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Theorem slwsubg 16114
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 16112 . 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 1004 1  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720    C_ wss 3333   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Primecprime 13768   ↾s cress 14180  SubGrpcsubg 15680   pGrp cpgp 16035   pSyl cslw 16036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  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 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-subg 15683  df-slw 16040
This theorem is referenced by:  slwpgp  16117  subgslw  16120  slwhash  16128  fislw  16129  sylow2  16130  sylow3lem1  16131  sylow3lem2  16132  sylow3lem3  16133  sylow3lem4  16134  sylow3lem5  16135  sylow3lem6  16136
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