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Theorem lagsubg2 14928
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 14917 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  .~  Er  X
)
51, 4syl 16 . . 3  |-  ( ph  ->  .~  Er  X )
6 lagsubg.4 . . 3  |-  ( ph  ->  X  e.  Fin )
75, 6qshash 12533 . 2  |-  ( ph  ->  ( # `  X
)  =  sum_ x  e.  ( X /.  .~  ) ( # `  x
) )
82, 3eqgen 14920 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  ( X /.  .~  ) )  ->  Y  ~~  x )
91, 8sylan 458 . . . 4  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  Y  ~~  x )
102subgss 14872 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
111, 10syl 16 . . . . . . 7  |-  ( ph  ->  Y  C_  X )
12 ssfi 7265 . . . . . . 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 6901 . . . . . . . 8  |-  ( ph  ->  ( X /.  .~  )  C_  ~P X )
1716sselda 3291 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  e.  ~P X )
1817elpwid 3751 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  C_  X
)
19 ssfi 7265 . . . . . 6  |-  ( ( X  e.  Fin  /\  x  C_  X )  ->  x  e.  Fin )
2015, 18, 19syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  e.  Fin )
21 hashen 11558 . . . . 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 12424 . 2  |-  ( ph  -> 
sum_ x  e.  ( X /.  .~  ) (
# `  Y )  =  sum_ x  e.  ( X /.  .~  )
( # `  x ) )
25 pwfi 7337 . . . . 5  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
266, 25sylib 189 . . . 4  |-  ( ph  ->  ~P X  e.  Fin )
27 ssfi 7265 . . . 4  |-  ( ( ~P X  e.  Fin  /\  ( X /.  .~  )  C_  ~P X )  ->  ( X /.  .~  )  e.  Fin )
2826, 16, 27syl2anc 643 . . 3  |-  ( ph  ->  ( X /.  .~  )  e.  Fin )
29 hashcl 11566 . . . . 5  |-  ( Y  e.  Fin  ->  ( # `
 Y )  e. 
NN0 )
3013, 29syl 16 . . . 4  |-  ( ph  ->  ( # `  Y
)  e.  NN0 )
3130nn0cnd 10208 . . 3  |-  ( ph  ->  ( # `  Y
)  e.  CC )
32 fsumconst 12500 . . 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 2425 1  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /.  .~  ) )  x.  ( # `  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3263   ~Pcpw 3742   class class class wbr 4153   ` cfv 5394  (class class class)co 6020    Er wer 6838   /.cqs 6840    ~~ cen 7042   Fincfn 7045   CCcc 8921    x. cmul 8928   NN0cn0 10153   #chash 11545   sum_csu 12406   Basecbs 13396  SubGrpcsubg 14865   ~QG cqg 14867
This theorem is referenced by:  lagsubg  14929  orbsta2  15018  sylow2blem3  15183  sylow3lem3  15190  sylow3lem4  15191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-disj 4124  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-ec 6843  df-qs 6847  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-subg 14868  df-eqg 14870
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