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Theorem slwispgp 14938
Description: Defining property of a Sylow  P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypothesis
Ref Expression
slwispgp.1  |-  S  =  ( Gs  K )
Assertion
Ref Expression
slwispgp  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )
)  ->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) )

Proof of Theorem slwispgp
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 isslw 14935 . . 3  |-  ( 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 ) ) )
21simp3bi 972 . 2  |-  ( H  e.  ( P pSyl  G
)  ->  A. k  e.  (SubGrp `  G )
( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k ) )
3 sseq2 3213 . . . . 5  |-  ( k  =  K  ->  ( H  C_  k  <->  H  C_  K
) )
4 oveq2 5882 . . . . . . 7  |-  ( k  =  K  ->  ( Gs  k )  =  ( Gs  K ) )
5 slwispgp.1 . . . . . . 7  |-  S  =  ( Gs  K )
64, 5syl6eqr 2346 . . . . . 6  |-  ( k  =  K  ->  ( Gs  k )  =  S )
76breq2d 4051 . . . . 5  |-  ( k  =  K  ->  ( P pGrp  ( Gs  k )  <->  P pGrp  S ) )
83, 7anbi12d 691 . . . 4  |-  ( k  =  K  ->  (
( H  C_  k  /\  P pGrp  ( Gs  k
) )  <->  ( H  C_  K  /\  P pGrp  S
) ) )
9 eqeq2 2305 . . . 4  |-  ( k  =  K  ->  ( H  =  k  <->  H  =  K ) )
108, 9bibi12d 312 . . 3  |-  ( k  =  K  ->  (
( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k )  <->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) ) )
1110rspccva 2896 . 2  |-  ( ( A. k  e.  (SubGrp `  G ) ( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k )  /\  K  e.  (SubGrp `  G ) )  -> 
( ( H  C_  K  /\  P pGrp  S )  <-> 
H  =  K ) )
122, 11sylan 457 1  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )
)  ->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Primecprime 12774   ↾s cress 13165  SubGrpcsubg 14631   pGrp cpgp 14858   pSyl cslw 14859
This theorem is referenced by:  slwpss  14939  slwpgp  14940  subgslw  14943  slwhash  14951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-subg 14634  df-slw 14863
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