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Theorem slwispgp 15174
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 15171 . . 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 3315 . . . . 5  |-  ( k  =  K  ->  ( H  C_  k  <->  H  C_  K
) )
4 oveq2 6030 . . . . . . 7  |-  ( k  =  K  ->  ( Gs  k )  =  ( Gs  K ) )
5 slwispgp.1 . . . . . . 7  |-  S  =  ( Gs  K )
64, 5syl6eqr 2439 . . . . . 6  |-  ( k  =  K  ->  ( Gs  k )  =  S )
76breq2d 4167 . . . . 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 2398 . . . 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 2996 . 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 1649    e. wcel 1717   A.wral 2651    C_ wss 3265   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Primecprime 13008   ↾s cress 13399  SubGrpcsubg 14867   pGrp cpgp 15094   pSyl cslw 15095
This theorem is referenced by:  slwpss  15175  slwpgp  15176  subgslw  15179  slwhash  15187
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-subg 14870  df-slw 15099
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