Step |
Hyp |
Ref |
Expression |
1 |
|
subgslw.1 |
|- H = ( G |`s S ) |
2 |
|
slwprm |
|- ( K e. ( P pSyl G ) -> P e. Prime ) |
3 |
2
|
3ad2ant2 |
|- ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> P e. Prime ) |
4 |
|
slwsubg |
|- ( K e. ( P pSyl G ) -> K e. ( SubGrp ` G ) ) |
5 |
4
|
3ad2ant2 |
|- ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> K e. ( SubGrp ` G ) ) |
6 |
|
simp3 |
|- ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> K C_ S ) |
7 |
1
|
subsubg |
|- ( S e. ( SubGrp ` G ) -> ( K e. ( SubGrp ` H ) <-> ( K e. ( SubGrp ` G ) /\ K C_ S ) ) ) |
8 |
7
|
3ad2ant1 |
|- ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> ( K e. ( SubGrp ` H ) <-> ( K e. ( SubGrp ` G ) /\ K C_ S ) ) ) |
9 |
5 6 8
|
mpbir2and |
|- ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> K e. ( SubGrp ` H ) ) |
10 |
1
|
oveq1i |
|- ( H |`s x ) = ( ( G |`s S ) |`s x ) |
11 |
|
simpl1 |
|- ( ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. ( SubGrp ` H ) ) -> S e. ( SubGrp ` G ) ) |
12 |
1
|
subsubg |
|- ( S e. ( SubGrp ` G ) -> ( x e. ( SubGrp ` H ) <-> ( x e. ( SubGrp ` G ) /\ x C_ S ) ) ) |
13 |
12
|
3ad2ant1 |
|- ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> ( x e. ( SubGrp ` H ) <-> ( x e. ( SubGrp ` G ) /\ x C_ S ) ) ) |
14 |
13
|
simplbda |
|- ( ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. ( SubGrp ` H ) ) -> x C_ S ) |
15 |
|
ressabs |
|- ( ( S e. ( SubGrp ` G ) /\ x C_ S ) -> ( ( G |`s S ) |`s x ) = ( G |`s x ) ) |
16 |
11 14 15
|
syl2anc |
|- ( ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. ( SubGrp ` H ) ) -> ( ( G |`s S ) |`s x ) = ( G |`s x ) ) |
17 |
10 16
|
eqtrid |
|- ( ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. ( SubGrp ` H ) ) -> ( H |`s x ) = ( G |`s x ) ) |
18 |
17
|
breq2d |
|- ( ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. ( SubGrp ` H ) ) -> ( P pGrp ( H |`s x ) <-> P pGrp ( G |`s x ) ) ) |
19 |
18
|
anbi2d |
|- ( ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. ( SubGrp ` H ) ) -> ( ( K C_ x /\ P pGrp ( H |`s x ) ) <-> ( K C_ x /\ P pGrp ( G |`s x ) ) ) ) |
20 |
|
simpl2 |
|- ( ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. ( SubGrp ` H ) ) -> K e. ( P pSyl G ) ) |
21 |
13
|
simprbda |
|- ( ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. ( SubGrp ` H ) ) -> x e. ( SubGrp ` G ) ) |
22 |
|
eqid |
|- ( G |`s x ) = ( G |`s x ) |
23 |
22
|
slwispgp |
|- ( ( K e. ( P pSyl G ) /\ x e. ( SubGrp ` G ) ) -> ( ( K C_ x /\ P pGrp ( G |`s x ) ) <-> K = x ) ) |
24 |
20 21 23
|
syl2anc |
|- ( ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. ( SubGrp ` H ) ) -> ( ( K C_ x /\ P pGrp ( G |`s x ) ) <-> K = x ) ) |
25 |
19 24
|
bitrd |
|- ( ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. ( SubGrp ` H ) ) -> ( ( K C_ x /\ P pGrp ( H |`s x ) ) <-> K = x ) ) |
26 |
25
|
ralrimiva |
|- ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> A. x e. ( SubGrp ` H ) ( ( K C_ x /\ P pGrp ( H |`s x ) ) <-> K = x ) ) |
27 |
|
isslw |
|- ( K e. ( P pSyl H ) <-> ( P e. Prime /\ K e. ( SubGrp ` H ) /\ A. x e. ( SubGrp ` H ) ( ( K C_ x /\ P pGrp ( H |`s x ) ) <-> K = x ) ) ) |
28 |
3 9 26 27
|
syl3anbrc |
|- ( ( S e. ( SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> K e. ( P pSyl H ) ) |