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Theorem isslw 16106
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 16034 . . 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 6303 . 2  |-  ( H  e.  ( P pSyl  G
)  ->  ( P  e.  Prime  /\  G  e.  Grp ) )
3 simp1 988 . . 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 15685 . . . 4  |-  ( H  e.  (SubGrp `  G
)  ->  G  e.  Grp )
543ad2ant2 1010 . . 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 532 . 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 461 . . . . . . . . 9  |-  ( ( p  =  P  /\  g  =  G )  ->  g  =  G )
87fveq2d 5694 . . . . . . . 8  |-  ( ( p  =  P  /\  g  =  G )  ->  (SubGrp `  g )  =  (SubGrp `  G )
)
9 simpl 457 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  g  =  G )  ->  p  =  P )
107oveq1d 6105 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  g  =  G )  ->  ( gs  k )  =  ( Gs  k ) )
119, 10breq12d 4304 . . . . . . . . . . 11  |-  ( ( p  =  P  /\  g  =  G )  ->  ( p pGrp  ( gs  k )  <->  P pGrp  ( Gs  k
) ) )
1211anbi2d 703 . . . . . . . . . 10  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( h  C_  k  /\  p pGrp  ( gs  k ) )  <->  ( h  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
1312bibi1d 319 . . . . . . . . 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 2930 . . . . . . . 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 2966 . . . . . . 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 5700 . . . . . . . 8  |-  (SubGrp `  G )  e.  _V
1716rabex 4442 . . . . . . 7  |-  { h  e.  (SubGrp `  G )  |  A. k  e.  (SubGrp `  G ) ( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
h  =  k ) }  e.  _V
1815, 1, 17ovmpt2a 6220 . . . . . 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 2509 . . . . 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 3376 . . . . . . . . 9  |-  ( h  =  H  ->  (
h  C_  k  <->  H  C_  k
) )
2120anbi1d 704 . . . . . . . 8  |-  ( h  =  H  ->  (
( h  C_  k  /\  P pGrp  ( Gs  k
) )  <->  ( H  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
22 eqeq1 2448 . . . . . . . 8  |-  ( h  =  H  ->  (
h  =  k  <->  H  =  k ) )
2321, 22bibi12d 321 . . . . . . 7  |-  ( h  =  H  ->  (
( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <->  h  =  k )  <->  ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) ) )
2423ralbidv 2734 . . . . . 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 3116 . . . . 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 261 . . . 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 457 . . . . 5  |-  ( ( P  e.  Prime  /\  G  e.  Grp )  ->  P  e.  Prime )
2827biantrurd 508 . . . 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 253 . . 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 969 . . 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 263 . 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 353 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   {crab 2718    C_ wss 3327   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   Primecprime 13762   ↾s cress 14174   Grpcgrp 15409  SubGrpcsubg 15674   pGrp cpgp 16029   pSyl cslw 16030
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 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-subg 15677  df-slw 16034
This theorem is referenced by:  slwprm  16107  slwsubg  16108  slwispgp  16109  pgpssslw  16112  subgslw  16114  fislw  16123
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