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Theorem lagsubg2 14640
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 14629 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  .~  Er  X
)
51, 4syl 17 . . 3  |-  ( ph  ->  .~  Er  X )
6 lagsubg.4 . . 3  |-  ( ph  ->  X  e.  Fin )
75, 6qshash 12250 . 2  |-  ( ph  ->  ( # `  X
)  =  sum_ x  e.  ( X /.  .~  ) ( # `  x
) )
82, 3eqgen 14632 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  ( X /.  .~  ) )  ->  Y  ~~  x )
91, 8sylan 459 . . . 4  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  Y  ~~  x )
102subgss 14584 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
111, 10syl 17 . . . . . . 7  |-  ( ph  ->  Y  C_  X )
12 ssfi 7051 . . . . . . 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 6688 . . . . . . . 8  |-  ( ph  ->  ( X /.  .~  )  C_  ~P X )
1716sselda 3155 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  e.  ~P X )
18 elpwi 3607 . . . . . . 7  |-  ( x  e.  ~P X  ->  x  C_  X )
1917, 18syl 17 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  C_  X
)
20 ssfi 7051 . . . . . 6  |-  ( ( X  e.  Fin  /\  x  C_  X )  ->  x  e.  Fin )
2115, 19, 20syl2anc 645 . . . . 5  |-  ( (
ph  /\  x  e.  ( X /.  .~  )
)  ->  x  e.  Fin )
22 hashen 11312 . . . . 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 12141 . 2  |-  ( ph  -> 
sum_ x  e.  ( X /.  .~  ) (
# `  Y )  =  sum_ x  e.  ( X /.  .~  )
( # `  x ) )
26 pwfi 7119 . . . . 5  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
276, 26sylib 190 . . . 4  |-  ( ph  ->  ~P X  e.  Fin )
28 ssfi 7051 . . . 4  |-  ( ( ~P X  e.  Fin  /\  ( X /.  .~  )  C_  ~P X )  ->  ( X /.  .~  )  e.  Fin )
2927, 16, 28syl2anc 645 . . 3  |-  ( ph  ->  ( X /.  .~  )  e.  Fin )
30 hashcl 11316 . . . . 5  |-  ( Y  e.  Fin  ->  ( # `
 Y )  e. 
NN0 )
3113, 30syl 17 . . . 4  |-  ( ph  ->  ( # `  Y
)  e.  NN0 )
3231nn0cnd 9987 . . 3  |-  ( ph  ->  ( # `  Y
)  e.  CC )
33 fsumconst 12217 . . 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 2296 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 3127   ~Pcpw 3599   class class class wbr 3997   ` cfv 4673  (class class class)co 5792    Er wer 6625   /.cqs 6627    ~~ cen 6828   Fincfn 6831   CCcc 8703    x. cmul 8710   NN0cn0 9932   #chash 11303   sum_csu 12123   Basecbs 13110  SubGrpcsubg 14577   ~QG cqg 14579
This theorem is referenced by:  lagsubg  14641  orbsta2  14730  sylow2blem3  14895  sylow3lem3  14902  sylow3lem4  14903
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-disj 3968  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-ec 6630  df-qs 6634  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9933  df-z 9992  df-uz 10198  df-rp 10322  df-fz 10749  df-fzo 10837  df-seq 11013  df-exp 11071  df-hash 11304  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-clim 11927  df-sum 12124  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-0g 13366  df-mnd 14329  df-grp 14451  df-minusg 14452  df-subg 14580  df-eqg 14582
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