Step |
Hyp |
Ref |
Expression |
1 |
|
df-slw |
|- pSyl = ( p e. Prime , g e. Grp |-> { h e. ( SubGrp ` g ) | A. k e. ( SubGrp ` g ) ( ( h C_ k /\ p pGrp ( g |`s k ) ) <-> h = k ) } ) |
2 |
1
|
elmpocl |
|- ( H e. ( P pSyl G ) -> ( P e. Prime /\ G e. Grp ) ) |
3 |
|
simp1 |
|- ( ( P e. Prime /\ H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) -> P e. Prime ) |
4 |
|
subgrcl |
|- ( H e. ( SubGrp ` G ) -> G e. Grp ) |
5 |
4
|
3ad2ant2 |
|- ( ( P e. Prime /\ H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) -> G e. Grp ) |
6 |
3 5
|
jca |
|- ( ( P e. Prime /\ H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) -> ( P e. Prime /\ G e. Grp ) ) |
7 |
|
simpr |
|- ( ( p = P /\ g = G ) -> g = G ) |
8 |
7
|
fveq2d |
|- ( ( p = P /\ g = G ) -> ( SubGrp ` g ) = ( SubGrp ` G ) ) |
9 |
|
simpl |
|- ( ( p = P /\ g = G ) -> p = P ) |
10 |
7
|
oveq1d |
|- ( ( p = P /\ g = G ) -> ( g |`s k ) = ( G |`s k ) ) |
11 |
9 10
|
breq12d |
|- ( ( p = P /\ g = G ) -> ( p pGrp ( g |`s k ) <-> P pGrp ( G |`s k ) ) ) |
12 |
11
|
anbi2d |
|- ( ( p = P /\ g = G ) -> ( ( h C_ k /\ p pGrp ( g |`s k ) ) <-> ( h C_ k /\ P pGrp ( G |`s k ) ) ) ) |
13 |
12
|
bibi1d |
|- ( ( p = P /\ g = G ) -> ( ( ( h C_ k /\ p pGrp ( g |`s k ) ) <-> h = k ) <-> ( ( h C_ k /\ P pGrp ( G |`s k ) ) <-> h = k ) ) ) |
14 |
8 13
|
raleqbidv |
|- ( ( p = P /\ g = G ) -> ( A. k e. ( SubGrp ` g ) ( ( h C_ k /\ p pGrp ( g |`s k ) ) <-> h = k ) <-> A. k e. ( SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G |`s k ) ) <-> h = k ) ) ) |
15 |
8 14
|
rabeqbidv |
|- ( ( p = P /\ g = G ) -> { h e. ( SubGrp ` g ) | A. k e. ( SubGrp ` g ) ( ( h C_ k /\ p pGrp ( g |`s k ) ) <-> h = k ) } = { h e. ( SubGrp ` G ) | A. k e. ( SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G |`s k ) ) <-> h = k ) } ) |
16 |
|
fvex |
|- ( SubGrp ` G ) e. _V |
17 |
16
|
rabex |
|- { h e. ( SubGrp ` G ) | A. k e. ( SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G |`s k ) ) <-> h = k ) } e. _V |
18 |
15 1 17
|
ovmpoa |
|- ( ( P e. Prime /\ G e. Grp ) -> ( P pSyl G ) = { h e. ( SubGrp ` G ) | A. k e. ( SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G |`s k ) ) <-> h = k ) } ) |
19 |
18
|
eleq2d |
|- ( ( P e. Prime /\ G e. Grp ) -> ( H e. ( P pSyl G ) <-> H e. { h e. ( SubGrp ` G ) | A. k e. ( SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G |`s k ) ) <-> h = k ) } ) ) |
20 |
|
cleq1lem |
|- ( h = H -> ( ( h C_ k /\ P pGrp ( G |`s k ) ) <-> ( H C_ k /\ P pGrp ( G |`s k ) ) ) ) |
21 |
|
eqeq1 |
|- ( h = H -> ( h = k <-> H = k ) ) |
22 |
20 21
|
bibi12d |
|- ( h = H -> ( ( ( h C_ k /\ P pGrp ( G |`s k ) ) <-> h = k ) <-> ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) ) |
23 |
22
|
ralbidv |
|- ( h = H -> ( A. k e. ( SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G |`s k ) ) <-> h = k ) <-> A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) ) |
24 |
23
|
elrab |
|- ( H e. { h e. ( SubGrp ` G ) | A. k e. ( SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G |`s k ) ) <-> h = k ) } <-> ( H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) ) |
25 |
19 24
|
bitrdi |
|- ( ( P e. Prime /\ G e. Grp ) -> ( H e. ( P pSyl G ) <-> ( H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) ) ) |
26 |
|
simpl |
|- ( ( P e. Prime /\ G e. Grp ) -> P e. Prime ) |
27 |
26
|
biantrurd |
|- ( ( P e. Prime /\ G e. Grp ) -> ( ( H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) <-> ( P e. Prime /\ ( H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) ) ) ) |
28 |
25 27
|
bitrd |
|- ( ( P e. Prime /\ G e. Grp ) -> ( H e. ( P pSyl G ) <-> ( P e. Prime /\ ( H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) ) ) ) |
29 |
|
3anass |
|- ( ( P e. Prime /\ H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) <-> ( P e. Prime /\ ( H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) ) ) |
30 |
28 29
|
bitr4di |
|- ( ( P e. Prime /\ G e. Grp ) -> ( H e. ( P pSyl G ) <-> ( P e. Prime /\ H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) ) ) |
31 |
2 6 30
|
pm5.21nii |
|- ( H e. ( P pSyl G ) <-> ( P e. Prime /\ H e. ( SubGrp ` G ) /\ A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) ) ) |