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Theorem slwpgp 15210
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 2412 . . 3  |-  H  =  H
2 slwsubg 15207 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
3 slwpgp.1 . . . . 5  |-  S  =  ( Gs  H )
43slwispgp 15208 . . . 4  |-  ( ( H  e.  ( P pSyl 
G )  /\  H  e.  (SubGrp `  G )
)  ->  ( ( H  C_  H  /\  P pGrp  S )  <->  H  =  H
) )
52, 4mpdan 650 . . 3  |-  ( H  e.  ( P pSyl  G
)  ->  ( ( H  C_  H  /\  P pGrp  S )  <->  H  =  H
) )
61, 5mpbiri 225 . 2  |-  ( H  e.  ( P pSyl  G
)  ->  ( H  C_  H  /\  P pGrp  S
) )
76simprd 450 1  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3288   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   ↾s cress 13433  SubGrpcsubg 14901   pGrp cpgp 15128   pSyl cslw 15129
This theorem is referenced by:  slwhash  15221  sylow2  15223  sylow3lem6  15229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-subg 14904  df-slw 15133
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