| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dvun.j |
⊢ 𝐽 = ( 𝐾 ↾t 𝑆 ) |
| 2 |
|
dvun.k |
⊢ 𝐾 = ( TopOpen ‘ ℂfld ) |
| 3 |
|
dvun.s |
⊢ ( 𝜑 → 𝑆 ⊆ ℂ ) |
| 4 |
|
dvun.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℂ ) |
| 5 |
|
dvun.g |
⊢ ( 𝜑 → 𝐺 : 𝐵 ⟶ ℂ ) |
| 6 |
|
dvun.a |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑆 ) |
| 7 |
|
dvun.b |
⊢ ( 𝜑 → 𝐵 ⊆ 𝑆 ) |
| 8 |
|
dvun.d |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
| 9 |
|
dvun.n |
⊢ ( 𝜑 → ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ∪ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) = ( ( int ‘ 𝐽 ) ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
| 10 |
|
resundi |
⊢ ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ∪ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) ) = ( ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ∪ ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) ) |
| 11 |
9
|
reseq2d |
⊢ ( 𝜑 → ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ∪ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) ) = ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
| 12 |
10 11
|
eqtr3id |
⊢ ( 𝜑 → ( ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ∪ ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) ) = ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
| 13 |
4 5 8
|
fun2d |
⊢ ( 𝜑 → ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐵 ) ⟶ ℂ ) |
| 14 |
6 7
|
unssd |
⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ⊆ 𝑆 ) |
| 15 |
2 1
|
dvres |
⊢ ( ( ( 𝑆 ⊆ ℂ ∧ ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐵 ) ⟶ ℂ ) ∧ ( ( 𝐴 ∪ 𝐵 ) ⊆ 𝑆 ∧ 𝐴 ⊆ 𝑆 ) ) → ( 𝑆 D ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐴 ) ) = ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ) |
| 16 |
3 13 14 6 15
|
syl22anc |
⊢ ( 𝜑 → ( 𝑆 D ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐴 ) ) = ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ) |
| 17 |
4
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
| 18 |
5
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn 𝐵 ) |
| 19 |
|
fnunres1 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐴 ) = 𝐹 ) |
| 20 |
17 18 8 19
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐴 ) = 𝐹 ) |
| 21 |
20
|
oveq2d |
⊢ ( 𝜑 → ( 𝑆 D ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐴 ) ) = ( 𝑆 D 𝐹 ) ) |
| 22 |
16 21
|
eqtr3d |
⊢ ( 𝜑 → ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) = ( 𝑆 D 𝐹 ) ) |
| 23 |
2 1
|
dvres |
⊢ ( ( ( 𝑆 ⊆ ℂ ∧ ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐵 ) ⟶ ℂ ) ∧ ( ( 𝐴 ∪ 𝐵 ) ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆 ) ) → ( 𝑆 D ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐵 ) ) = ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) ) |
| 24 |
3 13 14 7 23
|
syl22anc |
⊢ ( 𝜑 → ( 𝑆 D ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐵 ) ) = ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) ) |
| 25 |
|
fnunres2 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐵 ) = 𝐺 ) |
| 26 |
17 18 8 25
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐵 ) = 𝐺 ) |
| 27 |
26
|
oveq2d |
⊢ ( 𝜑 → ( 𝑆 D ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐵 ) ) = ( 𝑆 D 𝐺 ) ) |
| 28 |
24 27
|
eqtr3d |
⊢ ( 𝜑 → ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) = ( 𝑆 D 𝐺 ) ) |
| 29 |
22 28
|
uneq12d |
⊢ ( 𝜑 → ( ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ∪ ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) ) = ( ( 𝑆 D 𝐹 ) ∪ ( 𝑆 D 𝐺 ) ) ) |
| 30 |
2 1
|
dvres |
⊢ ( ( ( 𝑆 ⊆ ℂ ∧ ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐵 ) ⟶ ℂ ) ∧ ( ( 𝐴 ∪ 𝐵 ) ⊆ 𝑆 ∧ ( 𝐴 ∪ 𝐵 ) ⊆ 𝑆 ) ) → ( 𝑆 D ( ( 𝐹 ∪ 𝐺 ) ↾ ( 𝐴 ∪ 𝐵 ) ) ) = ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
| 31 |
3 13 14 14 30
|
syl22anc |
⊢ ( 𝜑 → ( 𝑆 D ( ( 𝐹 ∪ 𝐺 ) ↾ ( 𝐴 ∪ 𝐵 ) ) ) = ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
| 32 |
13
|
ffnd |
⊢ ( 𝜑 → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) |
| 33 |
|
fnresdm |
⊢ ( ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) → ( ( 𝐹 ∪ 𝐺 ) ↾ ( 𝐴 ∪ 𝐵 ) ) = ( 𝐹 ∪ 𝐺 ) ) |
| 34 |
32 33
|
syl |
⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐺 ) ↾ ( 𝐴 ∪ 𝐵 ) ) = ( 𝐹 ∪ 𝐺 ) ) |
| 35 |
34
|
oveq2d |
⊢ ( 𝜑 → ( 𝑆 D ( ( 𝐹 ∪ 𝐺 ) ↾ ( 𝐴 ∪ 𝐵 ) ) ) = ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ) |
| 36 |
31 35
|
eqtr3d |
⊢ ( 𝜑 → ( ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ ( 𝐴 ∪ 𝐵 ) ) ) = ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ) |
| 37 |
12 29 36
|
3eqtr3d |
⊢ ( 𝜑 → ( ( 𝑆 D 𝐹 ) ∪ ( 𝑆 D 𝐺 ) ) = ( 𝑆 D ( 𝐹 ∪ 𝐺 ) ) ) |