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Theorem slwispgp 16108
Description: Defining property of a Sylow  P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypothesis
Ref Expression
slwispgp.1  |-  S  =  ( Gs  K )
Assertion
Ref Expression
slwispgp  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )
)  ->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) )

Proof of Theorem slwispgp
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 isslw 16105 . . 3  |-  ( H  e.  ( P pSyl  G
)  <->  ( P  e. 
Prime  /\  H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G
) ( ( H 
C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) ) )
21simp3bi 1005 . 2  |-  ( H  e.  ( P pSyl  G
)  ->  A. k  e.  (SubGrp `  G )
( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k ) )
3 sseq2 3376 . . . . 5  |-  ( k  =  K  ->  ( H  C_  k  <->  H  C_  K
) )
4 oveq2 6097 . . . . . . 7  |-  ( k  =  K  ->  ( Gs  k )  =  ( Gs  K ) )
5 slwispgp.1 . . . . . . 7  |-  S  =  ( Gs  K )
64, 5syl6eqr 2491 . . . . . 6  |-  ( k  =  K  ->  ( Gs  k )  =  S )
76breq2d 4302 . . . . 5  |-  ( k  =  K  ->  ( P pGrp  ( Gs  k )  <->  P pGrp  S ) )
83, 7anbi12d 710 . . . 4  |-  ( k  =  K  ->  (
( H  C_  k  /\  P pGrp  ( Gs  k
) )  <->  ( H  C_  K  /\  P pGrp  S
) ) )
9 eqeq2 2450 . . . 4  |-  ( k  =  K  ->  ( H  =  k  <->  H  =  K ) )
108, 9bibi12d 321 . . 3  |-  ( k  =  K  ->  (
( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k )  <->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) ) )
1110rspccva 3070 . 2  |-  ( ( A. k  e.  (SubGrp `  G ) ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k )  /\  K  e.  (SubGrp `  G ) )  -> 
( ( H  C_  K  /\  P pGrp  S )  <-> 
H  =  K ) )
122, 11sylan 471 1  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )
)  ->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713    C_ wss 3326   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   Primecprime 13761   ↾s cress 14173  SubGrpcsubg 15673   pGrp cpgp 16028   pSyl cslw 16029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-subg 15676  df-slw 16033
This theorem is referenced by:  slwpss  16109  slwpgp  16110  subgslw  16113  slwhash  16121
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