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Theorem slwsubg 14921
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 14919 . 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 971 1  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Primecprime 12758   ↾s cress 13149  SubGrpcsubg 14615   pGrp cpgp 14842   pSyl cslw 14843
This theorem is referenced by:  slwpgp  14924  subgslw  14927  slwhash  14935  fislw  14936  sylow2  14937  sylow3lem1  14938  sylow3lem2  14939  sylow3lem3  14940  sylow3lem4  14941  sylow3lem5  14942  sylow3lem6  14943
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
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