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Theorem isslw 14935
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 14863 . . 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 6077 . 2  |-  ( H  e.  ( P pSyl  G
)  ->  ( P  e.  Prime  /\  G  e.  Grp ) )
3 simp1 955 . . 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 14642 . . . 4  |-  ( H  e.  (SubGrp `  G
)  ->  G  e.  Grp )
543ad2ant2 977 . . 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 518 . 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 447 . . . . . . . . 9  |-  ( ( p  =  P  /\  g  =  G )  ->  g  =  G )
87fveq2d 5545 . . . . . . . 8  |-  ( ( p  =  P  /\  g  =  G )  ->  (SubGrp `  g )  =  (SubGrp `  G )
)
9 simpl 443 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  g  =  G )  ->  p  =  P )
107oveq1d 5889 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  g  =  G )  ->  ( gs  k )  =  ( Gs  k ) )
119, 10breq12d 4052 . . . . . . . . . . 11  |-  ( ( p  =  P  /\  g  =  G )  ->  ( p pGrp  ( gs  k )  <->  P pGrp  ( Gs  k
) ) )
1211anbi2d 684 . . . . . . . . . 10  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( h  C_  k  /\  p pGrp  ( gs  k ) )  <->  ( h  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
1312bibi1d 310 . . . . . . . . 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 2761 . . . . . . . 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 2796 . . . . . . 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 5555 . . . . . . . 8  |-  (SubGrp `  G )  e.  _V
1716rabex 4181 . . . . . . 7  |-  { h  e.  (SubGrp `  G )  |  A. k  e.  (SubGrp `  G ) ( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
h  =  k ) }  e.  _V
1815, 1, 17ovmpt2a 5994 . . . . . 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 2363 . . . . 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 3212 . . . . . . . . 9  |-  ( h  =  H  ->  (
h  C_  k  <->  H  C_  k
) )
2120anbi1d 685 . . . . . . . 8  |-  ( h  =  H  ->  (
( h  C_  k  /\  P pGrp  ( Gs  k
) )  <->  ( H  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
22 eqeq1 2302 . . . . . . . 8  |-  ( h  =  H  ->  (
h  =  k  <->  H  =  k ) )
2321, 22bibi12d 312 . . . . . . 7  |-  ( h  =  H  ->  (
( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <->  h  =  k )  <->  ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) ) )
2423ralbidv 2576 . . . . . 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 2936 . . . . 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 252 . . . 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 443 . . . . 5  |-  ( ( P  e.  Prime  /\  G  e.  Grp )  ->  P  e.  Prime )
2827biantrurd 494 . . . 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 244 . . 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 938 . . 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 254 . 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 342 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Primecprime 12774   ↾s cress 13165   Grpcgrp 14378  SubGrpcsubg 14631   pGrp cpgp 14858   pSyl cslw 14859
This theorem is referenced by:  slwprm  14936  slwsubg  14937  slwispgp  14938  pgpssslw  14941  subgslw  14943  fislw  14952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-subg 14634  df-slw 14863
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