Theorem ome0

Description: The outer measure of the empty set is 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypothesis	ome0.1	⊢ ( 𝜑 → 𝑂 ∈ OutMeas )
Assertion	ome0	⊢ ( 𝜑 → ( 𝑂 ‘ ∅ ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		ome0.1	⊢ ( 𝜑 → 𝑂 ∈ OutMeas )
2		isome	⊢ ( 𝑂 ∈ OutMeas → ( 𝑂 ∈ OutMeas ↔ ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑥 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑦 ∈ 𝒫 𝑥 ( 𝑂 ‘ 𝑦 ) ≤ ( 𝑂 ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝒫 dom 𝑂 ( 𝑥 ≼ ω → ( 𝑂 ‘ ∪ 𝑥 ) ≤ ( Σ^ ‘ ( 𝑂 ↾ 𝑥 ) ) ) ) ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝑂 ∈ OutMeas ↔ ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑥 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑦 ∈ 𝒫 𝑥 ( 𝑂 ‘ 𝑦 ) ≤ ( 𝑂 ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝒫 dom 𝑂 ( 𝑥 ≼ ω → ( 𝑂 ‘ ∪ 𝑥 ) ≤ ( Σ^ ‘ ( 𝑂 ↾ 𝑥 ) ) ) ) ) )
4	1 3	mpbid	⊢ ( 𝜑 → ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑥 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑦 ∈ 𝒫 𝑥 ( 𝑂 ‘ 𝑦 ) ≤ ( 𝑂 ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝒫 dom 𝑂 ( 𝑥 ≼ ω → ( 𝑂 ‘ ∪ 𝑥 ) ≤ ( Σ^ ‘ ( 𝑂 ↾ 𝑥 ) ) ) ) )
5	4	simplld	⊢ ( 𝜑 → ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) )
6	5	simprd	⊢ ( 𝜑 → ( 𝑂 ‘ ∅ ) = 0 )