Metamath Proof Explorer


Theorem totprob

Description: Law of total probability. (Contributed by Thierry Arnoux, 25-Dec-2016)

Ref Expression
Assertion totprob ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( 𝐵 = dom 𝑃𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏𝐵 𝑏 ) ) ) → ( 𝑃𝐴 ) = Σ* 𝑏𝐵 ( 𝑃 ‘ ( 𝑏𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 simp1 ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( 𝐵 = dom 𝑃𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏𝐵 𝑏 ) ) ) → 𝑃 ∈ Prob )
2 simp2 ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( 𝐵 = dom 𝑃𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏𝐵 𝑏 ) ) ) → 𝐴 ∈ dom 𝑃 )
3 simp32 ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( 𝐵 = dom 𝑃𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏𝐵 𝑏 ) ) ) → 𝐵 ∈ 𝒫 dom 𝑃 )
4 simp31 ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( 𝐵 = dom 𝑃𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏𝐵 𝑏 ) ) ) → 𝐵 = dom 𝑃 )
5 simp33l ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( 𝐵 = dom 𝑃𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏𝐵 𝑏 ) ) ) → 𝐵 ≼ ω )
6 simp33r ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( 𝐵 = dom 𝑃𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏𝐵 𝑏 ) ) ) → Disj 𝑏𝐵 𝑏 )
7 id ( 𝑏 = 𝑐𝑏 = 𝑐 )
8 7 cbvdisjv ( Disj 𝑏𝐵 𝑏Disj 𝑐𝐵 𝑐 )
9 6 8 sylib ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( 𝐵 = dom 𝑃𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏𝐵 𝑏 ) ) ) → Disj 𝑐𝐵 𝑐 )
10 1 2 3 4 5 9 totprobd ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( 𝐵 = dom 𝑃𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏𝐵 𝑏 ) ) ) → ( 𝑃𝐴 ) = Σ* 𝑐𝐵 ( 𝑃 ‘ ( 𝑐𝐴 ) ) )
11 ineq1 ( 𝑏 = 𝑐 → ( 𝑏𝐴 ) = ( 𝑐𝐴 ) )
12 11 fveq2d ( 𝑏 = 𝑐 → ( 𝑃 ‘ ( 𝑏𝐴 ) ) = ( 𝑃 ‘ ( 𝑐𝐴 ) ) )
13 12 cbvesumv Σ* 𝑏𝐵 ( 𝑃 ‘ ( 𝑏𝐴 ) ) = Σ* 𝑐𝐵 ( 𝑃 ‘ ( 𝑐𝐴 ) )
14 10 13 eqtr4di ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( 𝐵 = dom 𝑃𝐵 ∈ 𝒫 dom 𝑃 ∧ ( 𝐵 ≼ ω ∧ Disj 𝑏𝐵 𝑏 ) ) ) → ( 𝑃𝐴 ) = Σ* 𝑏𝐵 ( 𝑃 ‘ ( 𝑏𝐴 ) ) )