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Theorem subgslw 14927
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 14920 . . 3  |-  ( K  e.  ( P pSyl  G
)  ->  P  e.  Prime )
213ad2ant2 977 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  P  e.  Prime )
3 slwsubg 14921 . . . 4  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
433ad2ant2 977 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  e.  (SubGrp `  G ) )
5 simp3 957 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  C_  S
)
6 subgslw.1 . . . . 5  |-  H  =  ( Gs  S )
76subsubg 14640 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( K  e.  (SubGrp `  H )  <->  ( K  e.  (SubGrp `  G )  /\  K  C_  S ) ) )
873ad2ant1 976 . . 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 888 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  e.  (SubGrp `  H ) )
106oveq1i 5868 . . . . . . 7  |-  ( Hs  x )  =  ( ( Gs  S )s  x )
11 simpl1 958 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  S  e.  (SubGrp `  G ) )
126subsubg 14640 . . . . . . . . . 10  |-  ( S  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  S ) ) )
13123ad2ant1 976 . . . . . . . . 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 607 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  x  C_  S
)
15 ressabs 13206 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  x  C_  S )  ->  (
( Gs  S )s  x )  =  ( Gs  x ) )
1611, 14, 15syl2anc 642 . . . . . . 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 2327 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  ( Hs  x )  =  ( Gs  x ) )
1817breq2d 4035 . . . . 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 684 . . . 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 959 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  K  e.  ( P pSyl  G ) )
2113simprbda 606 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  x  e.  (SubGrp `  G ) )
22 eqid 2283 . . . . . 6  |-  ( Gs  x )  =  ( Gs  x )
2322slwispgp 14922 . . . . 5  |-  ( ( K  e.  ( P pSyl 
G )  /\  x  e.  (SubGrp `  G )
)  ->  ( ( K  C_  x  /\  P pGrp  ( Gs  x ) )  <->  K  =  x ) )
2420, 21, 23syl2anc 642 . . . 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 244 . . 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 2626 . 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 14919 . 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 1136 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Primecprime 12758   ↾s cress 13149  SubGrpcsubg 14615   pGrp cpgp 14842   pSyl cslw 14843
This theorem is referenced by:  sylow3lem6  14943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-nn 9747  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-subg 14618  df-slw 14847
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