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Theorem subgslw 16838
Description: A Sylow subgroup that is contained in a larger subgroup is also Sylow with respect to the subgroup. (The converse need not be true.) (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 16831 . . 3  |-  ( K  e.  ( P pSyl  G
)  ->  P  e.  Prime )
213ad2ant2 1016 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  P  e.  Prime )
3 slwsubg 16832 . . . 4  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
433ad2ant2 1016 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  e.  (SubGrp `  G ) )
5 simp3 996 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  C_  S
)
6 subgslw.1 . . . . 5  |-  H  =  ( Gs  S )
76subsubg 16426 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( K  e.  (SubGrp `  H )  <->  ( K  e.  (SubGrp `  G )  /\  K  C_  S ) ) )
873ad2ant1 1015 . . 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 920 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  e.  (SubGrp `  H ) )
106oveq1i 6280 . . . . . . 7  |-  ( Hs  x )  =  ( ( Gs  S )s  x )
11 simpl1 997 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  S  e.  (SubGrp `  G ) )
126subsubg 16426 . . . . . . . . . 10  |-  ( S  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  S ) ) )
13123ad2ant1 1015 . . . . . . . . 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 622 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  x  C_  S
)
15 ressabs 14785 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  x  C_  S )  ->  (
( Gs  S )s  x )  =  ( Gs  x ) )
1611, 14, 15syl2anc 659 . . . . . . 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 2507 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  ( Hs  x )  =  ( Gs  x ) )
1817breq2d 4451 . . . . 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 701 . . . 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 998 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  K  e.  ( P pSyl  G ) )
2113simprbda 621 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  x  e.  (SubGrp `  G ) )
22 eqid 2454 . . . . . 6  |-  ( Gs  x )  =  ( Gs  x )
2322slwispgp 16833 . . . . 5  |-  ( ( K  e.  ( P pSyl 
G )  /\  x  e.  (SubGrp `  G )
)  ->  ( ( K  C_  x  /\  P pGrp  ( Gs  x ) )  <->  K  =  x ) )
2420, 21, 23syl2anc 659 . . . 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 253 . . 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 2868 . 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 16830 . 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 1178 1  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  e.  ( P pSyl  H )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Primecprime 14304   ↾s cress 14720  SubGrpcsubg 16397   pGrp cpgp 16753   pSyl cslw 16754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-nn 10532  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-subg 16400  df-slw 16758
This theorem is referenced by:  sylow3lem6  16854
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