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Theorem isslw 15169
Description: The property of being a Sylow subgroup. A Sylow  P-subgroup is a  P-group which has no proper supersets that are also  P-groups. (Contributed by Mario Carneiro, 16-Jan-2015.)
Assertion
Ref Expression
isslw  |-  ( 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 ) ) )
Distinct variable groups:    k, G    k, H    P, k

Proof of Theorem isslw
Dummy variables  g  h  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-slw 15097 . . 3  |- pSyl  =  ( p  e.  Prime ,  g  e.  Grp  |->  { h  e.  (SubGrp `  g )  |  A. k  e.  (SubGrp `  g ) ( ( h  C_  k  /\  p pGrp  ( gs  k ) )  <-> 
h  =  k ) } )
21elmpt2cl 6227 . 2  |-  ( H  e.  ( P pSyl  G
)  ->  ( P  e.  Prime  /\  G  e.  Grp ) )
3 simp1 957 . . 3  |-  ( ( P  e.  Prime  /\  H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G ) ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) )  ->  P  e.  Prime )
4 subgrcl 14876 . . . 4  |-  ( H  e.  (SubGrp `  G
)  ->  G  e.  Grp )
543ad2ant2 979 . . 3  |-  ( ( P  e.  Prime  /\  H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G ) ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) )  ->  G  e.  Grp )
63, 5jca 519 . 2  |-  ( ( P  e.  Prime  /\  H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G ) ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) )  ->  ( P  e.  Prime  /\  G  e.  Grp ) )
7 simpr 448 . . . . . . . . 9  |-  ( ( p  =  P  /\  g  =  G )  ->  g  =  G )
87fveq2d 5672 . . . . . . . 8  |-  ( ( p  =  P  /\  g  =  G )  ->  (SubGrp `  g )  =  (SubGrp `  G )
)
9 simpl 444 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  g  =  G )  ->  p  =  P )
107oveq1d 6035 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  g  =  G )  ->  ( gs  k )  =  ( Gs  k ) )
119, 10breq12d 4166 . . . . . . . . . . 11  |-  ( ( p  =  P  /\  g  =  G )  ->  ( p pGrp  ( gs  k )  <->  P pGrp  ( Gs  k
) ) )
1211anbi2d 685 . . . . . . . . . 10  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( h  C_  k  /\  p pGrp  ( gs  k ) )  <->  ( h  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
1312bibi1d 311 . . . . . . . . 9  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( ( h 
C_  k  /\  p pGrp  ( gs  k ) )  <-> 
h  =  k )  <-> 
( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <->  h  =  k ) ) )
148, 13raleqbidv 2859 . . . . . . . 8  |-  ( ( p  =  P  /\  g  =  G )  ->  ( A. k  e.  (SubGrp `  g )
( ( h  C_  k  /\  p pGrp  ( gs  k ) )  <->  h  =  k )  <->  A. k  e.  (SubGrp `  G )
( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <->  h  =  k ) ) )
158, 14rabeqbidv 2894 . . . . . . 7  |-  ( ( p  =  P  /\  g  =  G )  ->  { h  e.  (SubGrp `  g )  |  A. k  e.  (SubGrp `  g
) ( ( h 
C_  k  /\  p pGrp  ( gs  k ) )  <-> 
h  =  k ) }  =  { h  e.  (SubGrp `  G )  |  A. k  e.  (SubGrp `  G ) ( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
h  =  k ) } )
16 fvex 5682 . . . . . . . 8  |-  (SubGrp `  G )  e.  _V
1716rabex 4295 . . . . . . 7  |-  { h  e.  (SubGrp `  G )  |  A. k  e.  (SubGrp `  G ) ( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
h  =  k ) }  e.  _V
1815, 1, 17ovmpt2a 6143 . . . . . 6  |-  ( ( P  e.  Prime  /\  G  e.  Grp )  ->  ( P pSyl  G )  =  {
h  e.  (SubGrp `  G )  |  A. k  e.  (SubGrp `  G
) ( ( h 
C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
h  =  k ) } )
1918eleq2d 2454 . . . . 5  |-  ( ( 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  ( Gs  k ) )  <-> 
h  =  k ) } ) )
20 sseq1 3312 . . . . . . . . 9  |-  ( h  =  H  ->  (
h  C_  k  <->  H  C_  k
) )
2120anbi1d 686 . . . . . . . 8  |-  ( h  =  H  ->  (
( h  C_  k  /\  P pGrp  ( Gs  k
) )  <->  ( H  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
22 eqeq1 2393 . . . . . . . 8  |-  ( h  =  H  ->  (
h  =  k  <->  H  =  k ) )
2321, 22bibi12d 313 . . . . . . 7  |-  ( h  =  H  ->  (
( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <->  h  =  k )  <->  ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) ) )
2423ralbidv 2669 . . . . . 6  |-  ( h  =  H  ->  ( A. k  e.  (SubGrp `  G ) ( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
h  =  k )  <->  A. k  e.  (SubGrp `  G ) ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) ) )
2524elrab 3035 . . . . 5  |-  ( H  e.  { h  e.  (SubGrp `  G )  |  A. k  e.  (SubGrp `  G ) ( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
h  =  k ) }  <->  ( H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G ) ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) ) )
2619, 25syl6bb 253 . . . 4  |-  ( ( 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  ( Gs  k ) )  <-> 
H  =  k ) ) ) )
27 simpl 444 . . . . 5  |-  ( ( P  e.  Prime  /\  G  e.  Grp )  ->  P  e.  Prime )
2827biantrurd 495 . . . 4  |-  ( ( P  e.  Prime  /\  G  e.  Grp )  ->  (
( H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G
) ( ( H 
C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) )  <->  ( P  e. 
Prime  /\  ( H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G ) ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) ) ) ) )
2926, 28bitrd 245 . . 3  |-  ( ( 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  ( Gs  k ) )  <-> 
H  =  k ) ) ) ) )
30 3anass 940 . . 3  |-  ( ( P  e.  Prime  /\  H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G ) ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) )  <->  ( P  e. 
Prime  /\  ( H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G ) ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) ) ) )
3129, 30syl6bbr 255 . 2  |-  ( ( 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  ( Gs  k ) )  <-> 
H  =  k ) ) ) )
322, 6, 31pm5.21nii 343 1  |-  ( 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 ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653    C_ wss 3263   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Primecprime 13006   ↾s cress 13397   Grpcgrp 14612  SubGrpcsubg 14865   pGrp cpgp 15092   pSyl cslw 15093
This theorem is referenced by:  slwprm  15170  slwsubg  15171  slwispgp  15172  pgpssslw  15175  subgslw  15177  fislw  15186
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-subg 14868  df-slw 15097
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