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Theorem slwsubg 16503
Description: A Sylow  P-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
Assertion
Ref Expression
slwsubg  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )

Proof of Theorem slwsubg
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 isslw 16501 . 2  |-  ( 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 ) ) )
21simp2bi 1012 1  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817    C_ wss 3481   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Primecprime 14093   ↾s cress 14508  SubGrpcsubg 16067   pGrp cpgp 16424   pSyl cslw 16425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-subg 16070  df-slw 16429
This theorem is referenced by:  slwpgp  16506  subgslw  16509  slwhash  16517  fislw  16518  sylow2  16519  sylow3lem1  16520  sylow3lem2  16521  sylow3lem3  16522  sylow3lem4  16523  sylow3lem5  16524  sylow3lem6  16525
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