Step |
Hyp |
Ref |
Expression |
1 |
|
nconnsubb.2 |
|- ( ph -> J e. ( TopOn ` X ) ) |
2 |
|
nconnsubb.3 |
|- ( ph -> A C_ X ) |
3 |
|
nconnsubb.4 |
|- ( ph -> U e. J ) |
4 |
|
nconnsubb.5 |
|- ( ph -> V e. J ) |
5 |
|
nconnsubb.6 |
|- ( ph -> ( U i^i A ) =/= (/) ) |
6 |
|
nconnsubb.7 |
|- ( ph -> ( V i^i A ) =/= (/) ) |
7 |
|
nconnsubb.8 |
|- ( ph -> ( ( U i^i V ) i^i A ) = (/) ) |
8 |
|
nconnsubb.9 |
|- ( ph -> A C_ ( U u. V ) ) |
9 |
|
connsuba |
|- ( ( J e. ( TopOn ` X ) /\ A C_ X ) -> ( ( J |`t A ) e. Conn <-> A. x e. J A. y e. J ( ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) -> ( ( x u. y ) i^i A ) =/= A ) ) ) |
10 |
1 2 9
|
syl2anc |
|- ( ph -> ( ( J |`t A ) e. Conn <-> A. x e. J A. y e. J ( ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) -> ( ( x u. y ) i^i A ) =/= A ) ) ) |
11 |
5 6 7
|
3jca |
|- ( ph -> ( ( U i^i A ) =/= (/) /\ ( V i^i A ) =/= (/) /\ ( ( U i^i V ) i^i A ) = (/) ) ) |
12 |
|
ineq1 |
|- ( x = U -> ( x i^i A ) = ( U i^i A ) ) |
13 |
12
|
neeq1d |
|- ( x = U -> ( ( x i^i A ) =/= (/) <-> ( U i^i A ) =/= (/) ) ) |
14 |
|
ineq1 |
|- ( x = U -> ( x i^i y ) = ( U i^i y ) ) |
15 |
14
|
ineq1d |
|- ( x = U -> ( ( x i^i y ) i^i A ) = ( ( U i^i y ) i^i A ) ) |
16 |
15
|
eqeq1d |
|- ( x = U -> ( ( ( x i^i y ) i^i A ) = (/) <-> ( ( U i^i y ) i^i A ) = (/) ) ) |
17 |
13 16
|
3anbi13d |
|- ( x = U -> ( ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) <-> ( ( U i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( U i^i y ) i^i A ) = (/) ) ) ) |
18 |
|
uneq1 |
|- ( x = U -> ( x u. y ) = ( U u. y ) ) |
19 |
18
|
ineq1d |
|- ( x = U -> ( ( x u. y ) i^i A ) = ( ( U u. y ) i^i A ) ) |
20 |
19
|
neeq1d |
|- ( x = U -> ( ( ( x u. y ) i^i A ) =/= A <-> ( ( U u. y ) i^i A ) =/= A ) ) |
21 |
17 20
|
imbi12d |
|- ( x = U -> ( ( ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) -> ( ( x u. y ) i^i A ) =/= A ) <-> ( ( ( U i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( U i^i y ) i^i A ) = (/) ) -> ( ( U u. y ) i^i A ) =/= A ) ) ) |
22 |
|
ineq1 |
|- ( y = V -> ( y i^i A ) = ( V i^i A ) ) |
23 |
22
|
neeq1d |
|- ( y = V -> ( ( y i^i A ) =/= (/) <-> ( V i^i A ) =/= (/) ) ) |
24 |
|
ineq2 |
|- ( y = V -> ( U i^i y ) = ( U i^i V ) ) |
25 |
24
|
ineq1d |
|- ( y = V -> ( ( U i^i y ) i^i A ) = ( ( U i^i V ) i^i A ) ) |
26 |
25
|
eqeq1d |
|- ( y = V -> ( ( ( U i^i y ) i^i A ) = (/) <-> ( ( U i^i V ) i^i A ) = (/) ) ) |
27 |
23 26
|
3anbi23d |
|- ( y = V -> ( ( ( U i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( U i^i y ) i^i A ) = (/) ) <-> ( ( U i^i A ) =/= (/) /\ ( V i^i A ) =/= (/) /\ ( ( U i^i V ) i^i A ) = (/) ) ) ) |
28 |
|
sseqin2 |
|- ( A C_ ( U u. y ) <-> ( ( U u. y ) i^i A ) = A ) |
29 |
28
|
necon3bbii |
|- ( -. A C_ ( U u. y ) <-> ( ( U u. y ) i^i A ) =/= A ) |
30 |
|
uneq2 |
|- ( y = V -> ( U u. y ) = ( U u. V ) ) |
31 |
30
|
sseq2d |
|- ( y = V -> ( A C_ ( U u. y ) <-> A C_ ( U u. V ) ) ) |
32 |
31
|
notbid |
|- ( y = V -> ( -. A C_ ( U u. y ) <-> -. A C_ ( U u. V ) ) ) |
33 |
29 32
|
bitr3id |
|- ( y = V -> ( ( ( U u. y ) i^i A ) =/= A <-> -. A C_ ( U u. V ) ) ) |
34 |
27 33
|
imbi12d |
|- ( y = V -> ( ( ( ( U i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( U i^i y ) i^i A ) = (/) ) -> ( ( U u. y ) i^i A ) =/= A ) <-> ( ( ( U i^i A ) =/= (/) /\ ( V i^i A ) =/= (/) /\ ( ( U i^i V ) i^i A ) = (/) ) -> -. A C_ ( U u. V ) ) ) ) |
35 |
21 34
|
rspc2v |
|- ( ( U e. J /\ V e. J ) -> ( A. x e. J A. y e. J ( ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) -> ( ( x u. y ) i^i A ) =/= A ) -> ( ( ( U i^i A ) =/= (/) /\ ( V i^i A ) =/= (/) /\ ( ( U i^i V ) i^i A ) = (/) ) -> -. A C_ ( U u. V ) ) ) ) |
36 |
3 4 35
|
syl2anc |
|- ( ph -> ( A. x e. J A. y e. J ( ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) -> ( ( x u. y ) i^i A ) =/= A ) -> ( ( ( U i^i A ) =/= (/) /\ ( V i^i A ) =/= (/) /\ ( ( U i^i V ) i^i A ) = (/) ) -> -. A C_ ( U u. V ) ) ) ) |
37 |
11 36
|
mpid |
|- ( ph -> ( A. x e. J A. y e. J ( ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) -> ( ( x u. y ) i^i A ) =/= A ) -> -. A C_ ( U u. V ) ) ) |
38 |
10 37
|
sylbid |
|- ( ph -> ( ( J |`t A ) e. Conn -> -. A C_ ( U u. V ) ) ) |
39 |
8 38
|
mt2d |
|- ( ph -> -. ( J |`t A ) e. Conn ) |