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Theorem sylow2alem1 15251
Description: Lemma for sylow2a 15253. 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 2959 . . . . . 6  |-  w  e. 
_V
2 simpr 448 . . . . . 6  |-  ( (
ph  /\  A  e.  Z )  ->  A  e.  Z )
3 elecg 6943 . . . . . 6  |-  ( ( w  e.  _V  /\  A  e.  Z )  ->  ( w  e.  [ A ]  .~  <->  A  .~  w ) )
41, 2, 3sylancr 645 . . . . 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 15084 . . . . . . 7  |-  ( A  .~  w  <->  ( A  e.  Y  /\  w  e.  Y  /\  E. k  e.  X  ( k  .(+)  A )  =  w ) )
76simp3bi 974 . . . . . 6  |-  ( A  .~  w  ->  E. k  e.  X  ( k  .(+)  A )  =  w )
8 oveq2 6089 . . . . . . . . . . . . . 14  |-  ( u  =  A  ->  (
h  .(+)  u )  =  ( h  .(+)  A ) )
9 id 20 . . . . . . . . . . . . . 14  |-  ( u  =  A  ->  u  =  A )
108, 9eqeq12d 2450 . . . . . . . . . . . . 13  |-  ( u  =  A  ->  (
( h  .(+)  u )  =  u  <->  ( h  .(+) 
A )  =  A ) )
1110ralbidv 2725 . . . . . . . . . . . 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 3094 . . . . . . . . . . 11  |-  ( A  e.  Z  <->  ( A  e.  Y  /\  A. h  e.  X  ( h  .(+) 
A )  =  A ) )
142, 13sylib 189 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  Z )  ->  ( A  e.  Y  /\  A. h  e.  X  ( h  .(+)  A )  =  A ) )
1514simprd 450 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  Z )  ->  A. h  e.  X  ( h  .(+) 
A )  =  A )
16 oveq1 6088 . . . . . . . . . . 11  |-  ( h  =  k  ->  (
h  .(+)  A )  =  ( k  .(+)  A ) )
1716eqeq1d 2444 . . . . . . . . . 10  |-  ( h  =  k  ->  (
( h  .(+)  A )  =  A  <->  ( k  .(+)  A )  =  A ) )
1817rspccva 3051 . . . . . . . . 9  |-  ( ( A. h  e.  X  ( h  .(+)  A )  =  A  /\  k  e.  X )  ->  (
k  .(+)  A )  =  A )
1915, 18sylan 458 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  Z )  /\  k  e.  X )  ->  (
k  .(+)  A )  =  A )
20 eqeq1 2442 . . . . . . . 8  |-  ( ( k  .(+)  A )  =  w  ->  ( ( k  .(+)  A )  =  A  <->  w  =  A
) )
2119, 20syl5ibcom 212 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  Z )  /\  k  e.  X )  ->  (
( k  .(+)  A )  =  w  ->  w  =  A ) )
2221rexlimdva 2830 . . . . . 6  |-  ( (
ph  /\  A  e.  Z )  ->  ( E. k  e.  X  ( k  .(+)  A )  =  w  ->  w  =  A ) )
237, 22syl5 30 . . . . 5  |-  ( (
ph  /\  A  e.  Z )  ->  ( A  .~  w  ->  w  =  A ) )
244, 23sylbid 207 . . . 4  |-  ( (
ph  /\  A  e.  Z )  ->  (
w  e.  [ A ]  .~  ->  w  =  A ) )
25 elsn 3829 . . . 4  |-  ( w  e.  { A }  <->  w  =  A )
2624, 25syl6ibr 219 . . 3  |-  ( (
ph  /\  A  e.  Z )  ->  (
w  e.  [ A ]  .~  ->  w  e.  { A } ) )
2726ssrdv 3354 . 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 15085 . . . . . . 7  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .~  Er  Y
)
3128, 30syl 16 . . . . . 6  |-  ( ph  ->  .~  Er  Y )
3231adantr 452 . . . . 5  |-  ( (
ph  /\  A  e.  Z )  ->  .~  Er  Y )
3314simpld 446 . . . . 5  |-  ( (
ph  /\  A  e.  Z )  ->  A  e.  Y )
3432, 33erref 6925 . . . 4  |-  ( (
ph  /\  A  e.  Z )  ->  A  .~  A )
35 elecg 6943 . . . . 5  |-  ( ( A  e.  Z  /\  A  e.  Z )  ->  ( A  e.  [ A ]  .~  <->  A  .~  A ) )
362, 35sylancom 649 . . . 4  |-  ( (
ph  /\  A  e.  Z )  ->  ( A  e.  [ A ]  .~  <->  A  .~  A ) )
3734, 36mpbird 224 . . 3  |-  ( (
ph  /\  A  e.  Z )  ->  A  e.  [ A ]  .~  )
3837snssd 3943 . 2  |-  ( (
ph  /\  A  e.  Z )  ->  { A }  C_  [ A ]  .~  )
3927, 38eqssd 3365 1  |-  ( (
ph  /\  A  e.  Z )  ->  [ A ]  .~  =  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    C_ wss 3320   {csn 3814   {cpr 3815   class class class wbr 4212   {copab 4265   ` cfv 5454  (class class class)co 6081    Er wer 6902   [cec 6903   Fincfn 7109   Basecbs 13469    GrpAct cga 15066   pGrp cpgp 15165
This theorem is referenced by:  sylow2alem2  15252  sylow2a  15253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-ec 6907  df-map 7020  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-ga 15067
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