Step |
Hyp |
Ref |
Expression |
1 |
|
simp2 |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) → 𝑇 ∈ ∪ ran sigAlgebra ) |
2 |
|
measfrge0 |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → 𝑀 : 𝑆 ⟶ ( 0 [,] +∞ ) ) |
3 |
2
|
3ad2ant1 |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) → 𝑀 : 𝑆 ⟶ ( 0 [,] +∞ ) ) |
4 |
|
simp3 |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) → 𝑇 ⊆ 𝑆 ) |
5 |
3 4
|
fssresd |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) → ( 𝑀 ↾ 𝑇 ) : 𝑇 ⟶ ( 0 [,] +∞ ) ) |
6 |
|
0elsiga |
⊢ ( 𝑇 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑇 ) |
7 |
|
fvres |
⊢ ( ∅ ∈ 𝑇 → ( ( 𝑀 ↾ 𝑇 ) ‘ ∅ ) = ( 𝑀 ‘ ∅ ) ) |
8 |
1 6 7
|
3syl |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) → ( ( 𝑀 ↾ 𝑇 ) ‘ ∅ ) = ( 𝑀 ‘ ∅ ) ) |
9 |
|
measvnul |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → ( 𝑀 ‘ ∅ ) = 0 ) |
10 |
9
|
3ad2ant1 |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) → ( 𝑀 ‘ ∅ ) = 0 ) |
11 |
8 10
|
eqtrd |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) → ( ( 𝑀 ↾ 𝑇 ) ‘ ∅ ) = 0 ) |
12 |
|
simp11 |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → 𝑀 ∈ ( measures ‘ 𝑆 ) ) |
13 |
|
simp13 |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → 𝑇 ⊆ 𝑆 ) |
14 |
|
simp2 |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → 𝑥 ∈ 𝒫 𝑇 ) |
15 |
|
sspw |
⊢ ( 𝑇 ⊆ 𝑆 → 𝒫 𝑇 ⊆ 𝒫 𝑆 ) |
16 |
15
|
sselda |
⊢ ( ( 𝑇 ⊆ 𝑆 ∧ 𝑥 ∈ 𝒫 𝑇 ) → 𝑥 ∈ 𝒫 𝑆 ) |
17 |
13 14 16
|
syl2anc |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → 𝑥 ∈ 𝒫 𝑆 ) |
18 |
|
simp3 |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) |
19 |
|
measvun |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑆 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → ( 𝑀 ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( 𝑀 ‘ 𝑦 ) ) |
20 |
12 17 18 19
|
syl3anc |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → ( 𝑀 ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( 𝑀 ‘ 𝑦 ) ) |
21 |
1
|
3ad2ant1 |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → 𝑇 ∈ ∪ ran sigAlgebra ) |
22 |
|
simp3l |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → 𝑥 ≼ ω ) |
23 |
|
sigaclcu |
⊢ ( ( 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑇 ∧ 𝑥 ≼ ω ) → ∪ 𝑥 ∈ 𝑇 ) |
24 |
21 14 22 23
|
syl3anc |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → ∪ 𝑥 ∈ 𝑇 ) |
25 |
24
|
fvresd |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → ( ( 𝑀 ↾ 𝑇 ) ‘ ∪ 𝑥 ) = ( 𝑀 ‘ ∪ 𝑥 ) ) |
26 |
|
elpwi |
⊢ ( 𝑥 ∈ 𝒫 𝑇 → 𝑥 ⊆ 𝑇 ) |
27 |
26
|
sselda |
⊢ ( ( 𝑥 ∈ 𝒫 𝑇 ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ 𝑇 ) |
28 |
27
|
adantll |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ 𝑇 ) |
29 |
28
|
fvresd |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ) ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝑀 ↾ 𝑇 ) ‘ 𝑦 ) = ( 𝑀 ‘ 𝑦 ) ) |
30 |
29
|
esumeq2dv |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ) → Σ* 𝑦 ∈ 𝑥 ( ( 𝑀 ↾ 𝑇 ) ‘ 𝑦 ) = Σ* 𝑦 ∈ 𝑥 ( 𝑀 ‘ 𝑦 ) ) |
31 |
30
|
3adant3 |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → Σ* 𝑦 ∈ 𝑥 ( ( 𝑀 ↾ 𝑇 ) ‘ 𝑦 ) = Σ* 𝑦 ∈ 𝑥 ( 𝑀 ‘ 𝑦 ) ) |
32 |
20 25 31
|
3eqtr4d |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → ( ( 𝑀 ↾ 𝑇 ) ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( ( 𝑀 ↾ 𝑇 ) ‘ 𝑦 ) ) |
33 |
32
|
3expia |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑥 ∈ 𝒫 𝑇 ) → ( ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( ( 𝑀 ↾ 𝑇 ) ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( ( 𝑀 ↾ 𝑇 ) ‘ 𝑦 ) ) ) |
34 |
33
|
ralrimiva |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) → ∀ 𝑥 ∈ 𝒫 𝑇 ( ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( ( 𝑀 ↾ 𝑇 ) ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( ( 𝑀 ↾ 𝑇 ) ‘ 𝑦 ) ) ) |
35 |
5 11 34
|
3jca |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) → ( ( 𝑀 ↾ 𝑇 ) : 𝑇 ⟶ ( 0 [,] +∞ ) ∧ ( ( 𝑀 ↾ 𝑇 ) ‘ ∅ ) = 0 ∧ ∀ 𝑥 ∈ 𝒫 𝑇 ( ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( ( 𝑀 ↾ 𝑇 ) ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( ( 𝑀 ↾ 𝑇 ) ‘ 𝑦 ) ) ) ) |
36 |
|
ismeas |
⊢ ( 𝑇 ∈ ∪ ran sigAlgebra → ( ( 𝑀 ↾ 𝑇 ) ∈ ( measures ‘ 𝑇 ) ↔ ( ( 𝑀 ↾ 𝑇 ) : 𝑇 ⟶ ( 0 [,] +∞ ) ∧ ( ( 𝑀 ↾ 𝑇 ) ‘ ∅ ) = 0 ∧ ∀ 𝑥 ∈ 𝒫 𝑇 ( ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( ( 𝑀 ↾ 𝑇 ) ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( ( 𝑀 ↾ 𝑇 ) ‘ 𝑦 ) ) ) ) ) |
37 |
36
|
biimprd |
⊢ ( 𝑇 ∈ ∪ ran sigAlgebra → ( ( ( 𝑀 ↾ 𝑇 ) : 𝑇 ⟶ ( 0 [,] +∞ ) ∧ ( ( 𝑀 ↾ 𝑇 ) ‘ ∅ ) = 0 ∧ ∀ 𝑥 ∈ 𝒫 𝑇 ( ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( ( 𝑀 ↾ 𝑇 ) ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( ( 𝑀 ↾ 𝑇 ) ‘ 𝑦 ) ) ) → ( 𝑀 ↾ 𝑇 ) ∈ ( measures ‘ 𝑇 ) ) ) |
38 |
1 35 37
|
sylc |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆 ) → ( 𝑀 ↾ 𝑇 ) ∈ ( measures ‘ 𝑇 ) ) |