Theorem omef

Description: An outer measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	omef.o	⊢ ( 𝜑 → 𝑂 ∈ OutMeas )
	omef.x	⊢ 𝑋 = ∪ dom 𝑂
Assertion	omef	⊢ ( 𝜑 → 𝑂 : 𝒫 𝑋 ⟶ ( 0 [,] +∞ ) )

Proof

Step	Hyp	Ref	Expression
1		omef.o	⊢ ( 𝜑 → 𝑂 ∈ OutMeas )
2		omef.x	⊢ 𝑋 = ∪ dom 𝑂
3		isome	⊢ ( 𝑂 ∈ OutMeas → ( 𝑂 ∈ OutMeas ↔ ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂 ( 𝑦 ≼ ω → ( 𝑂 ‘ ∪ 𝑦 ) ≤ ( Σ^ ‘ ( 𝑂 ↾ 𝑦 ) ) ) ) ) )
4	1 3	syl	⊢ ( 𝜑 → ( 𝑂 ∈ OutMeas ↔ ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂 ( 𝑦 ≼ ω → ( 𝑂 ‘ ∪ 𝑦 ) ≤ ( Σ^ ‘ ( 𝑂 ↾ 𝑦 ) ) ) ) ) )
5	1 4	mpbid	⊢ ( 𝜑 → ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂 ( 𝑦 ≼ ω → ( 𝑂 ‘ ∪ 𝑦 ) ≤ ( Σ^ ‘ ( 𝑂 ↾ 𝑦 ) ) ) ) )
6	5	simplld	⊢ ( 𝜑 → ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) )
7	6	simplld	⊢ ( 𝜑 → 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) )
8	2	pweqi	⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
9		simp-4r	⊢ ( ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂 ( 𝑦 ≼ ω → ( 𝑂 ‘ ∪ 𝑦 ) ≤ ( Σ^ ‘ ( 𝑂 ↾ 𝑦 ) ) ) ) → dom 𝑂 = 𝒫 ∪ dom 𝑂 )
10	5 9	syl	⊢ ( 𝜑 → dom 𝑂 = 𝒫 ∪ dom 𝑂 )
11	8 10	eqtr4id	⊢ ( 𝜑 → 𝒫 𝑋 = dom 𝑂 )
12	11	feq2d	⊢ ( 𝜑 → ( 𝑂 : 𝒫 𝑋 ⟶ ( 0 [,] +∞ ) ↔ 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ) )
13	7 12	mpbird	⊢ ( 𝜑 → 𝑂 : 𝒫 𝑋 ⟶ ( 0 [,] +∞ ) )