MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lagsubg2 Unicode version

Theorem lagsubg2 14513
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
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 14502 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  .~  Er  X
)
51, 4syl 17 . . 3  |-  ( ph  ->  .~  Er  X )
6 lagsubg.4 . . 3  |-  ( ph  ->  X  e.  Fin )
75, 6qshash 12162 . 2  |-  ( ph  ->  ( # `  X
)  =  sum_ x  e.  ( X /.  .~  ) ( # `  x
) )
82, 3eqgen 14505 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  ( X /.  .~  ) )  ->  Y  ~~  x )
91, 8sylan 459 . . . 4  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  Y  ~~  x )
102subgss 14457 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
111, 10syl 17 . . . . . . 7  |-  ( ph  ->  Y  C_  X )
12 ssfi 6968 . . . . . . 7  |-  ( ( X  e.  Fin  /\  Y  C_  X )  ->  Y  e.  Fin )
136, 11, 12syl2anc 645 . . . . . 6  |-  ( ph  ->  Y  e.  Fin )
1413adantr 453 . . . . 5  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  Y  e.  Fin )
156adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  X  e.  Fin )
165qsss 6606 . . . . . . . 8  |-  ( ph  ->  ( X /.  .~  )  C_  ~P X )
1716sselda 3103 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  e.  ~P X )
18 elpwi 3538 . . . . . . 7  |-  ( x  e.  ~P X  ->  x  C_  X )
1917, 18syl 17 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  C_  X
)
20 ssfi 6968 . . . . . 6  |-  ( ( X  e.  Fin  /\  x  C_  X )  ->  x  e.  Fin )
2115, 19, 20syl2anc 645 . . . . 5  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  e.  Fin )
22 hashen 11224 . . . . 5  |-  ( ( Y  e.  Fin  /\  x  e.  Fin )  ->  ( ( # `  Y
)  =  ( # `  x )  <->  Y  ~~  x ) )
2314, 21, 22syl2anc 645 . . . 4  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  ( ( # `
 Y )  =  ( # `  x
)  <->  Y  ~~  x ) )
249, 23mpbird 225 . . 3  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  ( # `  Y
)  =  ( # `  x ) )
2524sumeq2dv 12053 . 2  |-  ( ph  -> 
sum_ x  e.  ( X /.  .~  ) (
# `  Y )  =  sum_ x  e.  ( X /.  .~  )
( # `  x ) )
26 pwfi 7035 . . . . 5  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
276, 26sylib 190 . . . 4  |-  ( ph  ->  ~P X  e.  Fin )
28 ssfi 6968 . . . 4  |-  ( ( ~P X  e.  Fin  /\  ( X /.  .~  )  C_  ~P X )  ->  ( X /.  .~  )  e.  Fin )
2927, 16, 28syl2anc 645 . . 3  |-  ( ph  ->  ( X /.  .~  )  e.  Fin )
30 hashcl 11228 . . . . 5  |-  ( Y  e.  Fin  ->  ( # `
 Y )  e. 
NN0 )
3113, 30syl 17 . . . 4  |-  ( ph  ->  ( # `  Y
)  e.  NN0 )
3231nn0cnd 9899 . . 3  |-  ( ph  ->  ( # `  Y
)  e.  CC )
33 fsumconst 12129 . . 3  |-  ( ( ( X /.  .~  )  e.  Fin  /\  ( # `
 Y )  e.  CC )  ->  sum_ x  e.  ( X /.  .~  ) ( # `  Y
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  Y
) ) )
3429, 32, 33syl2anc 645 . 2  |-  ( ph  -> 
sum_ x  e.  ( X /.  .~  ) (
# `  Y )  =  ( ( # `  ( X /.  .~  ) )  x.  ( # `
 Y ) ) )
357, 25, 343eqtr2d 2291 1  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    C_ wss 3078   ~Pcpw 3530   class class class wbr 3920   ` cfv 4592  (class class class)co 5710    Er wer 6543   /.cqs 6545    ~~ cen 6746   Fincfn 6749   CCcc 8615    x. cmul 8622   NN0cn0 9844   #chash 11215   sum_csu 12035   Basecbs 13022  SubGrpcsubg 14450   ~QG cqg 14452
This theorem is referenced by:  lagsubg  14514  orbsta2  14603  sylow2blem3  14768  sylow3lem3  14775  sylow3lem4  14776
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-disj 3892  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-ec 6548  df-qs 6552  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-fz 10661  df-fzo 10749  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-0g 13278  df-mnd 14202  df-grp 14324  df-minusg 14325  df-subg 14453  df-eqg 14455
  Copyright terms: Public domain W3C validator