Theorem prob01

Description: A probability is an element of [ 0 , 1 ]. First axiom of Kolmogorov. (Contributed by Thierry Arnoux, 25-Dec-2016)

	Ref	Expression
Assertion	prob01	⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → ( 𝑃 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) )

Proof

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 ) )