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Theorem slwpgp 16773
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 2396 . . 3  |-  H  =  H
2 slwsubg 16770 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
3 slwpgp.1 . . . . 5  |-  S  =  ( Gs  H )
43slwispgp 16771 . . . 4  |-  ( ( H  e.  ( P pSyl 
G )  /\  H  e.  (SubGrp `  G )
)  ->  ( ( H  C_  H  /\  P pGrp  S )  <->  H  =  H
) )
52, 4mpdan 666 . . 3  |-  ( H  e.  ( P pSyl  G
)  ->  ( ( H  C_  H  /\  P pGrp  S )  <->  H  =  H
) )
61, 5mpbiri 233 . 2  |-  ( H  e.  ( P pSyl  G
)  ->  ( H  C_  H  /\  P pGrp  S
) )
76simprd 461 1  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836    C_ wss 3406   class class class wbr 4384   ` cfv 5513  (class class class)co 6218   ↾s cress 14658  SubGrpcsubg 16335   pGrp cpgp 16691   pSyl cslw 16692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-subg 16338  df-slw 16696
This theorem is referenced by:  slwhash  16784  sylow2  16786  sylow3lem6  16792
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