Step |
Hyp |
Ref |
Expression |
1 |
|
domprobmeas |
⊢ ( 𝑃 ∈ Prob → 𝑃 ∈ ( measures ‘ dom 𝑃 ) ) |
2 |
|
measvxrge0 |
⊢ ( ( 𝑃 ∈ ( measures ‘ dom 𝑃 ) ∧ 𝐴 ∈ dom 𝑃 ) → ( 𝑃 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
3 |
1 2
|
sylan |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → ( 𝑃 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
4 |
|
elxrge0 |
⊢ ( ( 𝑃 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝑃 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑃 ‘ 𝐴 ) ) ) |
5 |
3 4
|
sylib |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → ( ( 𝑃 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑃 ‘ 𝐴 ) ) ) |
6 |
1
|
adantr |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → 𝑃 ∈ ( measures ‘ dom 𝑃 ) ) |
7 |
|
simpr |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → 𝐴 ∈ dom 𝑃 ) |
8 |
|
measbase |
⊢ ( 𝑃 ∈ ( measures ‘ dom 𝑃 ) → dom 𝑃 ∈ ∪ ran sigAlgebra ) |
9 |
|
unielsiga |
⊢ ( dom 𝑃 ∈ ∪ ran sigAlgebra → ∪ dom 𝑃 ∈ dom 𝑃 ) |
10 |
6 8 9
|
3syl |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → ∪ dom 𝑃 ∈ dom 𝑃 ) |
11 |
|
elssuni |
⊢ ( 𝐴 ∈ dom 𝑃 → 𝐴 ⊆ ∪ dom 𝑃 ) |
12 |
11
|
adantl |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → 𝐴 ⊆ ∪ dom 𝑃 ) |
13 |
6 7 10 12
|
measssd |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → ( 𝑃 ‘ 𝐴 ) ≤ ( 𝑃 ‘ ∪ dom 𝑃 ) ) |
14 |
|
probtot |
⊢ ( 𝑃 ∈ Prob → ( 𝑃 ‘ ∪ dom 𝑃 ) = 1 ) |
15 |
14
|
breq2d |
⊢ ( 𝑃 ∈ Prob → ( ( 𝑃 ‘ 𝐴 ) ≤ ( 𝑃 ‘ ∪ dom 𝑃 ) ↔ ( 𝑃 ‘ 𝐴 ) ≤ 1 ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → ( ( 𝑃 ‘ 𝐴 ) ≤ ( 𝑃 ‘ ∪ dom 𝑃 ) ↔ ( 𝑃 ‘ 𝐴 ) ≤ 1 ) ) |
17 |
13 16
|
mpbid |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → ( 𝑃 ‘ 𝐴 ) ≤ 1 ) |
18 |
|
df-3an |
⊢ ( ( ( 𝑃 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑃 ‘ 𝐴 ) ∧ ( 𝑃 ‘ 𝐴 ) ≤ 1 ) ↔ ( ( ( 𝑃 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑃 ‘ 𝐴 ) ) ∧ ( 𝑃 ‘ 𝐴 ) ≤ 1 ) ) |
19 |
5 17 18
|
sylanbrc |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → ( ( 𝑃 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑃 ‘ 𝐴 ) ∧ ( 𝑃 ‘ 𝐴 ) ≤ 1 ) ) |
20 |
|
0xr |
⊢ 0 ∈ ℝ* |
21 |
|
1xr |
⊢ 1 ∈ ℝ* |
22 |
|
elicc1 |
⊢ ( ( 0 ∈ ℝ* ∧ 1 ∈ ℝ* ) → ( ( 𝑃 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) ↔ ( ( 𝑃 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑃 ‘ 𝐴 ) ∧ ( 𝑃 ‘ 𝐴 ) ≤ 1 ) ) ) |
23 |
20 21 22
|
mp2an |
⊢ ( ( 𝑃 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) ↔ ( ( 𝑃 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑃 ‘ 𝐴 ) ∧ ( 𝑃 ‘ 𝐴 ) ≤ 1 ) ) |
24 |
19 23
|
sylibr |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → ( 𝑃 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) ) |