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Theorem sylow2alem1 14944
Description: Lemma for sylow2a 14946. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
Hypotheses
Ref Expression
sylow2a.x  |-  X  =  ( Base `  G
)
sylow2a.m  |-  ( ph  -> 
.(+)  e.  ( G  GrpAct  Y ) )
sylow2a.p  |-  ( ph  ->  P pGrp  G )
sylow2a.f  |-  ( ph  ->  X  e.  Fin )
sylow2a.y  |-  ( ph  ->  Y  e.  Fin )
sylow2a.z  |-  Z  =  { u  e.  Y  |  A. h  e.  X  ( h  .(+)  u )  =  u }
sylow2a.r  |-  .~  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  Y  /\  E. g  e.  X  (
g  .(+)  x )  =  y ) }
Assertion
Ref Expression
sylow2alem1  |-  ( (
ph  /\  A  e.  Z )  ->  [ A ]  .~  =  { A } )
Distinct variable groups:    .~ , h    g, h, u, x, y, A   
g, G, x, y    .(+) , g, h, u, x, y    g, X, h, u, x, y    ph, h    g, Y, h, u, x, y
Allowed substitution hints:    ph( x, y, u, g)    P( x, y, u, g, h)    .~ ( x, y, u, g)    G( u, h)    Z( x, y, u, g, h)

Proof of Theorem sylow2alem1
Dummy variables  k  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2804 . . . . . 6  |-  w  e. 
_V
2 simpr 447 . . . . . 6  |-  ( (
ph  /\  A  e.  Z )  ->  A  e.  Z )
3 elecg 6714 . . . . . 6  |-  ( ( w  e.  _V  /\  A  e.  Z )  ->  ( w  e.  [ A ]  .~  <->  A  .~  w ) )
41, 2, 3sylancr 644 . . . . 5  |-  ( (
ph  /\  A  e.  Z )  ->  (
w  e.  [ A ]  .~  <->  A  .~  w
) )
5 sylow2a.r . . . . . . . 8  |-  .~  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  Y  /\  E. g  e.  X  (
g  .(+)  x )  =  y ) }
65gaorb 14777 . . . . . . 7  |-  ( A  .~  w  <->  ( A  e.  Y  /\  w  e.  Y  /\  E. k  e.  X  ( k  .(+)  A )  =  w ) )
76simp3bi 972 . . . . . 6  |-  ( A  .~  w  ->  E. k  e.  X  ( k  .(+)  A )  =  w )
8 oveq2 5882 . . . . . . . . . . . . . 14  |-  ( u  =  A  ->  (
h  .(+)  u )  =  ( h  .(+)  A ) )
9 id 19 . . . . . . . . . . . . . 14  |-  ( u  =  A  ->  u  =  A )
108, 9eqeq12d 2310 . . . . . . . . . . . . 13  |-  ( u  =  A  ->  (
( h  .(+)  u )  =  u  <->  ( h  .(+) 
A )  =  A ) )
1110ralbidv 2576 . . . . . . . . . . . 12  |-  ( u  =  A  ->  ( A. h  e.  X  ( h  .(+)  u )  =  u  <->  A. h  e.  X  ( h  .(+) 
A )  =  A ) )
12 sylow2a.z . . . . . . . . . . . 12  |-  Z  =  { u  e.  Y  |  A. h  e.  X  ( h  .(+)  u )  =  u }
1311, 12elrab2 2938 . . . . . . . . . . 11  |-  ( A  e.  Z  <->  ( A  e.  Y  /\  A. h  e.  X  ( h  .(+) 
A )  =  A ) )
142, 13sylib 188 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  Z )  ->  ( A  e.  Y  /\  A. h  e.  X  ( h  .(+)  A )  =  A ) )
1514simprd 449 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  Z )  ->  A. h  e.  X  ( h  .(+) 
A )  =  A )
16 oveq1 5881 . . . . . . . . . . 11  |-  ( h  =  k  ->  (
h  .(+)  A )  =  ( k  .(+)  A ) )
1716eqeq1d 2304 . . . . . . . . . 10  |-  ( h  =  k  ->  (
( h  .(+)  A )  =  A  <->  ( k  .(+)  A )  =  A ) )
1817rspccva 2896 . . . . . . . . 9  |-  ( ( A. h  e.  X  ( h  .(+)  A )  =  A  /\  k  e.  X )  ->  (
k  .(+)  A )  =  A )
1915, 18sylan 457 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  Z )  /\  k  e.  X )  ->  (
k  .(+)  A )  =  A )
20 eqeq1 2302 . . . . . . . 8  |-  ( ( k  .(+)  A )  =  w  ->  ( ( k  .(+)  A )  =  A  <->  w  =  A
) )
2119, 20syl5ibcom 211 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  Z )  /\  k  e.  X )  ->  (
( k  .(+)  A )  =  w  ->  w  =  A ) )
2221rexlimdva 2680 . . . . . 6  |-  ( (
ph  /\  A  e.  Z )  ->  ( E. k  e.  X  ( k  .(+)  A )  =  w  ->  w  =  A ) )
237, 22syl5 28 . . . . 5  |-  ( (
ph  /\  A  e.  Z )  ->  ( A  .~  w  ->  w  =  A ) )
244, 23sylbid 206 . . . 4  |-  ( (
ph  /\  A  e.  Z )  ->  (
w  e.  [ A ]  .~  ->  w  =  A ) )
25 elsn 3668 . . . 4  |-  ( w  e.  { A }  <->  w  =  A )
2624, 25syl6ibr 218 . . 3  |-  ( (
ph  /\  A  e.  Z )  ->  (
w  e.  [ A ]  .~  ->  w  e.  { A } ) )
2726ssrdv 3198 . 2  |-  ( (
ph  /\  A  e.  Z )  ->  [ A ]  .~  C_  { A } )
28 sylow2a.m . . . . . . 7  |-  ( ph  -> 
.(+)  e.  ( G  GrpAct  Y ) )
29 sylow2a.x . . . . . . . 8  |-  X  =  ( Base `  G
)
305, 29gaorber 14778 . . . . . . 7  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .~  Er  Y
)
3128, 30syl 15 . . . . . 6  |-  ( ph  ->  .~  Er  Y )
3231adantr 451 . . . . 5  |-  ( (
ph  /\  A  e.  Z )  ->  .~  Er  Y )
3314simpld 445 . . . . 5  |-  ( (
ph  /\  A  e.  Z )  ->  A  e.  Y )
3432, 33erref 6696 . . . 4  |-  ( (
ph  /\  A  e.  Z )  ->  A  .~  A )
35 elecg 6714 . . . . 5  |-  ( ( A  e.  Z  /\  A  e.  Z )  ->  ( A  e.  [ A ]  .~  <->  A  .~  A ) )
362, 35sylancom 648 . . . 4  |-  ( (
ph  /\  A  e.  Z )  ->  ( A  e.  [ A ]  .~  <->  A  .~  A ) )
3734, 36mpbird 223 . . 3  |-  ( (
ph  /\  A  e.  Z )  ->  A  e.  [ A ]  .~  )
3837snssd 3776 . 2  |-  ( (
ph  /\  A  e.  Z )  ->  { A }  C_  [ A ]  .~  )
3927, 38eqssd 3209 1  |-  ( (
ph  /\  A  e.  Z )  ->  [ A ]  .~  =  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   {csn 3653   {cpr 3654   class class class wbr 4039   {copab 4092   ` cfv 5271  (class class class)co 5874    Er wer 6673   [cec 6674   Fincfn 6879   Basecbs 13164    GrpAct cga 14759   pGrp cpgp 14858
This theorem is referenced by:  sylow2alem2  14945  sylow2a  14946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-ec 6678  df-map 6790  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-ga 14760
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