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Theorem isslw 15234
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 15162 . . 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 6280 . 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 14941 . . . 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 5724 . . . . . . . 8  |-  ( ( p  =  P  /\  g  =  G )  ->  (SubGrp `  g )  =  (SubGrp `  G )
)
9 simpl 444 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  g  =  G )  ->  p  =  P )
107oveq1d 6088 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  g  =  G )  ->  ( gs  k )  =  ( Gs  k ) )
119, 10breq12d 4217 . . . . . . . . . . 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 2908 . . . . . . . 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 2943 . . . . . . 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 5734 . . . . . . . 8  |-  (SubGrp `  G )  e.  _V
1716rabex 4346 . . . . . . 7  |-  { h  e.  (SubGrp `  G )  |  A. k  e.  (SubGrp `  G ) ( ( h  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
h  =  k ) }  e.  _V
1815, 1, 17ovmpt2a 6196 . . . . . 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 2502 . . . . 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 3361 . . . . . . . . 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 2441 . . . . . . . 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 2717 . . . . . 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 3084 . . . . 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 1652    e. wcel 1725   A.wral 2697   {crab 2701    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Primecprime 13071   ↾s cress 13462   Grpcgrp 14677  SubGrpcsubg 14930   pGrp cpgp 15157   pSyl cslw 15158
This theorem is referenced by:  slwprm  15235  slwsubg  15236  slwispgp  15237  pgpssslw  15240  subgslw  15242  fislw  15251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-subg 14933  df-slw 15162
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