Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( ∪ 𝐵 = ∪ dom 𝑃 ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏 ∈ 𝐵 𝑏 ) ) ) → 𝑃 ∈ Prob ) |
2 |
|
simp2 |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( ∪ 𝐵 = ∪ dom 𝑃 ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏 ∈ 𝐵 𝑏 ) ) ) → 𝐴 ∈ dom 𝑃 ) |
3 |
|
simp32 |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( ∪ 𝐵 = ∪ dom 𝑃 ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏 ∈ 𝐵 𝑏 ) ) ) → 𝐵 ∈ 𝒫 dom 𝑃 ) |
4 |
|
simp31 |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( ∪ 𝐵 = ∪ dom 𝑃 ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏 ∈ 𝐵 𝑏 ) ) ) → ∪ 𝐵 = ∪ dom 𝑃 ) |
5 |
|
simp33l |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( ∪ 𝐵 = ∪ dom 𝑃 ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏 ∈ 𝐵 𝑏 ) ) ) → 𝐵 ≼ ω ) |
6 |
|
simp33r |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( ∪ 𝐵 = ∪ dom 𝑃 ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏 ∈ 𝐵 𝑏 ) ) ) → Disj 𝑏 ∈ 𝐵 𝑏 ) |
7 |
|
id |
⊢ ( 𝑏 = 𝑐 → 𝑏 = 𝑐 ) |
8 |
7
|
cbvdisjv |
⊢ ( Disj 𝑏 ∈ 𝐵 𝑏 ↔ Disj 𝑐 ∈ 𝐵 𝑐 ) |
9 |
6 8
|
sylib |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( ∪ 𝐵 = ∪ dom 𝑃 ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏 ∈ 𝐵 𝑏 ) ) ) → Disj 𝑐 ∈ 𝐵 𝑐 ) |
10 |
1 2 3 4 5 9
|
totprobd |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( ∪ 𝐵 = ∪ dom 𝑃 ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏 ∈ 𝐵 𝑏 ) ) ) → ( 𝑃 ‘ 𝐴 ) = Σ* 𝑐 ∈ 𝐵 ( 𝑃 ‘ ( 𝑐 ∩ 𝐴 ) ) ) |
11 |
|
ineq1 |
⊢ ( 𝑏 = 𝑐 → ( 𝑏 ∩ 𝐴 ) = ( 𝑐 ∩ 𝐴 ) ) |
12 |
11
|
fveq2d |
⊢ ( 𝑏 = 𝑐 → ( 𝑃 ‘ ( 𝑏 ∩ 𝐴 ) ) = ( 𝑃 ‘ ( 𝑐 ∩ 𝐴 ) ) ) |
13 |
12
|
cbvesumv |
⊢ Σ* 𝑏 ∈ 𝐵 ( 𝑃 ‘ ( 𝑏 ∩ 𝐴 ) ) = Σ* 𝑐 ∈ 𝐵 ( 𝑃 ‘ ( 𝑐 ∩ 𝐴 ) ) |
14 |
10 13
|
eqtr4di |
⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( ∪ 𝐵 = ∪ dom 𝑃 ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏 ∈ 𝐵 𝑏 ) ) ) → ( 𝑃 ‘ 𝐴 ) = Σ* 𝑏 ∈ 𝐵 ( 𝑃 ‘ ( 𝑏 ∩ 𝐴 ) ) ) |