Step |
Hyp |
Ref |
Expression |
1 |
|
meaf.m |
⊢ ( 𝜑 → 𝑀 ∈ 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 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
|
simpld |
⊢ ( 𝜑 → ( ( 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑀 ∈ SAlg ) ∧ ( 𝑀 ‘ ∅ ) = 0 ) ) |
6 |
5
|
simplld |
⊢ ( 𝜑 → 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ) |
7 |
2
|
a1i |
⊢ ( 𝜑 → 𝑆 = dom 𝑀 ) |
8 |
7
|
feq2d |
⊢ ( 𝜑 → ( 𝑀 : 𝑆 ⟶ ( 0 [,] +∞ ) ↔ 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ) ) |
9 |
6 8
|
mpbird |
⊢ ( 𝜑 → 𝑀 : 𝑆 ⟶ ( 0 [,] +∞ ) ) |