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