ima0 6075	. . . . . . . 8 ⊢ (^◡𝑋 “ ∅) = ∅
31	30	fveq2i 6893	. . . . . . 7 ⊢ (𝑃‘(^◡𝑋 “ ∅)) = (𝑃‘∅)
32		probnul 33711	. . . . . . . 8 ⊢ (𝑃 ∈ Prob → (𝑃‘∅) = 0)
33	2, 32	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑃‘∅) = 0)
34	31, 33	eqtrid 2782	. . . . . 6 ⊢ (𝜑 → (𝑃‘(^◡𝑋 “ ∅)) = 0)
35	27, 29, 34	3eqtrd 2774	. . . . 5 ⊢ (𝜑 → (𝐷‘∅) = 0)
36	2, 4	rrvvf 33741	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ)
37	36	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝑋:∪ dom 𝑃⟶ℝ)
38	37	ffund 6720	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → Fun 𝑋)
39		unipreima 32136	. . . . . . . . . . 11 ⊢ (Fun 𝑋 → (^◡𝑋 “ ∪ 𝑥) = ∪ 𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎))
40	39	fveq2d 6894	. . . . . . . . . 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 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝑃 ∈ Prob)
43		domprobmeas 33707	. . . . . . . . . . 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 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ)
46		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎 𝑥 ≼ ω
47		nfdisj1 5126	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎Disj 𝑎 ∈ 𝑥 𝑎
48	46, 47	nfan 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)
49	45, 48	nfan 1900	. . . . . . . . . . 11 ⊢ Ⅎ𝑎((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎))
50		simplll 771	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → 𝜑)
51		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → 𝑎 ∈ 𝑥)
52		simpllr 772	. . . . . . . . . . . . . 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 582	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → 𝑎 ∈ 𝔅_ℝ)
55	2, 4	rrvfinvima 33747	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑎 ∈ 𝔅_ℝ (^◡𝑋 “ 𝑎) ∈ dom 𝑃)
56	55	r19.21bi 3246	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → (^◡𝑋 “ 𝑎) ∈ dom 𝑃)
57	50, 54, 56	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → (^◡𝑋 “ 𝑎) ∈ dom 𝑃)
58	57	ex 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝑎 ∈ 𝑥 → (^◡𝑋 “ 𝑎) ∈ dom 𝑃))
59	49, 58	ralrimi 3252	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → ∀𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎) ∈ dom 𝑃)
60		simprl 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝑥 ≼ ω)
61		simprr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → Disj 𝑎 ∈ 𝑥 𝑎)
62		disjpreima 32082	. . . . . . . . . . 11 ⊢ ((Fun 𝑋 ∧ Disj 𝑎 ∈ 𝑥 𝑎) → Disj 𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎))
63	38, 61, 62	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → Disj 𝑎 ∈ 𝑥 (^◡𝑋 “ 𝑎))
64		measvuni 33510	. . . . . . . . . 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 1372	. . . . . . . . 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 2770	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝑃‘(^◡𝑋 “ ∪ 𝑥)) = Σ^*𝑎 ∈ 𝑥(𝑃‘(^◡𝑋 “ 𝑎)))
67	4	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝑋 ∈ (rRndVar‘𝑃))
68	1	ad2antrr 722	. . . . . . . . 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 765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → 𝑥 ∈ 𝒫 𝔅_ℝ)
71		sigaclcu 33413	. . . . . . . . . 10 ⊢ ((𝔅_ℝ ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝔅_ℝ ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝔅_ℝ)
72	69, 70, 60, 71	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → ∪ 𝑥 ∈ 𝔅_ℝ)
73	42, 67, 68, 72	dstrvval 33767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝐷‘∪ 𝑥) = (𝑃‘(^◡𝑋 “ ∪ 𝑥)))
74	1, 9	fvmpt2d 7010	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝔅_ℝ) → (𝐷‘𝑎) = (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))
75	50, 54, 74	syl2anc 582	. . . . . . . . . . . 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 479	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → 𝑃 ∈ Prob)
77	67	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → 𝑋 ∈ (rRndVar‘𝑃))
78	76, 77, 54	orvcelval 33765	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → (𝑋∘_RV/𝑐 E 𝑎) = (^◡𝑋 “ 𝑎))
79	78	fveq2d 6894	. . . . . . . . . . . 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 2770	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) ∧ 𝑎 ∈ 𝑥) → (𝐷‘𝑎) = (𝑃‘(^◡𝑋 “ 𝑎)))
81	80	ex 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝑎 ∈ 𝑥 → (𝐷‘𝑎) = (𝑃‘(^◡𝑋 “ 𝑎))))
82	49, 81	ralrimi 3252	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → ∀𝑎 ∈ 𝑥 (𝐷‘𝑎) = (𝑃‘(^◡𝑋 “ 𝑎)))
83	49, 82	esumeq2d 33333	. . . . . . . 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 2780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) ∧ (𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎)) → (𝐷‘∪ 𝑥) = Σ^*𝑎 ∈ 𝑥(𝐷‘𝑎))
85	84	ex 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝔅_ℝ) → ((𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎) → (𝐷‘∪ 𝑥) = Σ^*𝑎 ∈ 𝑥(𝐷‘𝑎)))
86	85	ralrimiva 3144	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝔅_ℝ((𝑥 ≼ ω ∧ Disj 𝑎 ∈ 𝑥 𝑎) → (𝐷‘∪ 𝑥) = Σ^*𝑎 ∈ 𝑥(𝐷‘𝑎)))
87		ismeas 33495	. . . . . 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 1341	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (measures‘𝔅_ℝ))
90	1	dmeqd 5904	. . . . . 6 ⊢ (𝜑 → dom 𝐷 = dom (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))))
91	15	ralrimiva 3144	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ 𝔅_ℝ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) ∈ (0[,]+∞))
92		dmmptg 6240	. . . . . . 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 2770	. . . . 5 ⊢ (𝜑 → dom 𝐷 = 𝔅_ℝ)
95	94	fveq2d 6894	. . . 4 ⊢ (𝜑 → (measures‘dom 𝐷) = (measures‘𝔅_ℝ))
96	89, 95	eleqtrrd 2834	. . 3 ⊢ (𝜑 → 𝐷 ∈ (measures‘dom 𝐷))
97		measbasedom 33498	. . 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 33482	. . . . 5 ⊢ ∪ 𝔅_ℝ = ℝ
101	99, 100	eqtrdi 2786	. . . 4 ⊢ (𝜑 → ∪ dom 𝐷 = ℝ)
102	101	fveq2d 6894	. . 3 ⊢ (𝜑 → (𝐷‘∪ dom 𝐷) = (𝐷‘ℝ))
103		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → 𝑎 = ℝ)
104	103	oveq2d 7427	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑋∘_RV/𝑐 E 𝑎) = (𝑋∘_RV/𝑐 E ℝ))
105		baselsiga 33411	. . . . . . . . . 10 ⊢ (𝔅_ℝ ∈ (sigAlgebra‘ℝ) → ℝ ∈ 𝔅_ℝ)
106	20, 105	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ 𝔅_ℝ)
107	2, 4, 106	orvcelval 33765	. . . . . . . 8 ⊢ (𝜑 → (𝑋∘_RV/𝑐 E ℝ) = (^◡𝑋 “ ℝ))
108	107	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑋∘_RV/𝑐 E ℝ) = (^◡𝑋 “ ℝ))
109	104, 108	eqtrd 2770	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑋∘_RV/𝑐 E 𝑎) = (^◡𝑋 “ ℝ))
110	109	fveq2d 6894	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = (𝑃‘(^◡𝑋 “ ℝ)))
111		fimacnv 6738	. . . . . . . . 9 ⊢ (𝑋:∪ dom 𝑃⟶ℝ → (^◡𝑋 “ ℝ) = ∪ dom 𝑃)
112	36, 111	syl 17	. . . . . . . 8 ⊢ (𝜑 → (^◡𝑋 “ ℝ) = ∪ dom 𝑃)
113	112	fveq2d 6894	. . . . . . 7 ⊢ (𝜑 → (𝑃‘(^◡𝑋 “ ℝ)) = (𝑃‘∪ dom 𝑃))
114		probtot 33709	. . . . . . . 8 ⊢ (𝑃 ∈ Prob → (𝑃‘∪ dom 𝑃) = 1)
115	2, 114	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑃‘∪ dom 𝑃) = 1)
116	113, 115	eqtrd 2770	. . . . . 6 ⊢ (𝜑 → (𝑃‘(^◡𝑋 “ ℝ)) = 1)
117	116	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑃‘(^◡𝑋 “ ℝ)) = 1)
118	110, 117	eqtrd 2770	. . . 4 ⊢ ((𝜑 ∧ 𝑎 = ℝ) → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = 1)
119		1red 11219	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
120	1, 118, 106, 119	fvmptd 7004	. . 3 ⊢ (𝜑 → (𝐷‘ℝ) = 1)
121	102, 120	eqtrd 2770	. 2 ⊢ (𝜑 → (𝐷‘∪ dom 𝐷) = 1)
122		elprob 33706	. 2 ⊢ (𝐷 ∈ Prob ↔ (𝐷 ∈ ∪ ran measures ∧ (𝐷‘∪ dom 𝐷) = 1))
123	98, 121, 122	sylanbrc 581	1 ⊢ (𝜑 → 𝐷 ∈ Prob)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∅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 6536 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ωcom 7857 ≼ cdom 8939 ℝcr 11111 0cc0 11112 1c1 11113 +∞cpnf 11249 ℝ^*cxr 11251 ≤ cle 11253 [,]cicc 13331 Σ^*cesum 33323 sigAlgebracsiga 33404 𝔅_ℝcbrsiga 33477 measurescmeas 33491 Probcprb 33704 rRndVarcrrv 33737 ∘_RV/𝑐corvc 33752

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-ordt 17451 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-ps 18523 df-tsr 18524 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-subrng 20434 df-subrg 20459 df-abv 20568 df-lmod 20616 df-scaf 20617 df-sra 20930 df-rgmod 20931 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-tmd 23796 df-tgp 23797 df-tsms 23851 df-trg 23884 df-xms 24046 df-ms 24047 df-tms 24048 df-nm 24311 df-ngp 24312 df-nrg 24314 df-nlm 24315 df-ii 24617 df-cncf 24618 df-limc 25615 df-dv 25616 df-log 26301 df-esum 33324 df-siga 33405 df-sigagen 33435 df-brsiga 33478 df-meas 33492 df-mbfm 33546 df-prob 33705 df-rrv 33738 df-orvc 33753

This theorem is referenced by: (None)

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