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Theorem orbsta2 15046
Description: Relation between the size of the orbit and the size of the stabilizer of a point in a finite group action. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypotheses
Ref Expression
orbsta2.x  |-  X  =  ( Base `  G
)
orbsta2.h  |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }
orbsta2.r  |-  .~  =  ( G ~QG  H )
orbsta2.o  |-  O  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  Y  /\  E. g  e.  X  (
g  .(+)  x )  =  y ) }
Assertion
Ref Expression
orbsta2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  [ A ] O )  x.  ( # `
 H ) ) )
Distinct variable groups:    u, g, x, y,  .(+)    A, g, u, x, y    g, G, u, x, y    g, Y, x, y    .~ , g, x, y    x, H, y   
g, X, u, x, y
Allowed substitution hints:    .~ ( u)    H( u, g)    O( x, y, u, g)    Y( u)

Proof of Theorem orbsta2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 orbsta2.x . . 3  |-  X  =  ( Base `  G
)
2 orbsta2.r . . 3  |-  .~  =  ( G ~QG  H )
3 orbsta2.h . . . . 5  |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }
41, 3gastacl 15041 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  H  e.  (SubGrp `  G )
)
54adantr 452 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  H  e.  (SubGrp `  G ) )
6 simpr 448 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  X  e.  Fin )
71, 2, 5, 6lagsubg2 14956 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  H
) ) )
8 eqid 2404 . . . . . . 7  |-  ran  (
k  e.  X  |->  <. [ k ]  .~  ,  ( k  .(+)  A ) >. )  =  ran  ( k  e.  X  |-> 
<. [ k ]  .~  ,  ( k  .(+)  A ) >. )
9 orbsta2.o . . . . . . 7  |-  O  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  Y  /\  E. g  e.  X  (
g  .(+)  x )  =  y ) }
101, 3, 2, 8, 9orbsta 15045 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  ran  ( k  e.  X  |-> 
<. [ k ]  .~  ,  ( k  .(+)  A ) >. ) : ( X /.  .~  ) -1-1-onto-> [ A ] O )
1110adantr 452 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ran  ( k  e.  X  |->  <. [ k ]  .~  ,  ( k  .(+)  A ) >. ) : ( X /.  .~  ) -1-1-onto-> [ A ] O )
12 fvex 5701 . . . . . . . 8  |-  ( Base `  G )  e.  _V
131, 12eqeltri 2474 . . . . . . 7  |-  X  e. 
_V
1413qsex 6922 . . . . . 6  |-  ( X /.  .~  )  e. 
_V
1514f1oen 7087 . . . . 5  |-  ( ran  ( k  e.  X  |-> 
<. [ k ]  .~  ,  ( k  .(+)  A ) >. ) : ( X /.  .~  ) -1-1-onto-> [ A ] O  ->  ( X /.  .~  )  ~~  [ A ] O )
1611, 15syl 16 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  ~~  [ A ] O )
17 pwfi 7360 . . . . . . 7  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
186, 17sylib 189 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ~P X  e. 
Fin )
191, 2eqger 14945 . . . . . . . 8  |-  ( H  e.  (SubGrp `  G
)  ->  .~  Er  X
)
205, 19syl 16 . . . . . . 7  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  .~  Er  X
)
2120qsss 6924 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  C_  ~P X
)
22 ssfi 7288 . . . . . 6  |-  ( ( ~P X  e.  Fin  /\  ( X /.  .~  )  C_  ~P X )  ->  ( X /.  .~  )  e.  Fin )
2318, 21, 22syl2anc 643 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  e.  Fin )
2416ensymd 7117 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  [ A ] O  ~~  ( X /.  .~  ) )
25 enfii 7285 . . . . . 6  |-  ( ( ( X /.  .~  )  e.  Fin  /\  [ A ] O  ~~  ( X /.  .~  ) )  ->  [ A ] O  e.  Fin )
2623, 24, 25syl2anc 643 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  [ A ] O  e.  Fin )
27 hashen 11586 . . . . 5  |-  ( ( ( X /.  .~  )  e.  Fin  /\  [ A ] O  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  =  (
# `  [ A ] O )  <->  ( X /.  .~  )  ~~  [ A ] O ) )
2823, 26, 27syl2anc 643 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  =  (
# `  [ A ] O )  <->  ( X /.  .~  )  ~~  [ A ] O ) )
2916, 28mpbird 224 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  ( X /.  .~  ) )  =  ( # `  [ A ] O ) )
3029oveq1d 6055 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  x.  ( # `
 H ) )  =  ( ( # `  [ A ] O
)  x.  ( # `  H ) ) )
317, 30eqtrd 2436 1  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  [ A ] O )  x.  ( # `
 H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   {crab 2670   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   {cpr 3775   <.cop 3777   class class class wbr 4172   {copab 4225    e. cmpt 4226   ran crn 4838   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    Er wer 6861   [cec 6862   /.cqs 6863    ~~ cen 7065   Fincfn 7068    x. cmul 8951   #chash 11573   Basecbs 13424  SubGrpcsubg 14893   ~QG cqg 14895    GrpAct cga 15021
This theorem is referenced by:  sylow1lem5  15191  sylow2alem2  15207  sylow3lem3  15218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-subg 14896  df-eqg 14898  df-ga 15022
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