Step | Hyp | Ref
| Expression |
1 | | simpll 786 |
. . . 4
⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑀 ∈ (measures‘𝑆)) |
2 | | measbase 29587 |
. . . . . 6
⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran
sigAlgebra) |
3 | 2 | ad2antrr 758 |
. . . . 5
⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ ∪ ran
sigAlgebra) |
4 | | simpr 476 |
. . . . 5
⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆) |
5 | | simplr 788 |
. . . . 5
⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑆) |
6 | | inelsiga 29525 |
. . . . 5
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝑥 ∩ 𝐴) ∈ 𝑆) |
7 | 3, 4, 5, 6 | syl3anc 1318 |
. . . 4
⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑥 ∩ 𝐴) ∈ 𝑆) |
8 | | measvxrge0 29595 |
. . . 4
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑥 ∩ 𝐴) ∈ 𝑆) → (𝑀‘(𝑥 ∩ 𝐴)) ∈ (0[,]+∞)) |
9 | 1, 7, 8 | syl2anc 691 |
. . 3
⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑀‘(𝑥 ∩ 𝐴)) ∈ (0[,]+∞)) |
10 | | eqid 2610 |
. . 3
⊢ (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))) = (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))) |
11 | 9, 10 | fmptd 6292 |
. 2
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))):𝑆⟶(0[,]+∞)) |
12 | | eqidd 2611 |
. . 3
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))) = (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))) |
13 | | ineq1 3769 |
. . . . . . 7
⊢ (𝑥 = ∅ → (𝑥 ∩ 𝐴) = (∅ ∩ 𝐴)) |
14 | | 0in 3921 |
. . . . . . 7
⊢ (∅
∩ 𝐴) =
∅ |
15 | 13, 14 | syl6eq 2660 |
. . . . . 6
⊢ (𝑥 = ∅ → (𝑥 ∩ 𝐴) = ∅) |
16 | 15 | fveq2d 6107 |
. . . . 5
⊢ (𝑥 = ∅ → (𝑀‘(𝑥 ∩ 𝐴)) = (𝑀‘∅)) |
17 | 16 | adantl 481 |
. . . 4
⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 = ∅) → (𝑀‘(𝑥 ∩ 𝐴)) = (𝑀‘∅)) |
18 | | measvnul 29596 |
. . . . 5
⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0) |
19 | 18 | ad2antrr 758 |
. . . 4
⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 = ∅) → (𝑀‘∅) = 0) |
20 | 17, 19 | eqtrd 2644 |
. . 3
⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 = ∅) → (𝑀‘(𝑥 ∩ 𝐴)) = 0) |
21 | 2 | adantr 480 |
. . . 4
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → 𝑆 ∈ ∪ ran
sigAlgebra) |
22 | | 0elsiga 29504 |
. . . 4
⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑆) |
23 | 21, 22 | syl 17 |
. . 3
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → ∅ ∈ 𝑆) |
24 | | 0red 9920 |
. . 3
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → 0 ∈ ℝ) |
25 | 12, 20, 23, 24 | fvmptd 6197 |
. 2
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘∅) = 0) |
26 | | measinblem 29610 |
. . . . 5
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → (𝑀‘(∪ 𝑧 ∩ 𝐴)) = Σ*𝑦 ∈ 𝑧(𝑀‘(𝑦 ∩ 𝐴))) |
27 | | eqidd 2611 |
. . . . . 6
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))) = (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))) |
28 | | ineq1 3769 |
. . . . . . . 8
⊢ (𝑥 = ∪
𝑧 → (𝑥 ∩ 𝐴) = (∪ 𝑧 ∩ 𝐴)) |
29 | 28 | adantl 481 |
. . . . . . 7
⊢
(((((𝑀 ∈
(measures‘𝑆) ∧
𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) ∧ 𝑥 = ∪ 𝑧) → (𝑥 ∩ 𝐴) = (∪ 𝑧 ∩ 𝐴)) |
30 | 29 | fveq2d 6107 |
. . . . . 6
⊢
(((((𝑀 ∈
(measures‘𝑆) ∧
𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) ∧ 𝑥 = ∪ 𝑧) → (𝑀‘(𝑥 ∩ 𝐴)) = (𝑀‘(∪ 𝑧 ∩ 𝐴))) |
31 | | simplll 794 |
. . . . . . . 8
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → 𝑀 ∈ (measures‘𝑆)) |
32 | 31, 2 | syl 17 |
. . . . . . 7
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → 𝑆 ∈ ∪ ran
sigAlgebra) |
33 | | simplr 788 |
. . . . . . 7
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → 𝑧 ∈ 𝒫 𝑆) |
34 | | simprl 790 |
. . . . . . 7
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → 𝑧 ≼ ω) |
35 | | sigaclcu 29507 |
. . . . . . 7
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑧 ∈ 𝒫 𝑆 ∧ 𝑧 ≼ ω) → ∪ 𝑧
∈ 𝑆) |
36 | 32, 33, 34, 35 | syl3anc 1318 |
. . . . . 6
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → ∪ 𝑧 ∈ 𝑆) |
37 | | simpllr 795 |
. . . . . . . 8
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → 𝐴 ∈ 𝑆) |
38 | | inelsiga 29525 |
. . . . . . . 8
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑧 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (∪ 𝑧 ∩ 𝐴) ∈ 𝑆) |
39 | 32, 36, 37, 38 | syl3anc 1318 |
. . . . . . 7
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → (∪ 𝑧 ∩ 𝐴) ∈ 𝑆) |
40 | | measvxrge0 29595 |
. . . . . . 7
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (∪ 𝑧
∩ 𝐴) ∈ 𝑆) → (𝑀‘(∪ 𝑧 ∩ 𝐴)) ∈ (0[,]+∞)) |
41 | 31, 39, 40 | syl2anc 691 |
. . . . . 6
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → (𝑀‘(∪ 𝑧 ∩ 𝐴)) ∈ (0[,]+∞)) |
42 | 27, 30, 36, 41 | fvmptd 6197 |
. . . . 5
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘∪ 𝑧) = (𝑀‘(∪ 𝑧 ∩ 𝐴))) |
43 | | eqidd 2611 |
. . . . . . . 8
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) → (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))) = (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))) |
44 | | ineq1 3769 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → (𝑥 ∩ 𝐴) = (𝑦 ∩ 𝐴)) |
45 | 44 | adantl 481 |
. . . . . . . . 9
⊢
(((((𝑀 ∈
(measures‘𝑆) ∧
𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) ∧ 𝑥 = 𝑦) → (𝑥 ∩ 𝐴) = (𝑦 ∩ 𝐴)) |
46 | 45 | fveq2d 6107 |
. . . . . . . 8
⊢
(((((𝑀 ∈
(measures‘𝑆) ∧
𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) ∧ 𝑥 = 𝑦) → (𝑀‘(𝑥 ∩ 𝐴)) = (𝑀‘(𝑦 ∩ 𝐴))) |
47 | | elpwi 4117 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝒫 𝑆 → 𝑧 ⊆ 𝑆) |
48 | 47 | ad2antlr 759 |
. . . . . . . . 9
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) → 𝑧 ⊆ 𝑆) |
49 | | simpr 476 |
. . . . . . . . 9
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ 𝑧) |
50 | 48, 49 | sseldd 3569 |
. . . . . . . 8
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ 𝑆) |
51 | | simplll 794 |
. . . . . . . . 9
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) → 𝑀 ∈ (measures‘𝑆)) |
52 | 51, 2 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) → 𝑆 ∈ ∪ ran
sigAlgebra) |
53 | | simpllr 795 |
. . . . . . . . . 10
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) → 𝐴 ∈ 𝑆) |
54 | | inelsiga 29525 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑦 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝑦 ∩ 𝐴) ∈ 𝑆) |
55 | 52, 50, 53, 54 | syl3anc 1318 |
. . . . . . . . 9
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) → (𝑦 ∩ 𝐴) ∈ 𝑆) |
56 | | measvxrge0 29595 |
. . . . . . . . 9
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑦 ∩ 𝐴) ∈ 𝑆) → (𝑀‘(𝑦 ∩ 𝐴)) ∈ (0[,]+∞)) |
57 | 51, 55, 56 | syl2anc 691 |
. . . . . . . 8
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) → (𝑀‘(𝑦 ∩ 𝐴)) ∈ (0[,]+∞)) |
58 | 43, 46, 50, 57 | fvmptd 6197 |
. . . . . . 7
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑧) → ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘𝑦) = (𝑀‘(𝑦 ∩ 𝐴))) |
59 | 58 | esumeq2dv 29427 |
. . . . . 6
⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) → Σ*𝑦 ∈ 𝑧((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘𝑦) = Σ*𝑦 ∈ 𝑧(𝑀‘(𝑦 ∩ 𝐴))) |
60 | 59 | adantr 480 |
. . . . 5
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → Σ*𝑦 ∈ 𝑧((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘𝑦) = Σ*𝑦 ∈ 𝑧(𝑀‘(𝑦 ∩ 𝐴))) |
61 | 26, 42, 60 | 3eqtr4d 2654 |
. . . 4
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) ∧ (𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦)) → ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘∪ 𝑧) = Σ*𝑦 ∈ 𝑧((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘𝑦)) |
62 | 61 | ex 449 |
. . 3
⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ 𝒫 𝑆) → ((𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦) → ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘∪ 𝑧) = Σ*𝑦 ∈ 𝑧((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘𝑦))) |
63 | 62 | ralrimiva 2949 |
. 2
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → ∀𝑧 ∈ 𝒫 𝑆((𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦) → ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘∪ 𝑧) = Σ*𝑦 ∈ 𝑧((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘𝑦))) |
64 | | ismeas 29589 |
. . 3
⊢ (𝑆 ∈ ∪ ran sigAlgebra → ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))) ∈ (measures‘𝑆) ↔ ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))):𝑆⟶(0[,]+∞) ∧ ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘∅) = 0 ∧ ∀𝑧 ∈ 𝒫 𝑆((𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦) → ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘∪ 𝑧) = Σ*𝑦 ∈ 𝑧((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘𝑦))))) |
65 | 21, 64 | syl 17 |
. 2
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))) ∈ (measures‘𝑆) ↔ ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))):𝑆⟶(0[,]+∞) ∧ ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘∅) = 0 ∧ ∀𝑧 ∈ 𝒫 𝑆((𝑧 ≼ ω ∧ Disj 𝑦 ∈ 𝑧 𝑦) → ((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘∪ 𝑧) = Σ*𝑦 ∈ 𝑧((𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴)))‘𝑦))))) |
66 | 11, 25, 63, 65 | mpbir3and 1238 |
1
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))) ∈ (measures‘𝑆)) |