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Theorem slwpgp 14924
Description: A Sylow  P-subgroup is a  P-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypothesis
Ref Expression
slwpgp.1  |-  S  =  ( Gs  H )
Assertion
Ref Expression
slwpgp  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  S )

Proof of Theorem slwpgp
StepHypRef Expression
1 eqid 2283 . . 3  |-  H  =  H
2 slwsubg 14921 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
3 slwpgp.1 . . . . 5  |-  S  =  ( Gs  H )
43slwispgp 14922 . . . 4  |-  ( ( H  e.  ( P pSyl 
G )  /\  H  e.  (SubGrp `  G )
)  ->  ( ( H  C_  H  /\  P pGrp  S )  <->  H  =  H
) )
52, 4mpdan 649 . . 3  |-  ( H  e.  ( P pSyl  G
)  ->  ( ( H  C_  H  /\  P pGrp  S )  <->  H  =  H
) )
61, 5mpbiri 224 . 2  |-  ( H  e.  ( P pSyl  G
)  ->  ( H  C_  H  /\  P pGrp  S
) )
76simprd 449 1  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   ↾s cress 13149  SubGrpcsubg 14615   pGrp cpgp 14842   pSyl cslw 14843
This theorem is referenced by:  slwhash  14935  sylow2  14937  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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