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Theorem lagsubg2 14989
Description: Lagrange's theorem for finite groups. Call the "order" of a group the cardinal number of the basic set of the group, and "index of a subgroup" the cardinal number of the set of left (or right, this is the same) cosets of this subgroup. Then the order of the group is the (cardinal) product of the order of any of its subgroups by the index of this subgroup. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
lagsubg.1  |-  X  =  ( Base `  G
)
lagsubg.2  |-  .~  =  ( G ~QG  Y )
lagsubg.3  |-  ( ph  ->  Y  e.  (SubGrp `  G ) )
lagsubg.4  |-  ( ph  ->  X  e.  Fin )
Assertion
Ref Expression
lagsubg2  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  Y
) ) )

Proof of Theorem lagsubg2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lagsubg.3 . . . 4  |-  ( ph  ->  Y  e.  (SubGrp `  G ) )
2 lagsubg.1 . . . . 5  |-  X  =  ( Base `  G
)
3 lagsubg.2 . . . . 5  |-  .~  =  ( G ~QG  Y )
42, 3eqger 14978 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  .~  Er  X
)
51, 4syl 16 . . 3  |-  ( ph  ->  .~  Er  X )
6 lagsubg.4 . . 3  |-  ( ph  ->  X  e.  Fin )
75, 6qshash 12594 . 2  |-  ( ph  ->  ( # `  X
)  =  sum_ x  e.  ( X /.  .~  ) ( # `  x
) )
82, 3eqgen 14981 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  ( X /.  .~  ) )  ->  Y  ~~  x )
91, 8sylan 458 . . . 4  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  Y  ~~  x )
102subgss 14933 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
111, 10syl 16 . . . . . . 7  |-  ( ph  ->  Y  C_  X )
12 ssfi 7320 . . . . . . 7  |-  ( ( X  e.  Fin  /\  Y  C_  X )  ->  Y  e.  Fin )
136, 11, 12syl2anc 643 . . . . . 6  |-  ( ph  ->  Y  e.  Fin )
1413adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  Y  e.  Fin )
156adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  X  e.  Fin )
165qsss 6956 . . . . . . . 8  |-  ( ph  ->  ( X /.  .~  )  C_  ~P X )
1716sselda 3340 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  e.  ~P X )
1817elpwid 3800 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  C_  X
)
19 ssfi 7320 . . . . . 6  |-  ( ( X  e.  Fin  /\  x  C_  X )  ->  x  e.  Fin )
2015, 18, 19syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  e.  Fin )
21 hashen 11619 . . . . 5  |-  ( ( Y  e.  Fin  /\  x  e.  Fin )  ->  ( ( # `  Y
)  =  ( # `  x )  <->  Y  ~~  x ) )
2214, 20, 21syl2anc 643 . . . 4  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  ( ( # `
 Y )  =  ( # `  x
)  <->  Y  ~~  x ) )
239, 22mpbird 224 . . 3  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  ( # `  Y
)  =  ( # `  x ) )
2423sumeq2dv 12485 . 2  |-  ( ph  -> 
sum_ x  e.  ( X /.  .~  ) (
# `  Y )  =  sum_ x  e.  ( X /.  .~  )
( # `  x ) )
25 pwfi 7393 . . . . 5  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
266, 25sylib 189 . . . 4  |-  ( ph  ->  ~P X  e.  Fin )
27 ssfi 7320 . . . 4  |-  ( ( ~P X  e.  Fin  /\  ( X /.  .~  )  C_  ~P X )  ->  ( X /.  .~  )  e.  Fin )
2826, 16, 27syl2anc 643 . . 3  |-  ( ph  ->  ( X /.  .~  )  e.  Fin )
29 hashcl 11627 . . . . 5  |-  ( Y  e.  Fin  ->  ( # `
 Y )  e. 
NN0 )
3013, 29syl 16 . . . 4  |-  ( ph  ->  ( # `  Y
)  e.  NN0 )
3130nn0cnd 10265 . . 3  |-  ( ph  ->  ( # `  Y
)  e.  CC )
32 fsumconst 12561 . . 3  |-  ( ( ( X /.  .~  )  e.  Fin  /\  ( # `
 Y )  e.  CC )  ->  sum_ x  e.  ( X /.  .~  ) ( # `  Y
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  Y
) ) )
3328, 31, 32syl2anc 643 . 2  |-  ( ph  -> 
sum_ x  e.  ( X /.  .~  ) (
# `  Y )  =  ( ( # `  ( X /.  .~  ) )  x.  ( # `
 Y ) ) )
347, 24, 333eqtr2d 2473 1  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   ~Pcpw 3791   class class class wbr 4204   ` cfv 5445  (class class class)co 6072    Er wer 6893   /.cqs 6895    ~~ cen 7097   Fincfn 7100   CCcc 8977    x. cmul 8984   NN0cn0 10210   #chash 11606   sum_csu 12467   Basecbs 13457  SubGrpcsubg 14926   ~QG cqg 14928
This theorem is referenced by:  lagsubg  14990  orbsta2  15079  sylow2blem3  15244  sylow3lem3  15251  sylow3lem4  15252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-ec 6898  df-qs 6902  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-sup 7437  df-oi 7468  df-card 7815  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-fz 11033  df-fzo 11124  df-seq 11312  df-exp 11371  df-hash 11607  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-clim 12270  df-sum 12468  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-0g 13715  df-mnd 14678  df-grp 14800  df-minusg 14801  df-subg 14929  df-eqg 14931
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