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Theorem subgslw 15242
Description: A Sylow subgroup that is contained in a larger subgroup is also Sylow with respect to the subgroup. (The converse may not be true, though.) (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypothesis
Ref Expression
subgslw.1  |-  H  =  ( Gs  S )
Assertion
Ref Expression
subgslw  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  e.  ( P pSyl  H )
)

Proof of Theorem subgslw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 slwprm 15235 . . 3  |-  ( K  e.  ( P pSyl  G
)  ->  P  e.  Prime )
213ad2ant2 979 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  P  e.  Prime )
3 slwsubg 15236 . . . 4  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
433ad2ant2 979 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  e.  (SubGrp `  G ) )
5 simp3 959 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  C_  S
)
6 subgslw.1 . . . . 5  |-  H  =  ( Gs  S )
76subsubg 14955 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( K  e.  (SubGrp `  H )  <->  ( K  e.  (SubGrp `  G )  /\  K  C_  S ) ) )
873ad2ant1 978 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  ( K  e.  (SubGrp `  H )  <->  ( K  e.  (SubGrp `  G )  /\  K  C_  S ) ) )
94, 5, 8mpbir2and 889 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  e.  (SubGrp `  H ) )
106oveq1i 6083 . . . . . . 7  |-  ( Hs  x )  =  ( ( Gs  S )s  x )
11 simpl1 960 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  S  e.  (SubGrp `  G ) )
126subsubg 14955 . . . . . . . . . 10  |-  ( S  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  S ) ) )
13123ad2ant1 978 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  ( x  e.  (SubGrp `  H )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  S ) ) )
1413simplbda 608 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  x  C_  S
)
15 ressabs 13519 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  x  C_  S )  ->  (
( Gs  S )s  x )  =  ( Gs  x ) )
1611, 14, 15syl2anc 643 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  ( ( Gs  S )s  x )  =  ( Gs  x ) )
1710, 16syl5eq 2479 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  ( Hs  x )  =  ( Gs  x ) )
1817breq2d 4216 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  ( P pGrp  ( Hs  x )  <->  P pGrp  ( Gs  x ) ) )
1918anbi2d 685 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  ( ( K 
C_  x  /\  P pGrp  ( Hs  x ) )  <->  ( K  C_  x  /\  P pGrp  ( Gs  x ) ) ) )
20 simpl2 961 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  K  e.  ( P pSyl  G ) )
2113simprbda 607 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  x  e.  (SubGrp `  G ) )
22 eqid 2435 . . . . . 6  |-  ( Gs  x )  =  ( Gs  x )
2322slwispgp 15237 . . . . 5  |-  ( ( K  e.  ( P pSyl 
G )  /\  x  e.  (SubGrp `  G )
)  ->  ( ( K  C_  x  /\  P pGrp  ( Gs  x ) )  <->  K  =  x ) )
2420, 21, 23syl2anc 643 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  ( ( K 
C_  x  /\  P pGrp  ( Gs  x ) )  <->  K  =  x ) )
2519, 24bitrd 245 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  ( ( K 
C_  x  /\  P pGrp  ( Hs  x ) )  <->  K  =  x ) )
2625ralrimiva 2781 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  A. x  e.  (SubGrp `  H )
( ( K  C_  x  /\  P pGrp  ( Hs  x ) )  <->  K  =  x ) )
27 isslw 15234 . 2  |-  ( K  e.  ( P pSyl  H
)  <->  ( P  e. 
Prime  /\  K  e.  (SubGrp `  H )  /\  A. x  e.  (SubGrp `  H
) ( ( K 
C_  x  /\  P pGrp  ( Hs  x ) )  <->  K  =  x ) ) )
282, 9, 26, 27syl3anbrc 1138 1  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  e.  ( P pSyl  H )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Primecprime 13071   ↾s cress 13462  SubGrpcsubg 14930   pGrp cpgp 15157   pSyl cslw 15158
This theorem is referenced by:  sylow3lem6  15258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rrecex 9054  ax-cnre 9055
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-nn 9993  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-subg 14933  df-slw 15162
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