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Theorem slwprm 17202
Description: Reverse closure for the first argument of a Sylow  P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 2-May-2015.)
Assertion
Ref Expression
slwprm  |-  ( H  e.  ( P pSyl  G
)  ->  P  e.  Prime )

Proof of Theorem slwprm
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 isslw 17201 . 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 ) ) )
21simp1bi 1020 1  |-  ( H  e.  ( P pSyl  G
)  ->  P  e.  Prime )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773    C_ wss 3433   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   Primecprime 14582   ↾s cress 15082  SubGrpcsubg 16763   pGrp cpgp 17120   pSyl cslw 17121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  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 5556  df-fun 5594  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-subg 16766  df-slw 17126
This theorem is referenced by:  subgslw  17209  slwhash  17217
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