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Theorem slwispgp 15235
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 15232 . . 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 974 . 2  |-  ( H  e.  ( P pSyl  G
)  ->  A. k  e.  (SubGrp `  G )
( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k ) )
3 sseq2 3362 . . . . 5  |-  ( k  =  K  ->  ( H  C_  k  <->  H  C_  K
) )
4 oveq2 6081 . . . . . . 7  |-  ( k  =  K  ->  ( Gs  k )  =  ( Gs  K ) )
5 slwispgp.1 . . . . . . 7  |-  S  =  ( Gs  K )
64, 5syl6eqr 2485 . . . . . 6  |-  ( k  =  K  ->  ( Gs  k )  =  S )
76breq2d 4216 . . . . 5  |-  ( k  =  K  ->  ( P pGrp  ( Gs  k )  <->  P pGrp  S ) )
83, 7anbi12d 692 . . . 4  |-  ( k  =  K  ->  (
( H  C_  k  /\  P pGrp  ( Gs  k
) )  <->  ( H  C_  K  /\  P pGrp  S
) ) )
9 eqeq2 2444 . . . 4  |-  ( k  =  K  ->  ( H  =  k  <->  H  =  K ) )
108, 9bibi12d 313 . . 3  |-  ( k  =  K  ->  (
( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k )  <->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) ) )
1110rspccva 3043 . 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 458 1  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )
)  ->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Primecprime 13069   ↾s cress 13460  SubGrpcsubg 14928   pGrp cpgp 15155   pSyl cslw 15156
This theorem is referenced by:  slwpss  15236  slwpgp  15237  subgslw  15240  slwhash  15248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-subg 14931  df-slw 15160
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