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Theorem orbsta2 16157
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 16152 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  H  e.  (SubGrp `  G )
)
54adantr 465 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  H  e.  (SubGrp `  G ) )
6 simpr 461 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  X  e.  Fin )
71, 2, 5, 6lagsubg2 16067 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  H
) ) )
8 eqid 2467 . . . . . . 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 16156 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  ran  ( k  e.  X  |-> 
<. [ k ]  .~  ,  ( k  .(+)  A ) >. ) : ( X /.  .~  ) -1-1-onto-> [ A ] O )
1110adantr 465 . . . . 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 5876 . . . . . . . 8  |-  ( Base `  G )  e.  _V
131, 12eqeltri 2551 . . . . . . 7  |-  X  e. 
_V
1413qsex 7370 . . . . . 6  |-  ( X /.  .~  )  e. 
_V
1514f1oen 7536 . . . . 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 7815 . . . . . . 7  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
186, 17sylib 196 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ~P X  e. 
Fin )
191, 2eqger 16056 . . . . . . . 8  |-  ( H  e.  (SubGrp `  G
)  ->  .~  Er  X
)
205, 19syl 16 . . . . . . 7  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  .~  Er  X
)
2120qsss 7372 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  C_  ~P X
)
22 ssfi 7740 . . . . . 6  |-  ( ( ~P X  e.  Fin  /\  ( X /.  .~  )  C_  ~P X )  ->  ( X /.  .~  )  e.  Fin )
2318, 21, 22syl2anc 661 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  e.  Fin )
2416ensymd 7566 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  [ A ] O  ~~  ( X /.  .~  ) )
25 enfii 7737 . . . . . 6  |-  ( ( ( X /.  .~  )  e.  Fin  /\  [ A ] O  ~~  ( X /.  .~  ) )  ->  [ A ] O  e.  Fin )
2623, 24, 25syl2anc 661 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  [ A ] O  e.  Fin )
27 hashen 12388 . . . . 5  |-  ( ( ( X /.  .~  )  e.  Fin  /\  [ A ] O  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  =  (
# `  [ A ] O )  <->  ( X /.  .~  )  ~~  [ A ] O ) )
2823, 26, 27syl2anc 661 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  =  (
# `  [ A ] O )  <->  ( X /.  .~  )  ~~  [ A ] O ) )
2916, 28mpbird 232 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  ( X /.  .~  ) )  =  ( # `  [ A ] O ) )
3029oveq1d 6299 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  x.  ( # `
 H ) )  =  ( ( # `  [ A ] O
)  x.  ( # `  H ) ) )
317, 30eqtrd 2508 1  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  [ A ] O )  x.  ( # `
 H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   {cpr 4029   <.cop 4033   class class class wbr 4447   {copab 4504    |-> cmpt 4505   ran crn 5000   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284    Er wer 7308   [cec 7309   /.cqs 7310    ~~ cen 7513   Fincfn 7516    x. cmul 9497   #chash 12373   Basecbs 14490  SubGrpcsubg 16000   ~QG cqg 16002    GrpAct cga 16132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-subg 16003  df-eqg 16005  df-ga 16133
This theorem is referenced by:  sylow1lem5  16428  sylow2alem2  16444  sylow3lem3  16455
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