Metamath Proof Explorer


Theorem dstrvprob

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

Ref Expression
Hypotheses dstrvprob.1 ( 𝜑𝑃 ∈ Prob )
dstrvprob.2 ( 𝜑𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
dstrvprob.3 ( 𝜑𝐷 = ( 𝑎 ∈ 𝔅 ↦ ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ) )
Assertion dstrvprob ( 𝜑𝐷 ∈ Prob )

Proof

Step Hyp Ref Expression
1 dstrvprob.1 ( 𝜑𝑃 ∈ Prob )
2 dstrvprob.2 ( 𝜑𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
3 dstrvprob.3 ( 𝜑𝐷 = ( 𝑎 ∈ 𝔅 ↦ ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ) )
4 1 adantr ( ( 𝜑𝑎 ∈ 𝔅 ) → 𝑃 ∈ Prob )
5 2 adantr ( ( 𝜑𝑎 ∈ 𝔅 ) → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
6 simpr ( ( 𝜑𝑎 ∈ 𝔅 ) → 𝑎 ∈ 𝔅 )
7 4 5 6 orvcelel ( ( 𝜑𝑎 ∈ 𝔅 ) → ( 𝑋RV/𝑐 E 𝑎 ) ∈ dom 𝑃 )
8 prob01 ( ( 𝑃 ∈ Prob ∧ ( 𝑋RV/𝑐 E 𝑎 ) ∈ dom 𝑃 ) → ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ( 0 [,] 1 ) )
9 4 7 8 syl2anc ( ( 𝜑𝑎 ∈ 𝔅 ) → ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ( 0 [,] 1 ) )
10 elunitrn ( ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ( 0 [,] 1 ) → ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ℝ )
11 10 rexrd ( ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ( 0 [,] 1 ) → ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ℝ* )
12 elunitge0 ( ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ( 0 [,] 1 ) → 0 ≤ ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) )
13 elxrge0 ( ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ℝ* ∧ 0 ≤ ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ) )
14 11 12 13 sylanbrc ( ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ( 0 [,] 1 ) → ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ( 0 [,] +∞ ) )
15 9 14 syl ( ( 𝜑𝑎 ∈ 𝔅 ) → ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ( 0 [,] +∞ ) )
16 3 15 fmpt3d ( 𝜑𝐷 : 𝔅 ⟶ ( 0 [,] +∞ ) )
17 simpr ( ( 𝜑𝑎 = ∅ ) → 𝑎 = ∅ )
18 17 oveq2d ( ( 𝜑𝑎 = ∅ ) → ( 𝑋RV/𝑐 E 𝑎 ) = ( 𝑋RV/𝑐 E ∅ ) )
19 18 fveq2d ( ( 𝜑𝑎 = ∅ ) → ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) = ( 𝑃 ‘ ( 𝑋RV/𝑐 E ∅ ) ) )
20 brsigarn 𝔅 ∈ ( sigAlgebra ‘ ℝ )
21 elrnsiga ( 𝔅 ∈ ( sigAlgebra ‘ ℝ ) → 𝔅 ran sigAlgebra )
22 0elsiga ( 𝔅 ran sigAlgebra → ∅ ∈ 𝔅 )
23 20 21 22 mp2b ∅ ∈ 𝔅
24 23 a1i ( 𝜑 → ∅ ∈ 𝔅 )
25 1 2 24 orvcelel ( 𝜑 → ( 𝑋RV/𝑐 E ∅ ) ∈ dom 𝑃 )
26 1 25 probvalrnd ( 𝜑 → ( 𝑃 ‘ ( 𝑋RV/𝑐 E ∅ ) ) ∈ ℝ )
27 3 19 24 26 fvmptd ( 𝜑 → ( 𝐷 ‘ ∅ ) = ( 𝑃 ‘ ( 𝑋RV/𝑐 E ∅ ) ) )
28 1 2 24 orvcelval ( 𝜑 → ( 𝑋RV/𝑐 E ∅ ) = ( 𝑋 “ ∅ ) )
29 28 fveq2d ( 𝜑 → ( 𝑃 ‘ ( 𝑋RV/𝑐 E ∅ ) ) = ( 𝑃 ‘ ( 𝑋 “ ∅ ) ) )
30 ima0 ( 𝑋 “ ∅ ) = ∅
31 30 fveq2i ( 𝑃 ‘ ( 𝑋 “ ∅ ) ) = ( 𝑃 ‘ ∅ )
32 probnul ( 𝑃 ∈ Prob → ( 𝑃 ‘ ∅ ) = 0 )
33 1 32 syl ( 𝜑 → ( 𝑃 ‘ ∅ ) = 0 )
34 31 33 syl5eq ( 𝜑 → ( 𝑃 ‘ ( 𝑋 “ ∅ ) ) = 0 )
35 27 29 34 3eqtrd ( 𝜑 → ( 𝐷 ‘ ∅ ) = 0 )
36 1 2 rrvvf ( 𝜑𝑋 : dom 𝑃 ⟶ ℝ )
37 36 ad2antrr ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → 𝑋 : dom 𝑃 ⟶ ℝ )
38 37 ffund ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → Fun 𝑋 )
39 unipreima ( Fun 𝑋 → ( 𝑋 𝑥 ) = 𝑎𝑥 ( 𝑋𝑎 ) )
40 39 fveq2d ( Fun 𝑋 → ( 𝑃 ‘ ( 𝑋 𝑥 ) ) = ( 𝑃 𝑎𝑥 ( 𝑋𝑎 ) ) )
41 38 40 syl ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → ( 𝑃 ‘ ( 𝑋 𝑥 ) ) = ( 𝑃 𝑎𝑥 ( 𝑋𝑎 ) ) )
42 1 ad2antrr ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → 𝑃 ∈ Prob )
43 domprobmeas ( 𝑃 ∈ Prob → 𝑃 ∈ ( measures ‘ dom 𝑃 ) )
44 42 43 syl ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → 𝑃 ∈ ( measures ‘ dom 𝑃 ) )
45 nfv 𝑎 ( 𝜑𝑥 ∈ 𝒫 𝔅 )
46 nfv 𝑎 𝑥 ≼ ω
47 nfdisj1 𝑎 Disj 𝑎𝑥 𝑎
48 46 47 nfan 𝑎 ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 )
49 45 48 nfan 𝑎 ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) )
50 simplll ( ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) ∧ 𝑎𝑥 ) → 𝜑 )
51 simpr ( ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) ∧ 𝑎𝑥 ) → 𝑎𝑥 )
52 simpllr ( ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) ∧ 𝑎𝑥 ) → 𝑥 ∈ 𝒫 𝔅 )
53 elelpwi ( ( 𝑎𝑥𝑥 ∈ 𝒫 𝔅 ) → 𝑎 ∈ 𝔅 )
54 51 52 53 syl2anc ( ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) ∧ 𝑎𝑥 ) → 𝑎 ∈ 𝔅 )
55 1 2 rrvfinvima ( 𝜑 → ∀ 𝑎 ∈ 𝔅 ( 𝑋𝑎 ) ∈ dom 𝑃 )
56 55 r19.21bi ( ( 𝜑𝑎 ∈ 𝔅 ) → ( 𝑋𝑎 ) ∈ dom 𝑃 )
57 50 54 56 syl2anc ( ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) ∧ 𝑎𝑥 ) → ( 𝑋𝑎 ) ∈ dom 𝑃 )
58 57 ex ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → ( 𝑎𝑥 → ( 𝑋𝑎 ) ∈ dom 𝑃 ) )
59 49 58 ralrimi ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → ∀ 𝑎𝑥 ( 𝑋𝑎 ) ∈ dom 𝑃 )
60 simprl ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → 𝑥 ≼ ω )
61 simprr ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → Disj 𝑎𝑥 𝑎 )
62 disjpreima ( ( Fun 𝑋Disj 𝑎𝑥 𝑎 ) → Disj 𝑎𝑥 ( 𝑋𝑎 ) )
63 38 61 62 syl2anc ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → Disj 𝑎𝑥 ( 𝑋𝑎 ) )
64 measvuni ( ( 𝑃 ∈ ( measures ‘ dom 𝑃 ) ∧ ∀ 𝑎𝑥 ( 𝑋𝑎 ) ∈ dom 𝑃 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 ( 𝑋𝑎 ) ) ) → ( 𝑃 𝑎𝑥 ( 𝑋𝑎 ) ) = Σ* 𝑎𝑥 ( 𝑃 ‘ ( 𝑋𝑎 ) ) )
65 44 59 60 63 64 syl112anc ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → ( 𝑃 𝑎𝑥 ( 𝑋𝑎 ) ) = Σ* 𝑎𝑥 ( 𝑃 ‘ ( 𝑋𝑎 ) ) )
66 41 65 eqtrd ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → ( 𝑃 ‘ ( 𝑋 𝑥 ) ) = Σ* 𝑎𝑥 ( 𝑃 ‘ ( 𝑋𝑎 ) ) )
67 2 ad2antrr ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
68 3 ad2antrr ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → 𝐷 = ( 𝑎 ∈ 𝔅 ↦ ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ) )
69 20 21 mp1i ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → 𝔅 ran sigAlgebra )
70 simplr ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → 𝑥 ∈ 𝒫 𝔅 )
71 sigaclcu ( ( 𝔅 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝔅𝑥 ≼ ω ) → 𝑥 ∈ 𝔅 )
72 69 70 60 71 syl3anc ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → 𝑥 ∈ 𝔅 )
73 42 67 68 72 dstrvval ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → ( 𝐷 𝑥 ) = ( 𝑃 ‘ ( 𝑋 𝑥 ) ) )
74 3 9 fvmpt2d ( ( 𝜑𝑎 ∈ 𝔅 ) → ( 𝐷𝑎 ) = ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) )
75 50 54 74 syl2anc ( ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) ∧ 𝑎𝑥 ) → ( 𝐷𝑎 ) = ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) )
76 42 adantr ( ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) ∧ 𝑎𝑥 ) → 𝑃 ∈ Prob )
77 67 adantr ( ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) ∧ 𝑎𝑥 ) → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
78 76 77 54 orvcelval ( ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) ∧ 𝑎𝑥 ) → ( 𝑋RV/𝑐 E 𝑎 ) = ( 𝑋𝑎 ) )
79 78 fveq2d ( ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) ∧ 𝑎𝑥 ) → ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) = ( 𝑃 ‘ ( 𝑋𝑎 ) ) )
80 75 79 eqtrd ( ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) ∧ 𝑎𝑥 ) → ( 𝐷𝑎 ) = ( 𝑃 ‘ ( 𝑋𝑎 ) ) )
81 80 ex ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → ( 𝑎𝑥 → ( 𝐷𝑎 ) = ( 𝑃 ‘ ( 𝑋𝑎 ) ) ) )
82 49 81 ralrimi ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → ∀ 𝑎𝑥 ( 𝐷𝑎 ) = ( 𝑃 ‘ ( 𝑋𝑎 ) ) )
83 49 82 esumeq2d ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → Σ* 𝑎𝑥 ( 𝐷𝑎 ) = Σ* 𝑎𝑥 ( 𝑃 ‘ ( 𝑋𝑎 ) ) )
84 66 73 83 3eqtr4d ( ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) ∧ ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) ) → ( 𝐷 𝑥 ) = Σ* 𝑎𝑥 ( 𝐷𝑎 ) )
85 84 ex ( ( 𝜑𝑥 ∈ 𝒫 𝔅 ) → ( ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) → ( 𝐷 𝑥 ) = Σ* 𝑎𝑥 ( 𝐷𝑎 ) ) )
86 85 ralrimiva ( 𝜑 → ∀ 𝑥 ∈ 𝒫 𝔅 ( ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) → ( 𝐷 𝑥 ) = Σ* 𝑎𝑥 ( 𝐷𝑎 ) ) )
87 ismeas ( 𝔅 ran sigAlgebra → ( 𝐷 ∈ ( measures ‘ 𝔅 ) ↔ ( 𝐷 : 𝔅 ⟶ ( 0 [,] +∞ ) ∧ ( 𝐷 ‘ ∅ ) = 0 ∧ ∀ 𝑥 ∈ 𝒫 𝔅 ( ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) → ( 𝐷 𝑥 ) = Σ* 𝑎𝑥 ( 𝐷𝑎 ) ) ) ) )
88 20 21 87 mp2b ( 𝐷 ∈ ( measures ‘ 𝔅 ) ↔ ( 𝐷 : 𝔅 ⟶ ( 0 [,] +∞ ) ∧ ( 𝐷 ‘ ∅ ) = 0 ∧ ∀ 𝑥 ∈ 𝒫 𝔅 ( ( 𝑥 ≼ ω ∧ Disj 𝑎𝑥 𝑎 ) → ( 𝐷 𝑥 ) = Σ* 𝑎𝑥 ( 𝐷𝑎 ) ) ) )
89 16 35 86 88 syl3anbrc ( 𝜑𝐷 ∈ ( measures ‘ 𝔅 ) )
90 3 dmeqd ( 𝜑 → dom 𝐷 = dom ( 𝑎 ∈ 𝔅 ↦ ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ) )
91 15 ralrimiva ( 𝜑 → ∀ 𝑎 ∈ 𝔅 ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ( 0 [,] +∞ ) )
92 dmmptg ( ∀ 𝑎 ∈ 𝔅 ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ∈ ( 0 [,] +∞ ) → dom ( 𝑎 ∈ 𝔅 ↦ ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ) = 𝔅 )
93 91 92 syl ( 𝜑 → dom ( 𝑎 ∈ 𝔅 ↦ ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) ) = 𝔅 )
94 90 93 eqtrd ( 𝜑 → dom 𝐷 = 𝔅 )
95 94 fveq2d ( 𝜑 → ( measures ‘ dom 𝐷 ) = ( measures ‘ 𝔅 ) )
96 89 95 eleqtrrd ( 𝜑𝐷 ∈ ( measures ‘ dom 𝐷 ) )
97 measbasedom ( 𝐷 ran measures ↔ 𝐷 ∈ ( measures ‘ dom 𝐷 ) )
98 96 97 sylibr ( 𝜑𝐷 ran measures )
99 94 unieqd ( 𝜑 dom 𝐷 = 𝔅 )
100 unibrsiga 𝔅 = ℝ
101 99 100 eqtrdi ( 𝜑 dom 𝐷 = ℝ )
102 101 fveq2d ( 𝜑 → ( 𝐷 dom 𝐷 ) = ( 𝐷 ‘ ℝ ) )
103 simpr ( ( 𝜑𝑎 = ℝ ) → 𝑎 = ℝ )
104 103 oveq2d ( ( 𝜑𝑎 = ℝ ) → ( 𝑋RV/𝑐 E 𝑎 ) = ( 𝑋RV/𝑐 E ℝ ) )
105 baselsiga ( 𝔅 ∈ ( sigAlgebra ‘ ℝ ) → ℝ ∈ 𝔅 )
106 20 105 mp1i ( 𝜑 → ℝ ∈ 𝔅 )
107 1 2 106 orvcelval ( 𝜑 → ( 𝑋RV/𝑐 E ℝ ) = ( 𝑋 “ ℝ ) )
108 107 adantr ( ( 𝜑𝑎 = ℝ ) → ( 𝑋RV/𝑐 E ℝ ) = ( 𝑋 “ ℝ ) )
109 104 108 eqtrd ( ( 𝜑𝑎 = ℝ ) → ( 𝑋RV/𝑐 E 𝑎 ) = ( 𝑋 “ ℝ ) )
110 109 fveq2d ( ( 𝜑𝑎 = ℝ ) → ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) = ( 𝑃 ‘ ( 𝑋 “ ℝ ) ) )
111 fimacnv ( 𝑋 : dom 𝑃 ⟶ ℝ → ( 𝑋 “ ℝ ) = dom 𝑃 )
112 36 111 syl ( 𝜑 → ( 𝑋 “ ℝ ) = dom 𝑃 )
113 112 fveq2d ( 𝜑 → ( 𝑃 ‘ ( 𝑋 “ ℝ ) ) = ( 𝑃 dom 𝑃 ) )
114 probtot ( 𝑃 ∈ Prob → ( 𝑃 dom 𝑃 ) = 1 )
115 1 114 syl ( 𝜑 → ( 𝑃 dom 𝑃 ) = 1 )
116 113 115 eqtrd ( 𝜑 → ( 𝑃 ‘ ( 𝑋 “ ℝ ) ) = 1 )
117 116 adantr ( ( 𝜑𝑎 = ℝ ) → ( 𝑃 ‘ ( 𝑋 “ ℝ ) ) = 1 )
118 110 117 eqtrd ( ( 𝜑𝑎 = ℝ ) → ( 𝑃 ‘ ( 𝑋RV/𝑐 E 𝑎 ) ) = 1 )
119 1red ( 𝜑 → 1 ∈ ℝ )
120 3 118 106 119 fvmptd ( 𝜑 → ( 𝐷 ‘ ℝ ) = 1 )
121 102 120 eqtrd ( 𝜑 → ( 𝐷 dom 𝐷 ) = 1 )
122 elprob ( 𝐷 ∈ Prob ↔ ( 𝐷 ran measures ∧ ( 𝐷 dom 𝐷 ) = 1 ) )
123 98 121 122 sylanbrc ( 𝜑𝐷 ∈ Prob )