Step |
Hyp |
Ref |
Expression |
1 |
|
meadjuni.m |
⊢ ( 𝜑 → 𝑀 ∈ Meas ) |
2 |
|
meadjuni.s |
⊢ 𝑆 = dom 𝑀 |
3 |
|
meadjuni.x |
⊢ ( 𝜑 → 𝑋 ⊆ 𝑆 ) |
4 |
|
meadjuni.cnb |
⊢ ( 𝜑 → 𝑋 ≼ ω ) |
5 |
|
meadjuni.dj |
⊢ ( 𝜑 → Disj 𝑥 ∈ 𝑋 𝑥 ) |
6 |
|
breq1 |
⊢ ( 𝑦 = 𝑋 → ( 𝑦 ≼ ω ↔ 𝑋 ≼ ω ) ) |
7 |
|
disjeq1 |
⊢ ( 𝑦 = 𝑋 → ( Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝑋 𝑥 ) ) |
8 |
6 7
|
anbi12d |
⊢ ( 𝑦 = 𝑋 → ( ( 𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥 ) ↔ ( 𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥 ) ) ) |
9 |
|
unieq |
⊢ ( 𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋 ) |
10 |
9
|
fveq2d |
⊢ ( 𝑦 = 𝑋 → ( 𝑀 ‘ ∪ 𝑦 ) = ( 𝑀 ‘ ∪ 𝑋 ) ) |
11 |
|
reseq2 |
⊢ ( 𝑦 = 𝑋 → ( 𝑀 ↾ 𝑦 ) = ( 𝑀 ↾ 𝑋 ) ) |
12 |
11
|
fveq2d |
⊢ ( 𝑦 = 𝑋 → ( Σ^ ‘ ( 𝑀 ↾ 𝑦 ) ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑋 ) ) ) |
13 |
10 12
|
eqeq12d |
⊢ ( 𝑦 = 𝑋 → ( ( 𝑀 ‘ ∪ 𝑦 ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑦 ) ) ↔ ( 𝑀 ‘ ∪ 𝑋 ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑋 ) ) ) ) |
14 |
8 13
|
imbi12d |
⊢ ( 𝑦 = 𝑋 → ( ( ( 𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥 ) → ( 𝑀 ‘ ∪ 𝑦 ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑦 ) ) ) ↔ ( ( 𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥 ) → ( 𝑀 ‘ ∪ 𝑋 ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑋 ) ) ) ) ) |
15 |
|
ismea |
⊢ ( 𝑀 ∈ Meas ↔ ( ( ( 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑀 ∈ SAlg ) ∧ ( 𝑀 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑀 ( ( 𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥 ) → ( 𝑀 ‘ ∪ 𝑦 ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑦 ) ) ) ) ) |
16 |
1 15
|
sylib |
⊢ ( 𝜑 → ( ( ( 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑀 ∈ SAlg ) ∧ ( 𝑀 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑀 ( ( 𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥 ) → ( 𝑀 ‘ ∪ 𝑦 ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑦 ) ) ) ) ) |
17 |
16
|
simprd |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝒫 dom 𝑀 ( ( 𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥 ) → ( 𝑀 ‘ ∪ 𝑦 ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑦 ) ) ) ) |
18 |
1 2
|
dmmeasal |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
19 |
18 3
|
ssexd |
⊢ ( 𝜑 → 𝑋 ∈ V ) |
20 |
3 2
|
sseqtrdi |
⊢ ( 𝜑 → 𝑋 ⊆ dom 𝑀 ) |
21 |
19 20
|
elpwd |
⊢ ( 𝜑 → 𝑋 ∈ 𝒫 dom 𝑀 ) |
22 |
14 17 21
|
rspcdva |
⊢ ( 𝜑 → ( ( 𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥 ) → ( 𝑀 ‘ ∪ 𝑋 ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑋 ) ) ) ) |
23 |
4 5 22
|
mp2and |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑋 ) = ( Σ^ ‘ ( 𝑀 ↾ 𝑋 ) ) ) |