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Theorem sylow2alem1 14928
Description: Lemma for sylow2a 14930. 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 2791 . . . . . 6  |-  w  e. 
_V
2 simpr 447 . . . . . 6  |-  ( (
ph  /\  A  e.  Z )  ->  A  e.  Z )
3 elecg 6698 . . . . . 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 14761 . . . . . . 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 5866 . . . . . . . . . . . . . 14  |-  ( u  =  A  ->  (
h  .(+)  u )  =  ( h  .(+)  A ) )
9 id 19 . . . . . . . . . . . . . 14  |-  ( u  =  A  ->  u  =  A )
108, 9eqeq12d 2297 . . . . . . . . . . . . 13  |-  ( u  =  A  ->  (
( h  .(+)  u )  =  u  <->  ( h  .(+) 
A )  =  A ) )
1110ralbidv 2563 . . . . . . . . . . . 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 2925 . . . . . . . . . . 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 5865 . . . . . . . . . . 11  |-  ( h  =  k  ->  (
h  .(+)  A )  =  ( k  .(+)  A ) )
1716eqeq1d 2291 . . . . . . . . . 10  |-  ( h  =  k  ->  (
( h  .(+)  A )  =  A  <->  ( k  .(+)  A )  =  A ) )
1817rspccva 2883 . . . . . . . . 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 2289 . . . . . . . 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 2667 . . . . . 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 3655 . . . 4  |-  ( w  e.  { A }  <->  w  =  A )
2624, 25syl6ibr 218 . . 3  |-  ( (
ph  /\  A  e.  Z )  ->  (
w  e.  [ A ]  .~  ->  w  e.  { A } ) )
2726ssrdv 3185 . 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 14762 . . . . . . 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 6680 . . . 4  |-  ( (
ph  /\  A  e.  Z )  ->  A  .~  A )
35 elecg 6698 . . . . 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 3760 . 2  |-  ( (
ph  /\  A  e.  Z )  ->  { A }  C_  [ A ]  .~  )
3927, 38eqssd 3196 1  |-  ( (
ph  /\  A  e.  Z )  ->  [ A ]  .~  =  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   {csn 3640   {cpr 3641   class class class wbr 4023   {copab 4076   ` cfv 5255  (class class class)co 5858    Er wer 6657   [cec 6658   Fincfn 6863   Basecbs 13148    GrpAct cga 14743   pGrp cpgp 14842
This theorem is referenced by:  sylow2alem2  14929  sylow2a  14930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-ec 6662  df-map 6774  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-ga 14744
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