Step |
Hyp |
Ref |
Expression |
1 |
|
subgtgp.h |
|- H = ( G |`s S ) |
2 |
1
|
submmnd |
|- ( S e. ( SubMnd ` G ) -> H e. Mnd ) |
3 |
2
|
adantl |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> H e. Mnd ) |
4 |
|
tmdtps |
|- ( G e. TopMnd -> G e. TopSp ) |
5 |
|
resstps |
|- ( ( G e. TopSp /\ S e. ( SubMnd ` G ) ) -> ( G |`s S ) e. TopSp ) |
6 |
4 5
|
sylan |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ( G |`s S ) e. TopSp ) |
7 |
1 6
|
eqeltrid |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> H e. TopSp ) |
8 |
|
eqid |
|- ( Base ` H ) = ( Base ` H ) |
9 |
|
eqid |
|- ( +g ` H ) = ( +g ` H ) |
10 |
|
eqid |
|- ( +f ` H ) = ( +f ` H ) |
11 |
8 9 10
|
plusffval |
|- ( +f ` H ) = ( x e. ( Base ` H ) , y e. ( Base ` H ) |-> ( x ( +g ` H ) y ) ) |
12 |
1
|
submbas |
|- ( S e. ( SubMnd ` G ) -> S = ( Base ` H ) ) |
13 |
12
|
adantl |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> S = ( Base ` H ) ) |
14 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
15 |
1 14
|
ressplusg |
|- ( S e. ( SubMnd ` G ) -> ( +g ` G ) = ( +g ` H ) ) |
16 |
15
|
adantl |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ( +g ` G ) = ( +g ` H ) ) |
17 |
16
|
oveqd |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) ) |
18 |
13 13 17
|
mpoeq123dv |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ( x e. S , y e. S |-> ( x ( +g ` G ) y ) ) = ( x e. ( Base ` H ) , y e. ( Base ` H ) |-> ( x ( +g ` H ) y ) ) ) |
19 |
11 18
|
eqtr4id |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ( +f ` H ) = ( x e. S , y e. S |-> ( x ( +g ` G ) y ) ) ) |
20 |
|
eqid |
|- ( ( TopOpen ` G ) |`t S ) = ( ( TopOpen ` G ) |`t S ) |
21 |
|
eqid |
|- ( TopOpen ` G ) = ( TopOpen ` G ) |
22 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
23 |
21 22
|
tmdtopon |
|- ( G e. TopMnd -> ( TopOpen ` G ) e. ( TopOn ` ( Base ` G ) ) ) |
24 |
23
|
adantr |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ( TopOpen ` G ) e. ( TopOn ` ( Base ` G ) ) ) |
25 |
22
|
submss |
|- ( S e. ( SubMnd ` G ) -> S C_ ( Base ` G ) ) |
26 |
25
|
adantl |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> S C_ ( Base ` G ) ) |
27 |
|
eqid |
|- ( +f ` G ) = ( +f ` G ) |
28 |
22 14 27
|
plusffval |
|- ( +f ` G ) = ( x e. ( Base ` G ) , y e. ( Base ` G ) |-> ( x ( +g ` G ) y ) ) |
29 |
21 27
|
tmdcn |
|- ( G e. TopMnd -> ( +f ` G ) e. ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) Cn ( TopOpen ` G ) ) ) |
30 |
28 29
|
eqeltrrid |
|- ( G e. TopMnd -> ( x e. ( Base ` G ) , y e. ( Base ` G ) |-> ( x ( +g ` G ) y ) ) e. ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) Cn ( TopOpen ` G ) ) ) |
31 |
30
|
adantr |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ( x e. ( Base ` G ) , y e. ( Base ` G ) |-> ( x ( +g ` G ) y ) ) e. ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) Cn ( TopOpen ` G ) ) ) |
32 |
20 24 26 20 24 26 31
|
cnmpt2res |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ( x e. S , y e. S |-> ( x ( +g ` G ) y ) ) e. ( ( ( ( TopOpen ` G ) |`t S ) tX ( ( TopOpen ` G ) |`t S ) ) Cn ( TopOpen ` G ) ) ) |
33 |
19 32
|
eqeltrd |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ( +f ` H ) e. ( ( ( ( TopOpen ` G ) |`t S ) tX ( ( TopOpen ` G ) |`t S ) ) Cn ( TopOpen ` G ) ) ) |
34 |
8 10
|
mndplusf |
|- ( H e. Mnd -> ( +f ` H ) : ( ( Base ` H ) X. ( Base ` H ) ) --> ( Base ` H ) ) |
35 |
|
frn |
|- ( ( +f ` H ) : ( ( Base ` H ) X. ( Base ` H ) ) --> ( Base ` H ) -> ran ( +f ` H ) C_ ( Base ` H ) ) |
36 |
3 34 35
|
3syl |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ran ( +f ` H ) C_ ( Base ` H ) ) |
37 |
36 13
|
sseqtrrd |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ran ( +f ` H ) C_ S ) |
38 |
|
cnrest2 |
|- ( ( ( TopOpen ` G ) e. ( TopOn ` ( Base ` G ) ) /\ ran ( +f ` H ) C_ S /\ S C_ ( Base ` G ) ) -> ( ( +f ` H ) e. ( ( ( ( TopOpen ` G ) |`t S ) tX ( ( TopOpen ` G ) |`t S ) ) Cn ( TopOpen ` G ) ) <-> ( +f ` H ) e. ( ( ( ( TopOpen ` G ) |`t S ) tX ( ( TopOpen ` G ) |`t S ) ) Cn ( ( TopOpen ` G ) |`t S ) ) ) ) |
39 |
24 37 26 38
|
syl3anc |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ( ( +f ` H ) e. ( ( ( ( TopOpen ` G ) |`t S ) tX ( ( TopOpen ` G ) |`t S ) ) Cn ( TopOpen ` G ) ) <-> ( +f ` H ) e. ( ( ( ( TopOpen ` G ) |`t S ) tX ( ( TopOpen ` G ) |`t S ) ) Cn ( ( TopOpen ` G ) |`t S ) ) ) ) |
40 |
33 39
|
mpbid |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> ( +f ` H ) e. ( ( ( ( TopOpen ` G ) |`t S ) tX ( ( TopOpen ` G ) |`t S ) ) Cn ( ( TopOpen ` G ) |`t S ) ) ) |
41 |
1 21
|
resstopn |
|- ( ( TopOpen ` G ) |`t S ) = ( TopOpen ` H ) |
42 |
10 41
|
istmd |
|- ( H e. TopMnd <-> ( H e. Mnd /\ H e. TopSp /\ ( +f ` H ) e. ( ( ( ( TopOpen ` G ) |`t S ) tX ( ( TopOpen ` G ) |`t S ) ) Cn ( ( TopOpen ` G ) |`t S ) ) ) ) |
43 |
3 7 40 42
|
syl3anbrc |
|- ( ( G e. TopMnd /\ S e. ( SubMnd ` G ) ) -> H e. TopMnd ) |