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Theorem orbsta2 14861
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 14856 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  H  e.  (SubGrp `  G )
)
54adantr 451 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  H  e.  (SubGrp `  G ) )
6 simpr 447 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  X  e.  Fin )
71, 2, 5, 6lagsubg2 14771 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  H
) ) )
8 eqid 2358 . . . . . . 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 14860 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  ran  ( k  e.  X  |-> 
<. [ k ]  .~  ,  ( k  .(+)  A ) >. ) : ( X /.  .~  ) -1-1-onto-> [ A ] O )
1110adantr 451 . . . . 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 5619 . . . . . . . 8  |-  ( Base `  G )  e.  _V
131, 12eqeltri 2428 . . . . . . 7  |-  X  e. 
_V
1413qsex 6802 . . . . . 6  |-  ( X /.  .~  )  e. 
_V
1514f1oen 6967 . . . . 5  |-  ( ran  ( k  e.  X  |-> 
<. [ k ]  .~  ,  ( k  .(+)  A ) >. ) : ( X /.  .~  ) -1-1-onto-> [ A ] O  ->  ( X /.  .~  )  ~~  [ A ] O )
1611, 15syl 15 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  ~~  [ A ] O )
17 pwfi 7238 . . . . . . 7  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
186, 17sylib 188 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ~P X  e. 
Fin )
191, 2eqger 14760 . . . . . . . 8  |-  ( H  e.  (SubGrp `  G
)  ->  .~  Er  X
)
205, 19syl 15 . . . . . . 7  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  .~  Er  X
)
2120qsss 6804 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  C_  ~P X
)
22 ssfi 7168 . . . . . 6  |-  ( ( ~P X  e.  Fin  /\  ( X /.  .~  )  C_  ~P X )  ->  ( X /.  .~  )  e.  Fin )
2318, 21, 22syl2anc 642 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  e.  Fin )
24 ensym 6995 . . . . . . 7  |-  ( ( X /.  .~  )  ~~  [ A ] O  ->  [ A ] O  ~~  ( X /.  .~  ) )
2516, 24syl 15 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  [ A ] O  ~~  ( X /.  .~  ) )
26 enfii 7165 . . . . . 6  |-  ( ( ( X /.  .~  )  e.  Fin  /\  [ A ] O  ~~  ( X /.  .~  ) )  ->  [ A ] O  e.  Fin )
2723, 25, 26syl2anc 642 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  [ A ] O  e.  Fin )
28 hashen 11436 . . . . 5  |-  ( ( ( X /.  .~  )  e.  Fin  /\  [ A ] O  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  =  (
# `  [ A ] O )  <->  ( X /.  .~  )  ~~  [ A ] O ) )
2923, 27, 28syl2anc 642 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  =  (
# `  [ A ] O )  <->  ( X /.  .~  )  ~~  [ A ] O ) )
3016, 29mpbird 223 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  ( X /.  .~  ) )  =  ( # `  [ A ] O ) )
3130oveq1d 5957 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  x.  ( # `
 H ) )  =  ( ( # `  [ A ] O
)  x.  ( # `  H ) ) )
327, 31eqtrd 2390 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 176    /\ wa 358    = wceq 1642    e. wcel 1710   E.wrex 2620   {crab 2623   _Vcvv 2864    C_ wss 3228   ~Pcpw 3701   {cpr 3717   <.cop 3719   class class class wbr 4102   {copab 4155    e. cmpt 4156   ran crn 4769   -1-1-onto->wf1o 5333   ` cfv 5334  (class class class)co 5942    Er wer 6741   [cec 6742   /.cqs 6743    ~~ cen 6945   Fincfn 6948    x. cmul 8829   #chash 11427   Basecbs 13239  SubGrpcsubg 14708   ~QG cqg 14710    GrpAct cga 14836
This theorem is referenced by:  sylow1lem5  15006  sylow2alem2  15022  sylow3lem3  15033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-disj 4073  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-ec 6746  df-qs 6750  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-fz 10872  df-fzo 10960  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-sum 12250  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-0g 13497  df-mnd 14460  df-grp 14582  df-minusg 14583  df-subg 14711  df-eqg 14713  df-ga 14837
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