MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  orbsta2 Structured version   Unicode version

Theorem orbsta2 15096
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 15091 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  H  e.  (SubGrp `  G )
)
54adantr 453 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  H  e.  (SubGrp `  G ) )
6 simpr 449 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  X  e.  Fin )
71, 2, 5, 6lagsubg2 15006 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  H
) ) )
8 eqid 2438 . . . . . . 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 15095 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  ran  ( k  e.  X  |-> 
<. [ k ]  .~  ,  ( k  .(+)  A ) >. ) : ( X /.  .~  ) -1-1-onto-> [ A ] O )
1110adantr 453 . . . . 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 5745 . . . . . . . 8  |-  ( Base `  G )  e.  _V
131, 12eqeltri 2508 . . . . . . 7  |-  X  e. 
_V
1413qsex 6966 . . . . . 6  |-  ( X /.  .~  )  e. 
_V
1514f1oen 7131 . . . . 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 7405 . . . . . . 7  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
186, 17sylib 190 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ~P X  e. 
Fin )
191, 2eqger 14995 . . . . . . . 8  |-  ( H  e.  (SubGrp `  G
)  ->  .~  Er  X
)
205, 19syl 16 . . . . . . 7  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  .~  Er  X
)
2120qsss 6968 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  C_  ~P X
)
22 ssfi 7332 . . . . . 6  |-  ( ( ~P X  e.  Fin  /\  ( X /.  .~  )  C_  ~P X )  ->  ( X /.  .~  )  e.  Fin )
2318, 21, 22syl2anc 644 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( X /.  .~  )  e.  Fin )
2416ensymd 7161 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  [ A ] O  ~~  ( X /.  .~  ) )
25 enfii 7329 . . . . . 6  |-  ( ( ( X /.  .~  )  e.  Fin  /\  [ A ] O  ~~  ( X /.  .~  ) )  ->  [ A ] O  e.  Fin )
2623, 24, 25syl2anc 644 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  [ A ] O  e.  Fin )
27 hashen 11636 . . . . 5  |-  ( ( ( X /.  .~  )  e.  Fin  /\  [ A ] O  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  =  (
# `  [ A ] O )  <->  ( X /.  .~  )  ~~  [ A ] O ) )
2823, 26, 27syl2anc 644 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( ( # `  ( X /.  .~  ) )  =  (
# `  [ A ] O )  <->  ( X /.  .~  )  ~~  [ A ] O ) )
2916, 28mpbird 225 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  ( X /.  .~  ) )  =  ( # `  [ A ] O ) )
3029oveq1d 6099 . 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   {crab 2711   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   {cpr 3817   <.cop 3819   class class class wbr 4215   {copab 4268    e. cmpt 4269   ran crn 4882   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084    Er wer 6905   [cec 6906   /.cqs 6907    ~~ cen 7109   Fincfn 7112    x. cmul 9000   #chash 11623   Basecbs 13474  SubGrpcsubg 14943   ~QG cqg 14945    GrpAct cga 15071
This theorem is referenced by:  sylow1lem5  15241  sylow2alem2  15257  sylow3lem3  15268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-disj 4186  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-ec 6910  df-qs 6914  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818  df-subg 14946  df-eqg 14948  df-ga 15072
  Copyright terms: Public domain W3C validator