Step |
Hyp |
Ref |
Expression |
1 |
|
inundif |
⊢ ( ( 𝐴 ∩ 𝐵 ) ∪ ( 𝐴 ∖ 𝐵 ) ) = 𝐴 |
2 |
1
|
fveq2i |
⊢ ( 𝑃 ‘ ( ( 𝐴 ∩ 𝐵 ) ∪ ( 𝐴 ∖ 𝐵 ) ) ) = ( 𝑃 ‘ 𝐴 ) |
3 |
|
simp1 |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → 𝑃 ∈ Prob ) |
4 |
|
domprobsiga |
⊢ ( 𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra ) |
5 |
|
inelsiga |
⊢ ( ( dom 𝑃 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( 𝐴 ∩ 𝐵 ) ∈ dom 𝑃 ) |
6 |
4 5
|
syl3an1 |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( 𝐴 ∩ 𝐵 ) ∈ dom 𝑃 ) |
7 |
|
difelsiga |
⊢ ( ( dom 𝑃 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( 𝐴 ∖ 𝐵 ) ∈ dom 𝑃 ) |
8 |
4 7
|
syl3an1 |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( 𝐴 ∖ 𝐵 ) ∈ dom 𝑃 ) |
9 |
|
inindif |
⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐴 ∖ 𝐵 ) ) = ∅ |
10 |
|
probun |
⊢ ( ( 𝑃 ∈ Prob ∧ ( 𝐴 ∩ 𝐵 ) ∈ dom 𝑃 ∧ ( 𝐴 ∖ 𝐵 ) ∈ dom 𝑃 ) → ( ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐴 ∖ 𝐵 ) ) = ∅ → ( 𝑃 ‘ ( ( 𝐴 ∩ 𝐵 ) ∪ ( 𝐴 ∖ 𝐵 ) ) ) = ( ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) + ( 𝑃 ‘ ( 𝐴 ∖ 𝐵 ) ) ) ) ) |
11 |
9 10
|
mpi |
⊢ ( ( 𝑃 ∈ Prob ∧ ( 𝐴 ∩ 𝐵 ) ∈ dom 𝑃 ∧ ( 𝐴 ∖ 𝐵 ) ∈ dom 𝑃 ) → ( 𝑃 ‘ ( ( 𝐴 ∩ 𝐵 ) ∪ ( 𝐴 ∖ 𝐵 ) ) ) = ( ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) + ( 𝑃 ‘ ( 𝐴 ∖ 𝐵 ) ) ) ) |
12 |
3 6 8 11
|
syl3anc |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( 𝑃 ‘ ( ( 𝐴 ∩ 𝐵 ) ∪ ( 𝐴 ∖ 𝐵 ) ) ) = ( ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) + ( 𝑃 ‘ ( 𝐴 ∖ 𝐵 ) ) ) ) |
13 |
2 12
|
eqtr3id |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( 𝑃 ‘ 𝐴 ) = ( ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) + ( 𝑃 ‘ ( 𝐴 ∖ 𝐵 ) ) ) ) |
14 |
13
|
oveq1d |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( ( 𝑃 ‘ 𝐴 ) − ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) ) = ( ( ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) + ( 𝑃 ‘ ( 𝐴 ∖ 𝐵 ) ) ) − ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) ) ) |
15 |
|
unitsscn |
⊢ ( 0 [,] 1 ) ⊆ ℂ |
16 |
|
prob01 |
⊢ ( ( 𝑃 ∈ Prob ∧ ( 𝐴 ∩ 𝐵 ) ∈ dom 𝑃 ) → ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) ∈ ( 0 [,] 1 ) ) |
17 |
3 6 16
|
syl2anc |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) ∈ ( 0 [,] 1 ) ) |
18 |
15 17
|
sselid |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) ∈ ℂ ) |
19 |
|
prob01 |
⊢ ( ( 𝑃 ∈ Prob ∧ ( 𝐴 ∖ 𝐵 ) ∈ dom 𝑃 ) → ( 𝑃 ‘ ( 𝐴 ∖ 𝐵 ) ) ∈ ( 0 [,] 1 ) ) |
20 |
3 8 19
|
syl2anc |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( 𝑃 ‘ ( 𝐴 ∖ 𝐵 ) ) ∈ ( 0 [,] 1 ) ) |
21 |
15 20
|
sselid |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( 𝑃 ‘ ( 𝐴 ∖ 𝐵 ) ) ∈ ℂ ) |
22 |
18 21
|
pncan2d |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( ( ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) + ( 𝑃 ‘ ( 𝐴 ∖ 𝐵 ) ) ) − ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) ) = ( 𝑃 ‘ ( 𝐴 ∖ 𝐵 ) ) ) |
23 |
14 22
|
eqtr2d |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) → ( 𝑃 ‘ ( 𝐴 ∖ 𝐵 ) ) = ( ( 𝑃 ‘ 𝐴 ) − ( 𝑃 ‘ ( 𝐴 ∩ 𝐵 ) ) ) ) |