Step |
Hyp |
Ref |
Expression |
1 |
|
caragen0.o |
⊢ ( 𝜑 → 𝑂 ∈ OutMeas ) |
2 |
|
caragen0.s |
⊢ 𝑆 = ( CaraGen ‘ 𝑂 ) |
3 |
|
eqid |
⊢ ∪ dom 𝑂 = ∪ dom 𝑂 |
4 |
|
0elpw |
⊢ ∅ ∈ 𝒫 ∪ dom 𝑂 |
5 |
4
|
a1i |
⊢ ( 𝜑 → ∅ ∈ 𝒫 ∪ dom 𝑂 ) |
6 |
|
in0 |
⊢ ( 𝑎 ∩ ∅ ) = ∅ |
7 |
6
|
fveq2i |
⊢ ( 𝑂 ‘ ( 𝑎 ∩ ∅ ) ) = ( 𝑂 ‘ ∅ ) |
8 |
|
dif0 |
⊢ ( 𝑎 ∖ ∅ ) = 𝑎 |
9 |
8
|
fveq2i |
⊢ ( 𝑂 ‘ ( 𝑎 ∖ ∅ ) ) = ( 𝑂 ‘ 𝑎 ) |
10 |
7 9
|
oveq12i |
⊢ ( ( 𝑂 ‘ ( 𝑎 ∩ ∅ ) ) +𝑒 ( 𝑂 ‘ ( 𝑎 ∖ ∅ ) ) ) = ( ( 𝑂 ‘ ∅ ) +𝑒 ( 𝑂 ‘ 𝑎 ) ) |
11 |
10
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → ( ( 𝑂 ‘ ( 𝑎 ∩ ∅ ) ) +𝑒 ( 𝑂 ‘ ( 𝑎 ∖ ∅ ) ) ) = ( ( 𝑂 ‘ ∅ ) +𝑒 ( 𝑂 ‘ 𝑎 ) ) ) |
12 |
1
|
ome0 |
⊢ ( 𝜑 → ( 𝑂 ‘ ∅ ) = 0 ) |
13 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → ( 𝑂 ‘ ∅ ) = 0 ) |
14 |
13
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → ( ( 𝑂 ‘ ∅ ) +𝑒 ( 𝑂 ‘ 𝑎 ) ) = ( 0 +𝑒 ( 𝑂 ‘ 𝑎 ) ) ) |
15 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
16 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → 𝑂 ∈ OutMeas ) |
17 |
|
elpwi |
⊢ ( 𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂 ) |
18 |
17
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → 𝑎 ⊆ ∪ dom 𝑂 ) |
19 |
16 3 18
|
omecl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → ( 𝑂 ‘ 𝑎 ) ∈ ( 0 [,] +∞ ) ) |
20 |
15 19
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → ( 𝑂 ‘ 𝑎 ) ∈ ℝ* ) |
21 |
20
|
xaddid2d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → ( 0 +𝑒 ( 𝑂 ‘ 𝑎 ) ) = ( 𝑂 ‘ 𝑎 ) ) |
22 |
11 14 21
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → ( ( 𝑂 ‘ ( 𝑎 ∩ ∅ ) ) +𝑒 ( 𝑂 ‘ ( 𝑎 ∖ ∅ ) ) ) = ( 𝑂 ‘ 𝑎 ) ) |
23 |
1 3 2 5 22
|
carageneld |
⊢ ( 𝜑 → ∅ ∈ 𝑆 ) |