Theorem dmmeasal

Description: The domain of a measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	dmmeasal.m	⊢ ( 𝜑 → 𝑀 ∈ Meas )
	dmmeasal.s	⊢ 𝑆 = dom 𝑀
Assertion	dmmeasal	⊢ ( 𝜑 → 𝑆 ∈ SAlg )

Proof

Step	Hyp	Ref	Expression
1		dmmeasal.m	⊢ ( 𝜑 → 𝑀 ∈ Meas )
2		dmmeasal.s	⊢ 𝑆 = dom 𝑀
3		ismea	⊢ ( 𝑀 ∈ Meas ↔ ( ( ( 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑀 ∈ SAlg ) ∧ ( 𝑀 ‘ ∅ ) = 0 ) ∧ ∀ 𝑥 ∈ 𝒫 dom 𝑀 ( ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( 𝑀 ‘ ∪ 𝑥 ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑥 ) ) ) ) )
4	1 3	sylib	⊢ ( 𝜑 → ( ( ( 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑀 ∈ SAlg ) ∧ ( 𝑀 ‘ ∅ ) = 0 ) ∧ ∀ 𝑥 ∈ 𝒫 dom 𝑀 ( ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( 𝑀 ‘ ∪ 𝑥 ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑥 ) ) ) ) )
5	4	simplld	⊢ ( 𝜑 → ( 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑀 ∈ SAlg ) )
6	5	simprd	⊢ ( 𝜑 → dom 𝑀 ∈ SAlg )
7	2 6	eqeltrid	⊢ ( 𝜑 → 𝑆 ∈ SAlg )