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Metamath Proof Explorer

Theorem slwispgp

Description: Defining property of a Sylow P -subgroup. (Contributed by Mario Carneiro, 16-Jan-2015)

Ref Expression
Hypothesis slwispgp.1 
|- S = ( G |`s K )
Assertion slwispgp 
|- ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) ) -> ( ( H C_ K /\ P pGrp S ) <-> H = K ) )

Proof

Step Hyp Ref Expression
1 slwispgp.1 
 |-  S = ( G |`s K )
2 isslw 
 |-  ( 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 ) ) )
3 2 simp3bi 
 |-  ( H e. ( P pSyl G ) -> A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) )
4 sseq2 
 |-  ( k = K -> ( H C_ k <-> H C_ K ) )
5 oveq2 
 |-  ( k = K -> ( G |`s k ) = ( G |`s K ) )
6 5 1 eqtr4di 
 |-  ( k = K -> ( G |`s k ) = S )
7 6 breq2d 
 |-  ( k = K -> ( P pGrp ( G |`s k ) <-> P pGrp S ) )
8 4 7 anbi12d 
 |-  ( k = K -> ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> ( H C_ K /\ P pGrp S ) ) )
9 eqeq2 
 |-  ( k = K -> ( H = k <-> H = K ) )
10 8 9 bibi12d 
 |-  ( k = K -> ( ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) <-> ( ( H C_ K /\ P pGrp S ) <-> H = K ) ) )
11 10 rspccva 
 |-  ( ( A. k e. ( SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G |`s k ) ) <-> H = k ) /\ K e. ( SubGrp ` G ) ) -> ( ( H C_ K /\ P pGrp S ) <-> H = K ) )
12 3 11 sylan 
 |-  ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) ) -> ( ( H C_ K /\ P pGrp S ) <-> H = K ) )