MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subgslw Unicode version

Theorem subgslw 15170
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 15163 . . 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 15164 . . . 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 14883 . . . 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 6023 . . . . . . 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 14883 . . . . . . . . . 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 13447 . . . . . . . 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 2424 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  ( Hs  x )  =  ( Gs  x ) )
1817breq2d 4158 . . . . 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 2380 . . . . . 6  |-  ( Gs  x )  =  ( Gs  x )
2322slwispgp 15165 . . . . 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 2725 . 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 15162 . 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 1649    e. wcel 1717   A.wral 2642    C_ wss 3256   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Primecprime 12999   ↾s cress 13390  SubGrpcsubg 14858   pGrp cpgp 15085   pSyl cslw 15086
This theorem is referenced by:  sylow3lem6  15186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-i2m1 8984  ax-1ne0 8985  ax-rrecex 8988  ax-cnre 8989
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-recs 6562  df-rdg 6597  df-nn 9926  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-subg 14861  df-slw 15090
  Copyright terms: Public domain W3C validator