MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isslw Unicode version

Theorem isslw 14919
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 14847 . . 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 6061 . 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 14626 . . . 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 5529 . . . . . . . 8  |-  ( ( p  =  P  /\  g  =  G )  ->  (SubGrp `  g )  =  (SubGrp `  G )
)
9 simpl 443 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  g  =  G )  ->  p  =  P )
107oveq1d 5873 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  g  =  G )  ->  ( gs  k )  =  ( Gs  k ) )
119, 10breq12d 4036 . . . . . . . . . . 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 2748 . . . . . . . 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 2783 . . . . . . 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 5539 . . . . . . . 8  |-  (SubGrp `  G )  e.  _V
1716rabex 4165 . . . . . . 7  |-  { h  e.  (SubGrp `  G )  |  A. k  e.  (SubGrp `  G ) ( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
h  =  k ) }  e.  _V
1815, 1, 17ovmpt2a 5978 . . . . . 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 2350 . . . . 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 3199 . . . . . . . . 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 2289 . . . . . . . 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 2563 . . . . . 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 2923 . . . . 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 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Primecprime 12758   ↾s cress 13149   Grpcgrp 14362  SubGrpcsubg 14615   pGrp cpgp 14842   pSyl cslw 14843
This theorem is referenced by:  slwprm  14920  slwsubg  14921  slwispgp  14922  pgpssslw  14925  subgslw  14927  fislw  14936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-subg 14618  df-slw 14847
  Copyright terms: Public domain W3C validator