| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dvun.j |
|- J = ( K |`t S ) |
| 2 |
|
dvun.k |
|- K = ( TopOpen ` CCfld ) |
| 3 |
|
dvun.s |
|- ( ph -> S C_ CC ) |
| 4 |
|
dvun.f |
|- ( ph -> F : A --> CC ) |
| 5 |
|
dvun.g |
|- ( ph -> G : B --> CC ) |
| 6 |
|
dvun.a |
|- ( ph -> A C_ S ) |
| 7 |
|
dvun.b |
|- ( ph -> B C_ S ) |
| 8 |
|
dvun.d |
|- ( ph -> ( A i^i B ) = (/) ) |
| 9 |
|
dvun.n |
|- ( ph -> ( ( ( int ` J ) ` A ) u. ( ( int ` J ) ` B ) ) = ( ( int ` J ) ` ( A u. B ) ) ) |
| 10 |
|
resundi |
|- ( ( S _D ( F u. G ) ) |` ( ( ( int ` J ) ` A ) u. ( ( int ` J ) ` B ) ) ) = ( ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` A ) ) u. ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` B ) ) ) |
| 11 |
9
|
reseq2d |
|- ( ph -> ( ( S _D ( F u. G ) ) |` ( ( ( int ` J ) ` A ) u. ( ( int ` J ) ` B ) ) ) = ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` ( A u. B ) ) ) ) |
| 12 |
10 11
|
eqtr3id |
|- ( ph -> ( ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` A ) ) u. ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` B ) ) ) = ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` ( A u. B ) ) ) ) |
| 13 |
4 5 8
|
fun2d |
|- ( ph -> ( F u. G ) : ( A u. B ) --> CC ) |
| 14 |
6 7
|
unssd |
|- ( ph -> ( A u. B ) C_ S ) |
| 15 |
2 1
|
dvres |
|- ( ( ( S C_ CC /\ ( F u. G ) : ( A u. B ) --> CC ) /\ ( ( A u. B ) C_ S /\ A C_ S ) ) -> ( S _D ( ( F u. G ) |` A ) ) = ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` A ) ) ) |
| 16 |
3 13 14 6 15
|
syl22anc |
|- ( ph -> ( S _D ( ( F u. G ) |` A ) ) = ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` A ) ) ) |
| 17 |
4
|
ffnd |
|- ( ph -> F Fn A ) |
| 18 |
5
|
ffnd |
|- ( ph -> G Fn B ) |
| 19 |
|
fnunres1 |
|- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` A ) = F ) |
| 20 |
17 18 8 19
|
syl3anc |
|- ( ph -> ( ( F u. G ) |` A ) = F ) |
| 21 |
20
|
oveq2d |
|- ( ph -> ( S _D ( ( F u. G ) |` A ) ) = ( S _D F ) ) |
| 22 |
16 21
|
eqtr3d |
|- ( ph -> ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` A ) ) = ( S _D F ) ) |
| 23 |
2 1
|
dvres |
|- ( ( ( S C_ CC /\ ( F u. G ) : ( A u. B ) --> CC ) /\ ( ( A u. B ) C_ S /\ B C_ S ) ) -> ( S _D ( ( F u. G ) |` B ) ) = ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` B ) ) ) |
| 24 |
3 13 14 7 23
|
syl22anc |
|- ( ph -> ( S _D ( ( F u. G ) |` B ) ) = ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` B ) ) ) |
| 25 |
|
fnunres2 |
|- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` B ) = G ) |
| 26 |
17 18 8 25
|
syl3anc |
|- ( ph -> ( ( F u. G ) |` B ) = G ) |
| 27 |
26
|
oveq2d |
|- ( ph -> ( S _D ( ( F u. G ) |` B ) ) = ( S _D G ) ) |
| 28 |
24 27
|
eqtr3d |
|- ( ph -> ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` B ) ) = ( S _D G ) ) |
| 29 |
22 28
|
uneq12d |
|- ( ph -> ( ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` A ) ) u. ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` B ) ) ) = ( ( S _D F ) u. ( S _D G ) ) ) |
| 30 |
2 1
|
dvres |
|- ( ( ( S C_ CC /\ ( F u. G ) : ( A u. B ) --> CC ) /\ ( ( A u. B ) C_ S /\ ( A u. B ) C_ S ) ) -> ( S _D ( ( F u. G ) |` ( A u. B ) ) ) = ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` ( A u. B ) ) ) ) |
| 31 |
3 13 14 14 30
|
syl22anc |
|- ( ph -> ( S _D ( ( F u. G ) |` ( A u. B ) ) ) = ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` ( A u. B ) ) ) ) |
| 32 |
13
|
ffnd |
|- ( ph -> ( F u. G ) Fn ( A u. B ) ) |
| 33 |
|
fnresdm |
|- ( ( F u. G ) Fn ( A u. B ) -> ( ( F u. G ) |` ( A u. B ) ) = ( F u. G ) ) |
| 34 |
32 33
|
syl |
|- ( ph -> ( ( F u. G ) |` ( A u. B ) ) = ( F u. G ) ) |
| 35 |
34
|
oveq2d |
|- ( ph -> ( S _D ( ( F u. G ) |` ( A u. B ) ) ) = ( S _D ( F u. G ) ) ) |
| 36 |
31 35
|
eqtr3d |
|- ( ph -> ( ( S _D ( F u. G ) ) |` ( ( int ` J ) ` ( A u. B ) ) ) = ( S _D ( F u. G ) ) ) |
| 37 |
12 29 36
|
3eqtr3d |
|- ( ph -> ( ( S _D F ) u. ( S _D G ) ) = ( S _D ( F u. G ) ) ) |