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Theorem orbsta2 16919
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 16914 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  H  e.  (SubGrp `  G )
)
54adantr 466 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  H  e.  (SubGrp `  G ) )
6 simpr 462 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  X  e.  Fin )
71, 2, 5, 6lagsubg2 16829 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  H
) ) )
8 eqid 2429 . . . . . . 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 16918 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  ran  ( k  e.  X  |-> 
<. [ k ]  .~  ,  ( k  .(+)  A ) >. ) : ( X /.  .~  ) -1-1-onto-> [ A ] O )
1110adantr 466 . . . . 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 5891 . . . . . . . 8  |-  ( Base `  G )  e.  _V
131, 12eqeltri 2513 . . . . . . 7  |-  X  e. 
_V
1413qsex 7430 . . . . . 6  |-  ( X /.  .~  )  e. 
_V
1514f1oen 7597 . . . . 5  |-  ( ran  ( k  e.  X  |-> 
<. [ k ]  .~  ,  ( k  .(+)  A ) >. ) : ( X /.  .~  ) -1-1-onto-> [ A ] O  ->  ( X /.  .~  )  ~~  [ A ] O )
1611, 15syl 17 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  ~~  [ A ] O )
17 pwfi 7875 . . . . . . 7  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
186, 17sylib 199 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ~P X  e. 
Fin )
191, 2eqger 16818 . . . . . . . 8  |-  ( H  e.  (SubGrp `  G
)  ->  .~  Er  X
)
205, 19syl 17 . . . . . . 7  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  .~  Er  X
)
2120qsss 7432 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  C_  ~P X
)
22 ssfi 7798 . . . . . 6  |-  ( ( ~P X  e.  Fin  /\  ( X /.  .~  )  C_  ~P X )  ->  ( X /.  .~  )  e.  Fin )
2318, 21, 22syl2anc 665 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  e.  Fin )
2416ensymd 7627 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  [ A ] O  ~~  ( X /.  .~  ) )
25 enfii 7795 . . . . . 6  |-  ( ( ( X /.  .~  )  e.  Fin  /\  [ A ] O  ~~  ( X /.  .~  ) )  ->  [ A ] O  e.  Fin )
2623, 24, 25syl2anc 665 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  [ A ] O  e.  Fin )
27 hashen 12527 . . . . 5  |-  ( ( ( X /.  .~  )  e.  Fin  /\  [ A ] O  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  =  (
# `  [ A ] O )  <->  ( X /.  .~  )  ~~  [ A ] O ) )
2823, 26, 27syl2anc 665 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  =  (
# `  [ A ] O )  <->  ( X /.  .~  )  ~~  [ A ] O ) )
2916, 28mpbird 235 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  ( X /.  .~  ) )  =  ( # `  [ A ] O ) )
3029oveq1d 6320 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  x.  ( # `
 H ) )  =  ( ( # `  [ A ] O
)  x.  ( # `  H ) ) )
317, 30eqtrd 2470 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   {crab 2786   _Vcvv 3087    C_ wss 3442   ~Pcpw 3985   {cpr 4004   <.cop 4008   class class class wbr 4426   {copab 4483    |-> cmpt 4484   ran crn 4855   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305    Er wer 7368   [cec 7369   /.cqs 7370    ~~ cen 7574   Fincfn 7577    x. cmul 9543   #chash 12512   Basecbs 15084  SubGrpcsubg 16762   ~QG cqg 16764    GrpAct cga 16894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-ec 7373  df-qs 7377  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-subg 16765  df-eqg 16767  df-ga 16895
This theorem is referenced by:  sylow1lem5  17189  sylow2alem2  17205  sylow3lem3  17216
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