Step |
Hyp |
Ref |
Expression |
1 |
|
omeiunle.nph |
⊢ Ⅎ 𝑛 𝜑 |
2 |
|
omeiunle.ne |
⊢ Ⅎ 𝑛 𝐸 |
3 |
|
omeiunle.o |
⊢ ( 𝜑 → 𝑂 ∈ OutMeas ) |
4 |
|
omeiunle.x |
⊢ 𝑋 = ∪ dom 𝑂 |
5 |
|
omeiunle.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑁 ) |
6 |
|
omeiunle.e |
⊢ ( 𝜑 → 𝐸 : 𝑍 ⟶ 𝒫 𝑋 ) |
7 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
8 |
6
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐸 ‘ 𝑛 ) ∈ 𝒫 𝑋 ) |
9 |
|
elpwi |
⊢ ( ( 𝐸 ‘ 𝑛 ) ∈ 𝒫 𝑋 → ( 𝐸 ‘ 𝑛 ) ⊆ 𝑋 ) |
10 |
8 9
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐸 ‘ 𝑛 ) ⊆ 𝑋 ) |
11 |
10
|
ex |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 → ( 𝐸 ‘ 𝑛 ) ⊆ 𝑋 ) ) |
12 |
1 11
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ⊆ 𝑋 ) |
13 |
|
iunss |
⊢ ( ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ⊆ 𝑋 ↔ ∀ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ⊆ 𝑋 ) |
14 |
12 13
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ⊆ 𝑋 ) |
15 |
3 4 14
|
omecl |
⊢ ( 𝜑 → ( 𝑂 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ∈ ( 0 [,] +∞ ) ) |
16 |
7 15
|
sselid |
⊢ ( 𝜑 → ( 𝑂 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ∈ ℝ* ) |
17 |
6
|
ffnd |
⊢ ( 𝜑 → 𝐸 Fn 𝑍 ) |
18 |
5
|
fvexi |
⊢ 𝑍 ∈ V |
19 |
18
|
a1i |
⊢ ( 𝜑 → 𝑍 ∈ V ) |
20 |
|
fnex |
⊢ ( ( 𝐸 Fn 𝑍 ∧ 𝑍 ∈ V ) → 𝐸 ∈ V ) |
21 |
17 19 20
|
syl2anc |
⊢ ( 𝜑 → 𝐸 ∈ V ) |
22 |
|
rnexg |
⊢ ( 𝐸 ∈ V → ran 𝐸 ∈ V ) |
23 |
21 22
|
syl |
⊢ ( 𝜑 → ran 𝐸 ∈ V ) |
24 |
3 4
|
omef |
⊢ ( 𝜑 → 𝑂 : 𝒫 𝑋 ⟶ ( 0 [,] +∞ ) ) |
25 |
6
|
frnd |
⊢ ( 𝜑 → ran 𝐸 ⊆ 𝒫 𝑋 ) |
26 |
24 25
|
fssresd |
⊢ ( 𝜑 → ( 𝑂 ↾ ran 𝐸 ) : ran 𝐸 ⟶ ( 0 [,] +∞ ) ) |
27 |
23 26
|
sge0xrcl |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑂 ↾ ran 𝐸 ) ) ∈ ℝ* ) |
28 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑂 ∈ OutMeas ) |
29 |
28 4 10
|
omecl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ∈ ( 0 [,] +∞ ) ) |
30 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) |
31 |
1 29 30
|
fmptdf |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) : 𝑍 ⟶ ( 0 [,] +∞ ) ) |
32 |
19 31
|
sge0xrcl |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) ∈ ℝ* ) |
33 |
|
fvex |
⊢ ( 𝐸 ‘ 𝑛 ) ∈ V |
34 |
33
|
rgenw |
⊢ ∀ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ∈ V |
35 |
|
dfiun3g |
⊢ ( ∀ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ∈ V → ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) = ∪ ran ( 𝑛 ∈ 𝑍 ↦ ( 𝐸 ‘ 𝑛 ) ) ) |
36 |
34 35
|
ax-mp |
⊢ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) = ∪ ran ( 𝑛 ∈ 𝑍 ↦ ( 𝐸 ‘ 𝑛 ) ) |
37 |
36
|
a1i |
⊢ ( 𝜑 → ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) = ∪ ran ( 𝑛 ∈ 𝑍 ↦ ( 𝐸 ‘ 𝑛 ) ) ) |
38 |
6
|
feqmptd |
⊢ ( 𝜑 → 𝐸 = ( 𝑚 ∈ 𝑍 ↦ ( 𝐸 ‘ 𝑚 ) ) ) |
39 |
|
nfcv |
⊢ Ⅎ 𝑛 𝑚 |
40 |
2 39
|
nffv |
⊢ Ⅎ 𝑛 ( 𝐸 ‘ 𝑚 ) |
41 |
|
nfcv |
⊢ Ⅎ 𝑚 ( 𝐸 ‘ 𝑛 ) |
42 |
|
fveq2 |
⊢ ( 𝑚 = 𝑛 → ( 𝐸 ‘ 𝑚 ) = ( 𝐸 ‘ 𝑛 ) ) |
43 |
40 41 42
|
cbvmpt |
⊢ ( 𝑚 ∈ 𝑍 ↦ ( 𝐸 ‘ 𝑚 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝐸 ‘ 𝑛 ) ) |
44 |
43
|
a1i |
⊢ ( 𝜑 → ( 𝑚 ∈ 𝑍 ↦ ( 𝐸 ‘ 𝑚 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝐸 ‘ 𝑛 ) ) ) |
45 |
38 44
|
eqtrd |
⊢ ( 𝜑 → 𝐸 = ( 𝑛 ∈ 𝑍 ↦ ( 𝐸 ‘ 𝑛 ) ) ) |
46 |
45
|
rneqd |
⊢ ( 𝜑 → ran 𝐸 = ran ( 𝑛 ∈ 𝑍 ↦ ( 𝐸 ‘ 𝑛 ) ) ) |
47 |
46
|
unieqd |
⊢ ( 𝜑 → ∪ ran 𝐸 = ∪ ran ( 𝑛 ∈ 𝑍 ↦ ( 𝐸 ‘ 𝑛 ) ) ) |
48 |
37 47
|
eqtr4d |
⊢ ( 𝜑 → ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) = ∪ ran 𝐸 ) |
49 |
48
|
fveq2d |
⊢ ( 𝜑 → ( 𝑂 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) = ( 𝑂 ‘ ∪ ran 𝐸 ) ) |
50 |
|
fnrndomg |
⊢ ( 𝑍 ∈ V → ( 𝐸 Fn 𝑍 → ran 𝐸 ≼ 𝑍 ) ) |
51 |
19 17 50
|
sylc |
⊢ ( 𝜑 → ran 𝐸 ≼ 𝑍 ) |
52 |
5
|
uzct |
⊢ 𝑍 ≼ ω |
53 |
52
|
a1i |
⊢ ( 𝜑 → 𝑍 ≼ ω ) |
54 |
|
domtr |
⊢ ( ( ran 𝐸 ≼ 𝑍 ∧ 𝑍 ≼ ω ) → ran 𝐸 ≼ ω ) |
55 |
51 53 54
|
syl2anc |
⊢ ( 𝜑 → ran 𝐸 ≼ ω ) |
56 |
3 4 25 55
|
omeunile |
⊢ ( 𝜑 → ( 𝑂 ‘ ∪ ran 𝐸 ) ≤ ( Σ^ ‘ ( 𝑂 ↾ ran 𝐸 ) ) ) |
57 |
49 56
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝑂 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ≤ ( Σ^ ‘ ( 𝑂 ↾ ran 𝐸 ) ) ) |
58 |
|
ltweuz |
⊢ < We ( ℤ≥ ‘ 𝑁 ) |
59 |
|
weeq2 |
⊢ ( 𝑍 = ( ℤ≥ ‘ 𝑁 ) → ( < We 𝑍 ↔ < We ( ℤ≥ ‘ 𝑁 ) ) ) |
60 |
5 59
|
ax-mp |
⊢ ( < We 𝑍 ↔ < We ( ℤ≥ ‘ 𝑁 ) ) |
61 |
58 60
|
mpbir |
⊢ < We 𝑍 |
62 |
61
|
a1i |
⊢ ( 𝜑 → < We 𝑍 ) |
63 |
19 24 6 62
|
sge0resrn |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑂 ↾ ran 𝐸 ) ) ≤ ( Σ^ ‘ ( 𝑂 ∘ 𝐸 ) ) ) |
64 |
|
fcompt |
⊢ ( ( 𝑂 : 𝒫 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝐸 : 𝑍 ⟶ 𝒫 𝑋 ) → ( 𝑂 ∘ 𝐸 ) = ( 𝑚 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑚 ) ) ) ) |
65 |
|
nfcv |
⊢ Ⅎ 𝑛 𝑂 |
66 |
65 40
|
nffv |
⊢ Ⅎ 𝑛 ( 𝑂 ‘ ( 𝐸 ‘ 𝑚 ) ) |
67 |
|
nfcv |
⊢ Ⅎ 𝑚 ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) |
68 |
|
2fveq3 |
⊢ ( 𝑚 = 𝑛 → ( 𝑂 ‘ ( 𝐸 ‘ 𝑚 ) ) = ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) |
69 |
66 67 68
|
cbvmpt |
⊢ ( 𝑚 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑚 ) ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) |
70 |
69
|
a1i |
⊢ ( ( 𝑂 : 𝒫 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝐸 : 𝑍 ⟶ 𝒫 𝑋 ) → ( 𝑚 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑚 ) ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) |
71 |
64 70
|
eqtrd |
⊢ ( ( 𝑂 : 𝒫 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝐸 : 𝑍 ⟶ 𝒫 𝑋 ) → ( 𝑂 ∘ 𝐸 ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) |
72 |
24 6 71
|
syl2anc |
⊢ ( 𝜑 → ( 𝑂 ∘ 𝐸 ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) |
73 |
72
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑂 ∘ 𝐸 ) ) = ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) ) |
74 |
63 73
|
breqtrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑂 ↾ ran 𝐸 ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) ) |
75 |
16 27 32 57 74
|
xrletrd |
⊢ ( 𝜑 → ( 𝑂 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑂 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) ) |