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Theorem subgslw 16205
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 16198 . . 3  |-  ( K  e.  ( P pSyl  G
)  ->  P  e.  Prime )
213ad2ant2 1010 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  P  e.  Prime )
3 slwsubg 16199 . . . 4  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
433ad2ant2 1010 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  e.  (SubGrp `  G ) )
5 simp3 990 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  C_  S
)
6 subgslw.1 . . . . 5  |-  H  =  ( Gs  S )
76subsubg 15792 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( K  e.  (SubGrp `  H )  <->  ( K  e.  (SubGrp `  G )  /\  K  C_  S ) ) )
873ad2ant1 1009 . . 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 913 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  ->  K  e.  (SubGrp `  H ) )
106oveq1i 6186 . . . . . . 7  |-  ( Hs  x )  =  ( ( Gs  S )s  x )
11 simpl1 991 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  S  e.  (SubGrp `  G ) )
126subsubg 15792 . . . . . . . . . 10  |-  ( S  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  S ) ) )
13123ad2ant1 1009 . . . . . . . . 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 624 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  x  C_  S
)
15 ressabs 14324 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  x  C_  S )  ->  (
( Gs  S )s  x )  =  ( Gs  x ) )
1611, 14, 15syl2anc 661 . . . . . . 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 2502 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  ( Hs  x )  =  ( Gs  x ) )
1817breq2d 4388 . . . . 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 703 . . . 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 992 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  K  e.  ( P pSyl  G ) )
2113simprbda 623 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S
)  /\  x  e.  (SubGrp `  H ) )  ->  x  e.  (SubGrp `  G ) )
22 eqid 2450 . . . . . 6  |-  ( Gs  x )  =  ( Gs  x )
2322slwispgp 16200 . . . . 5  |-  ( ( K  e.  ( P pSyl 
G )  /\  x  e.  (SubGrp `  G )
)  ->  ( ( K  C_  x  /\  P pGrp  ( Gs  x ) )  <->  K  =  x ) )
2420, 21, 23syl2anc 661 . . . 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 2881 . 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 16197 . 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 1172 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 369    /\ w3a 965    = wceq 1370    e. wcel 1757   A.wral 2792    C_ wss 3412   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   Primecprime 13851   ↾s cress 14263  SubGrpcsubg 15763   pGrp cpgp 16120   pSyl cslw 16121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-i2m1 9437  ax-1ne0 9438  ax-rrecex 9441  ax-cnre 9442
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-recs 6918  df-rdg 6952  df-nn 10410  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-subg 15766  df-slw 16125
This theorem is referenced by:  sylow3lem6  16221
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