Step |
Hyp |
Ref |
Expression |
1 |
|
omessle.o |
⊢ ( 𝜑 → 𝑂 ∈ OutMeas ) |
2 |
|
omessle.x |
⊢ 𝑋 = ∪ dom 𝑂 |
3 |
|
omessle.b |
⊢ ( 𝜑 → 𝐵 ⊆ 𝑋 ) |
4 |
|
omessle.a |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) |
5 |
1 2
|
unidmex |
⊢ ( 𝜑 → 𝑋 ∈ V ) |
6 |
5 3
|
ssexd |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
7 |
6 4
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
8 |
|
elpwg |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) ) |
9 |
7 8
|
syl |
⊢ ( 𝜑 → ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) ) |
10 |
4 9
|
mpbird |
⊢ ( 𝜑 → 𝐴 ∈ 𝒫 𝐵 ) |
11 |
3 2
|
sseqtrdi |
⊢ ( 𝜑 → 𝐵 ⊆ ∪ dom 𝑂 ) |
12 |
|
elpwg |
⊢ ( 𝐵 ∈ V → ( 𝐵 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝐵 ⊆ ∪ dom 𝑂 ) ) |
13 |
6 12
|
syl |
⊢ ( 𝜑 → ( 𝐵 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝐵 ⊆ ∪ dom 𝑂 ) ) |
14 |
11 13
|
mpbird |
⊢ ( 𝜑 → 𝐵 ∈ 𝒫 ∪ dom 𝑂 ) |
15 |
|
isome |
⊢ ( 𝑂 ∈ OutMeas → ( 𝑂 ∈ OutMeas ↔ ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂 ( 𝑦 ≼ ω → ( 𝑂 ‘ ∪ 𝑦 ) ≤ ( Σ^ ‘ ( 𝑂 ↾ 𝑦 ) ) ) ) ) ) |
16 |
1 15
|
syl |
⊢ ( 𝜑 → ( 𝑂 ∈ OutMeas ↔ ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂 ( 𝑦 ≼ ω → ( 𝑂 ‘ ∪ 𝑦 ) ≤ ( Σ^ ‘ ( 𝑂 ↾ 𝑦 ) ) ) ) ) ) |
17 |
1 16
|
mpbid |
⊢ ( 𝜑 → ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂 ( 𝑦 ≼ ω → ( 𝑂 ‘ ∪ 𝑦 ) ≤ ( Σ^ ‘ ( 𝑂 ↾ 𝑦 ) ) ) ) ) |
18 |
17
|
simplrd |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ) |
19 |
|
pweq |
⊢ ( 𝑦 = 𝐵 → 𝒫 𝑦 = 𝒫 𝐵 ) |
20 |
19
|
raleqdv |
⊢ ( 𝑦 = 𝐵 → ( ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ↔ ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ) ) |
21 |
|
fveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝑂 ‘ 𝑦 ) = ( 𝑂 ‘ 𝐵 ) ) |
22 |
21
|
breq2d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ↔ ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝐵 ) ) ) |
23 |
22
|
ralbidv |
⊢ ( 𝑦 = 𝐵 → ( ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ↔ ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝐵 ) ) ) |
24 |
20 23
|
bitrd |
⊢ ( 𝑦 = 𝐵 → ( ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ↔ ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝐵 ) ) ) |
25 |
24
|
rspcva |
⊢ ( ( 𝐵 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀ 𝑦 ∈ 𝒫 ∪ dom 𝑂 ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝑦 ) ) → ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝐵 ) ) |
26 |
14 18 25
|
syl2anc |
⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝐵 ) ) |
27 |
|
fveq2 |
⊢ ( 𝑧 = 𝐴 → ( 𝑂 ‘ 𝑧 ) = ( 𝑂 ‘ 𝐴 ) ) |
28 |
27
|
breq1d |
⊢ ( 𝑧 = 𝐴 → ( ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝐵 ) ↔ ( 𝑂 ‘ 𝐴 ) ≤ ( 𝑂 ‘ 𝐵 ) ) ) |
29 |
28
|
rspcva |
⊢ ( ( 𝐴 ∈ 𝒫 𝐵 ∧ ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂 ‘ 𝑧 ) ≤ ( 𝑂 ‘ 𝐵 ) ) → ( 𝑂 ‘ 𝐴 ) ≤ ( 𝑂 ‘ 𝐵 ) ) |
30 |
10 26 29
|
syl2anc |
⊢ ( 𝜑 → ( 𝑂 ‘ 𝐴 ) ≤ ( 𝑂 ‘ 𝐵 ) ) |