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Theorem slwispgp 16757
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 16754 . . 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 1013 . 2  |-  ( H  e.  ( P pSyl  G
)  ->  A. k  e.  (SubGrp `  G )
( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k ) )
3 sseq2 3521 . . . . 5  |-  ( k  =  K  ->  ( H  C_  k  <->  H  C_  K
) )
4 oveq2 6304 . . . . . . 7  |-  ( k  =  K  ->  ( Gs  k )  =  ( Gs  K ) )
5 slwispgp.1 . . . . . . 7  |-  S  =  ( Gs  K )
64, 5syl6eqr 2516 . . . . . 6  |-  ( k  =  K  ->  ( Gs  k )  =  S )
76breq2d 4468 . . . . 5  |-  ( k  =  K  ->  ( P pGrp  ( Gs  k )  <->  P pGrp  S ) )
83, 7anbi12d 710 . . . 4  |-  ( k  =  K  ->  (
( H  C_  k  /\  P pGrp  ( Gs  k
) )  <->  ( H  C_  K  /\  P pGrp  S
) ) )
9 eqeq2 2472 . . . 4  |-  ( k  =  K  ->  ( H  =  k  <->  H  =  K ) )
108, 9bibi12d 321 . . 3  |-  ( k  =  K  ->  (
( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k )  <->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) ) )
1110rspccva 3209 . 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 471 1  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )
)  ->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Primecprime 14228   ↾s cress 14644  SubGrpcsubg 16321   pGrp cpgp 16677   pSyl cslw 16678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-subg 16324  df-slw 16682
This theorem is referenced by:  slwpss  16758  slwpgp  16759  subgslw  16762  slwhash  16770
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