Metamath Proof Explorer


Theorem meaf

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

Ref Expression
Hypotheses meaf.m ( 𝜑𝑀 ∈ Meas )
meaf.s 𝑆 = dom 𝑀
Assertion meaf ( 𝜑𝑀 : 𝑆 ⟶ ( 0 [,] +∞ ) )

Proof

Step Hyp Ref Expression
1 meaf.m ( 𝜑𝑀 ∈ Meas )
2 meaf.s 𝑆 = dom 𝑀
3 ismea ( 𝑀 ∈ Meas ↔ ( ( ( 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑀 ∈ SAlg ) ∧ ( 𝑀 ‘ ∅ ) = 0 ) ∧ ∀ 𝑥 ∈ 𝒫 dom 𝑀 ( ( 𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦 ) → ( 𝑀 𝑥 ) = ( Σ^ ‘ ( 𝑀𝑥 ) ) ) ) )
4 1 3 sylib ( 𝜑 → ( ( ( 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑀 ∈ SAlg ) ∧ ( 𝑀 ‘ ∅ ) = 0 ) ∧ ∀ 𝑥 ∈ 𝒫 dom 𝑀 ( ( 𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦 ) → ( 𝑀 𝑥 ) = ( Σ^ ‘ ( 𝑀𝑥 ) ) ) ) )
5 4 simpld ( 𝜑 → ( ( 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑀 ∈ SAlg ) ∧ ( 𝑀 ‘ ∅ ) = 0 ) )
6 5 simplld ( 𝜑𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) )
7 2 a1i ( 𝜑𝑆 = dom 𝑀 )
8 7 feq2d ( 𝜑 → ( 𝑀 : 𝑆 ⟶ ( 0 [,] +∞ ) ↔ 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ) )
9 6 8 mpbird ( 𝜑𝑀 : 𝑆 ⟶ ( 0 [,] +∞ ) )