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Theorem slwpgp 16234
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 2454 . . 3  |-  H  =  H
2 slwsubg 16231 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
3 slwpgp.1 . . . . 5  |-  S  =  ( Gs  H )
43slwispgp 16232 . . . 4  |-  ( ( H  e.  ( P pSyl 
G )  /\  H  e.  (SubGrp `  G )
)  ->  ( ( H  C_  H  /\  P pGrp  S )  <->  H  =  H
) )
52, 4mpdan 668 . . 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 463 1  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3437   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   ↾s cress 14294  SubGrpcsubg 15795   pGrp cpgp 16152   pSyl cslw 16153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-subg 15798  df-slw 16157
This theorem is referenced by:  slwhash  16245  sylow2  16247  sylow3lem6  16253
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