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Theorem lagsubg2 15733
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 15722 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  .~  Er  X
)
51, 4syl 16 . . 3  |-  ( ph  ->  .~  Er  X )
6 lagsubg.4 . . 3  |-  ( ph  ->  X  e.  Fin )
75, 6qshash 13282 . 2  |-  ( ph  ->  ( # `  X
)  =  sum_ x  e.  ( X /.  .~  ) ( # `  x
) )
82, 3eqgen 15725 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  ( X /.  .~  ) )  ->  Y  ~~  x )
91, 8sylan 471 . . . 4  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  Y  ~~  x )
102subgss 15673 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
111, 10syl 16 . . . . . . 7  |-  ( ph  ->  Y  C_  X )
12 ssfi 7525 . . . . . . 7  |-  ( ( X  e.  Fin  /\  Y  C_  X )  ->  Y  e.  Fin )
136, 11, 12syl2anc 661 . . . . . 6  |-  ( ph  ->  Y  e.  Fin )
1413adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  Y  e.  Fin )
156adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  X  e.  Fin )
165qsss 7153 . . . . . . . 8  |-  ( ph  ->  ( X /.  .~  )  C_  ~P X )
1716sselda 3351 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  e.  ~P X )
1817elpwid 3865 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  C_  X
)
19 ssfi 7525 . . . . . 6  |-  ( ( X  e.  Fin  /\  x  C_  X )  ->  x  e.  Fin )
2015, 18, 19syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  e.  Fin )
21 hashen 12110 . . . . 5  |-  ( ( Y  e.  Fin  /\  x  e.  Fin )  ->  ( ( # `  Y
)  =  ( # `  x )  <->  Y  ~~  x ) )
2214, 20, 21syl2anc 661 . . . 4  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  ( ( # `
 Y )  =  ( # `  x
)  <->  Y  ~~  x ) )
239, 22mpbird 232 . . 3  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  ( # `  Y
)  =  ( # `  x ) )
2423sumeq2dv 13172 . 2  |-  ( ph  -> 
sum_ x  e.  ( X /.  .~  ) (
# `  Y )  =  sum_ x  e.  ( X /.  .~  )
( # `  x ) )
25 pwfi 7598 . . . . 5  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
266, 25sylib 196 . . . 4  |-  ( ph  ->  ~P X  e.  Fin )
27 ssfi 7525 . . . 4  |-  ( ( ~P X  e.  Fin  /\  ( X /.  .~  )  C_  ~P X )  ->  ( X /.  .~  )  e.  Fin )
2826, 16, 27syl2anc 661 . . 3  |-  ( ph  ->  ( X /.  .~  )  e.  Fin )
29 hashcl 12118 . . . . 5  |-  ( Y  e.  Fin  ->  ( # `
 Y )  e. 
NN0 )
3013, 29syl 16 . . . 4  |-  ( ph  ->  ( # `  Y
)  e.  NN0 )
3130nn0cnd 10630 . . 3  |-  ( ph  ->  ( # `  Y
)  e.  CC )
32 fsumconst 13249 . . 3  |-  ( ( ( X /.  .~  )  e.  Fin  /\  ( # `
 Y )  e.  CC )  ->  sum_ x  e.  ( X /.  .~  ) ( # `  Y
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  Y
) ) )
3328, 31, 32syl2anc 661 . 2  |-  ( ph  -> 
sum_ x  e.  ( X /.  .~  ) (
# `  Y )  =  ( ( # `  ( X /.  .~  ) )  x.  ( # `
 Y ) ) )
347, 24, 333eqtr2d 2476 1  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3323   ~Pcpw 3855   class class class wbr 4287   ` cfv 5413  (class class class)co 6086    Er wer 7090   /.cqs 7092    ~~ cen 7299   Fincfn 7302   CCcc 9272    x. cmul 9279   NN0cn0 10571   #chash 12095   sum_csu 13155   Basecbs 14166  SubGrpcsubg 15666   ~QG cqg 15668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-ec 7095  df-qs 7099  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-0g 14372  df-mnd 15407  df-grp 15536  df-minusg 15537  df-subg 15669  df-eqg 15671
This theorem is referenced by:  lagsubg  15734  orbsta2  15823  sylow2blem3  16112  sylow3lem3  16119  sylow3lem4  16120
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