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Theorem sylow3lem4 14941
Description: Lemma for sylow3 14944, first part. The number of Sylow subgroups is a divisor of the size of  G reduced by the size of a Sylow subgroup of  G. (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypotheses
Ref Expression
sylow3.x  |-  X  =  ( Base `  G
)
sylow3.g  |-  ( ph  ->  G  e.  Grp )
sylow3.xf  |-  ( ph  ->  X  e.  Fin )
sylow3.p  |-  ( ph  ->  P  e.  Prime )
sylow3lem1.a  |-  .+  =  ( +g  `  G )
sylow3lem1.d  |-  .-  =  ( -g `  G )
sylow3lem1.m  |-  .(+)  =  ( x  e.  X , 
y  e.  ( P pSyl 
G )  |->  ran  (
z  e.  y  |->  ( ( x  .+  z
)  .-  x )
) )
sylow3lem2.k  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
sylow3lem2.h  |-  H  =  { u  e.  X  |  ( u  .(+)  K )  =  K }
sylow3lem2.n  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  K  <->  ( y  .+  x )  e.  K ) }
Assertion
Ref Expression
sylow3lem4  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
Distinct variable groups:    x, u, y, z,  .-    u,  .(+) , x, y, z    x, H, y    u, K, x, y, z    u, N, z    u, X, x, y, z    u, G, x, y, z    ph, u, x, y, z    u,  .+ , x, y, z    u, P, x, y, z
Allowed substitution hints:    H( z, u)    N( x, y)

Proof of Theorem sylow3lem4
StepHypRef Expression
1 sylow3.x . . 3  |-  X  =  ( Base `  G
)
2 sylow3.g . . 3  |-  ( ph  ->  G  e.  Grp )
3 sylow3.xf . . 3  |-  ( ph  ->  X  e.  Fin )
4 sylow3.p . . 3  |-  ( ph  ->  P  e.  Prime )
5 sylow3lem1.a . . 3  |-  .+  =  ( +g  `  G )
6 sylow3lem1.d . . 3  |-  .-  =  ( -g `  G )
7 sylow3lem1.m . . 3  |-  .(+)  =  ( x  e.  X , 
y  e.  ( P pSyl 
G )  |->  ran  (
z  e.  y  |->  ( ( x  .+  z
)  .-  x )
) )
8 sylow3lem2.k . . 3  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
9 sylow3lem2.h . . 3  |-  H  =  { u  e.  X  |  ( u  .(+)  K )  =  K }
10 sylow3lem2.n . . 3  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  K  <->  ( y  .+  x )  e.  K ) }
111, 2, 3, 4, 5, 6, 7, 8, 9, 10sylow3lem3 14940 . 2  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  =  ( # `  ( X /. ( G ~QG  N ) ) ) )
12 slwsubg 14921 . . . . . . . . . 10  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
138, 12syl 15 . . . . . . . . 9  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
14 eqid 2283 . . . . . . . . . . 11  |-  ( Gs  N )  =  ( Gs  N )
1510, 1, 5, 14nmznsg 14661 . . . . . . . . . 10  |-  ( K  e.  (SubGrp `  G
)  ->  K  e.  (NrmSGrp `  ( Gs  N ) ) )
16 nsgsubg 14649 . . . . . . . . . 10  |-  ( K  e.  (NrmSGrp `  ( Gs  N ) )  ->  K  e.  (SubGrp `  ( Gs  N ) ) )
1715, 16syl 15 . . . . . . . . 9  |-  ( K  e.  (SubGrp `  G
)  ->  K  e.  (SubGrp `  ( Gs  N ) ) )
1813, 17syl 15 . . . . . . . 8  |-  ( ph  ->  K  e.  (SubGrp `  ( Gs  N ) ) )
1910, 1, 5nmzsubg 14658 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
202, 19syl 15 . . . . . . . . . 10  |-  ( ph  ->  N  e.  (SubGrp `  G ) )
2114subgbas 14625 . . . . . . . . . 10  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  ( Gs  N
) ) )
2220, 21syl 15 . . . . . . . . 9  |-  ( ph  ->  N  =  ( Base `  ( Gs  N ) ) )
231subgss 14622 . . . . . . . . . . 11  |-  ( N  e.  (SubGrp `  G
)  ->  N  C_  X
)
2420, 23syl 15 . . . . . . . . . 10  |-  ( ph  ->  N  C_  X )
25 ssfi 7083 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  N  C_  X )  ->  N  e.  Fin )
263, 24, 25syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  N  e.  Fin )
2722, 26eqeltrrd 2358 . . . . . . . 8  |-  ( ph  ->  ( Base `  ( Gs  N ) )  e. 
Fin )
28 eqid 2283 . . . . . . . . 9  |-  ( Base `  ( Gs  N ) )  =  ( Base `  ( Gs  N ) )
2928lagsubg 14679 . . . . . . . 8  |-  ( ( K  e.  (SubGrp `  ( Gs  N ) )  /\  ( Base `  ( Gs  N
) )  e.  Fin )  ->  ( # `  K
)  ||  ( # `  ( Base `  ( Gs  N ) ) ) )
3018, 27, 29syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  ||  ( # `  ( Base `  ( Gs  N ) ) ) )
3122fveq2d 5529 . . . . . . 7  |-  ( ph  ->  ( # `  N
)  =  ( # `  ( Base `  ( Gs  N ) ) ) )
3230, 31breqtrrd 4049 . . . . . 6  |-  ( ph  ->  ( # `  K
)  ||  ( # `  N
) )
33 eqid 2283 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
3433subg0cl 14629 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  K
)
3513, 34syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  G
)  e.  K )
36 ne0i 3461 . . . . . . . . . 10  |-  ( ( 0g `  G )  e.  K  ->  K  =/=  (/) )
3735, 36syl 15 . . . . . . . . 9  |-  ( ph  ->  K  =/=  (/) )
381subgss 14622 . . . . . . . . . . . 12  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
3913, 38syl 15 . . . . . . . . . . 11  |-  ( ph  ->  K  C_  X )
40 ssfi 7083 . . . . . . . . . . 11  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
413, 39, 40syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Fin )
42 hashnncl 11354 . . . . . . . . . 10  |-  ( K  e.  Fin  ->  (
( # `  K )  e.  NN  <->  K  =/=  (/) ) )
4341, 42syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  K
)  e.  NN  <->  K  =/=  (/) ) )
4437, 43mpbird 223 . . . . . . . 8  |-  ( ph  ->  ( # `  K
)  e.  NN )
4544nnzd 10116 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
46 hashcl 11350 . . . . . . . . 9  |-  ( N  e.  Fin  ->  ( # `
 N )  e. 
NN0 )
4726, 46syl 15 . . . . . . . 8  |-  ( ph  ->  ( # `  N
)  e.  NN0 )
4847nn0zd 10115 . . . . . . 7  |-  ( ph  ->  ( # `  N
)  e.  ZZ )
49 pwfi 7151 . . . . . . . . . . 11  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
503, 49sylib 188 . . . . . . . . . 10  |-  ( ph  ->  ~P X  e.  Fin )
51 eqid 2283 . . . . . . . . . . . . 13  |-  ( G ~QG  N )  =  ( G ~QG  N )
521, 51eqger 14667 . . . . . . . . . . . 12  |-  ( N  e.  (SubGrp `  G
)  ->  ( G ~QG  N
)  Er  X )
5320, 52syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( G ~QG  N )  Er  X
)
5453qsss 6720 . . . . . . . . . 10  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  C_  ~P X )
55 ssfi 7083 . . . . . . . . . 10  |-  ( ( ~P X  e.  Fin  /\  ( X /. ( G ~QG  N ) )  C_  ~P X )  ->  ( X /. ( G ~QG  N ) )  e.  Fin )
5650, 54, 55syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  e. 
Fin )
57 hashcl 11350 . . . . . . . . 9  |-  ( ( X /. ( G ~QG  N ) )  e.  Fin  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  NN0 )
5856, 57syl 15 . . . . . . . 8  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  NN0 )
5958nn0zd 10115 . . . . . . 7  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  ZZ )
60 dvdscmul 12555 . . . . . . 7  |-  ( ( ( # `  K
)  e.  ZZ  /\  ( # `  N )  e.  ZZ  /\  ( # `
 ( X /. ( G ~QG  N ) ) )  e.  ZZ )  -> 
( ( # `  K
)  ||  ( # `  N
)  ->  ( ( # `
 ( X /. ( G ~QG  N ) ) )  x.  ( # `  K
) )  ||  (
( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) ) ) )
6145, 48, 59, 60syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  ||  ( # `  N
)  ->  ( ( # `
 ( X /. ( G ~QG  N ) ) )  x.  ( # `  K
) )  ||  (
( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) ) ) )
6232, 61mpd 14 . . . . 5  |-  ( ph  ->  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 K ) ) 
||  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  N
) ) )
63 hashcl 11350 . . . . . . . . 9  |-  ( X  e.  Fin  ->  ( # `
 X )  e. 
NN0 )
643, 63syl 15 . . . . . . . 8  |-  ( ph  ->  ( # `  X
)  e.  NN0 )
6564nn0cnd 10020 . . . . . . 7  |-  ( ph  ->  ( # `  X
)  e.  CC )
6644nncnd 9762 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  e.  CC )
6744nnne0d 9790 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  =/=  0 )
6865, 66, 67divcan1d 9537 . . . . . 6  |-  ( ph  ->  ( ( ( # `  X )  /  ( # `
 K ) )  x.  ( # `  K
) )  =  (
# `  X )
)
691, 51, 20, 3lagsubg2 14678 . . . . . 6  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  N
) ) )
7068, 69eqtrd 2315 . . . . 5  |-  ( ph  ->  ( ( ( # `  X )  /  ( # `
 K ) )  x.  ( # `  K
) )  =  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) ) )
7162, 70breqtrrd 4049 . . . 4  |-  ( ph  ->  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 K ) ) 
||  ( ( (
# `  X )  /  ( # `  K
) )  x.  ( # `
 K ) ) )
721lagsubg 14679 . . . . . . 7  |-  ( ( K  e.  (SubGrp `  G )  /\  X  e.  Fin )  ->  ( # `
 K )  ||  ( # `  X ) )
7313, 3, 72syl2anc 642 . . . . . 6  |-  ( ph  ->  ( # `  K
)  ||  ( # `  X
) )
7464nn0zd 10115 . . . . . . 7  |-  ( ph  ->  ( # `  X
)  e.  ZZ )
75 dvdsval2 12534 . . . . . . 7  |-  ( ( ( # `  K
)  e.  ZZ  /\  ( # `  K )  =/=  0  /\  ( # `
 X )  e.  ZZ )  ->  (
( # `  K ) 
||  ( # `  X
)  <->  ( ( # `  X )  /  ( # `
 K ) )  e.  ZZ ) )
7645, 67, 74, 75syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  ||  ( # `  X
)  <->  ( ( # `  X )  /  ( # `
 K ) )  e.  ZZ ) )
7773, 76mpbid 201 . . . . 5  |-  ( ph  ->  ( ( # `  X
)  /  ( # `  K ) )  e.  ZZ )
78 dvdsmulcr 12558 . . . . 5  |-  ( ( ( # `  ( X /. ( G ~QG  N ) ) )  e.  ZZ  /\  ( ( # `  X
)  /  ( # `  K ) )  e.  ZZ  /\  ( (
# `  K )  e.  ZZ  /\  ( # `  K )  =/=  0
) )  ->  (
( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 K ) ) 
||  ( ( (
# `  X )  /  ( # `  K
) )  x.  ( # `
 K ) )  <-> 
( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( # `  K
) ) ) )
7959, 77, 45, 67, 78syl112anc 1186 . . . 4  |-  ( ph  ->  ( ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  K
) )  ||  (
( ( # `  X
)  /  ( # `  K ) )  x.  ( # `  K
) )  <->  ( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( # `  K
) ) ) )
8071, 79mpbid 201 . . 3  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( # `  K
) ) )
811, 3, 8slwhash 14935 . . . 4  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
8281oveq2d 5874 . . 3  |-  ( ph  ->  ( ( # `  X
)  /  ( # `  K ) )  =  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
8380, 82breqtrd 4047 . 2  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) ) )
8411, 83eqbrtrd 4043 1  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    Er wer 6657   /.cqs 6659   Fincfn 6863   0cc0 8737    x. cmul 8742    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ^cexp 11104   #chash 11337    || cdivides 12531   Primecprime 12758    pCnt cpc 12889   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   -gcsg 14365  SubGrpcsubg 14615  NrmSGrpcnsg 14616   ~QG cqg 14617   pSyl cslw 14843
This theorem is referenced by:  sylow3  14944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-ga 14744  df-od 14844  df-pgp 14846  df-slw 14847
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