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Theorem sylow3lem4 15264
Description: Lemma for sylow3 15267, 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 15263 . 2  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  =  ( # `  ( X /. ( G ~QG  N ) ) ) )
12 slwsubg 15244 . . . . . . . . . 10  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
138, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
14 eqid 2436 . . . . . . . . . . 11  |-  ( Gs  N )  =  ( Gs  N )
1510, 1, 5, 14nmznsg 14984 . . . . . . . . . 10  |-  ( K  e.  (SubGrp `  G
)  ->  K  e.  (NrmSGrp `  ( Gs  N ) ) )
16 nsgsubg 14972 . . . . . . . . . 10  |-  ( K  e.  (NrmSGrp `  ( Gs  N ) )  ->  K  e.  (SubGrp `  ( Gs  N ) ) )
1715, 16syl 16 . . . . . . . . 9  |-  ( K  e.  (SubGrp `  G
)  ->  K  e.  (SubGrp `  ( Gs  N ) ) )
1813, 17syl 16 . . . . . . . 8  |-  ( ph  ->  K  e.  (SubGrp `  ( Gs  N ) ) )
1910, 1, 5nmzsubg 14981 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
202, 19syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  e.  (SubGrp `  G ) )
2114subgbas 14948 . . . . . . . . . 10  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  ( Gs  N
) ) )
2220, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  N  =  ( Base `  ( Gs  N ) ) )
231subgss 14945 . . . . . . . . . . 11  |-  ( N  e.  (SubGrp `  G
)  ->  N  C_  X
)
2420, 23syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  C_  X )
25 ssfi 7329 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  N  C_  X )  ->  N  e.  Fin )
263, 24, 25syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  N  e.  Fin )
2722, 26eqeltrrd 2511 . . . . . . . 8  |-  ( ph  ->  ( Base `  ( Gs  N ) )  e. 
Fin )
28 eqid 2436 . . . . . . . . 9  |-  ( Base `  ( Gs  N ) )  =  ( Base `  ( Gs  N ) )
2928lagsubg 15002 . . . . . . . 8  |-  ( ( K  e.  (SubGrp `  ( Gs  N ) )  /\  ( Base `  ( Gs  N
) )  e.  Fin )  ->  ( # `  K
)  ||  ( # `  ( Base `  ( Gs  N ) ) ) )
3018, 27, 29syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  ||  ( # `  ( Base `  ( Gs  N ) ) ) )
3122fveq2d 5732 . . . . . . 7  |-  ( ph  ->  ( # `  N
)  =  ( # `  ( Base `  ( Gs  N ) ) ) )
3230, 31breqtrrd 4238 . . . . . 6  |-  ( ph  ->  ( # `  K
)  ||  ( # `  N
) )
33 eqid 2436 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
3433subg0cl 14952 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  K
)
3513, 34syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  G
)  e.  K )
36 ne0i 3634 . . . . . . . . . 10  |-  ( ( 0g `  G )  e.  K  ->  K  =/=  (/) )
3735, 36syl 16 . . . . . . . . 9  |-  ( ph  ->  K  =/=  (/) )
381subgss 14945 . . . . . . . . . . . 12  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
3913, 38syl 16 . . . . . . . . . . 11  |-  ( ph  ->  K  C_  X )
40 ssfi 7329 . . . . . . . . . . 11  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
413, 39, 40syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Fin )
42 hashnncl 11645 . . . . . . . . . 10  |-  ( K  e.  Fin  ->  (
( # `  K )  e.  NN  <->  K  =/=  (/) ) )
4341, 42syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  K
)  e.  NN  <->  K  =/=  (/) ) )
4437, 43mpbird 224 . . . . . . . 8  |-  ( ph  ->  ( # `  K
)  e.  NN )
4544nnzd 10374 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
46 hashcl 11639 . . . . . . . . 9  |-  ( N  e.  Fin  ->  ( # `
 N )  e. 
NN0 )
4726, 46syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  N
)  e.  NN0 )
4847nn0zd 10373 . . . . . . 7  |-  ( ph  ->  ( # `  N
)  e.  ZZ )
49 pwfi 7402 . . . . . . . . . . 11  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
503, 49sylib 189 . . . . . . . . . 10  |-  ( ph  ->  ~P X  e.  Fin )
51 eqid 2436 . . . . . . . . . . . . 13  |-  ( G ~QG  N )  =  ( G ~QG  N )
521, 51eqger 14990 . . . . . . . . . . . 12  |-  ( N  e.  (SubGrp `  G
)  ->  ( G ~QG  N
)  Er  X )
5320, 52syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( G ~QG  N )  Er  X
)
5453qsss 6965 . . . . . . . . . 10  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  C_  ~P X )
55 ssfi 7329 . . . . . . . . . 10  |-  ( ( ~P X  e.  Fin  /\  ( X /. ( G ~QG  N ) )  C_  ~P X )  ->  ( X /. ( G ~QG  N ) )  e.  Fin )
5650, 54, 55syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  e. 
Fin )
57 hashcl 11639 . . . . . . . . 9  |-  ( ( X /. ( G ~QG  N ) )  e.  Fin  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  NN0 )
5856, 57syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  NN0 )
5958nn0zd 10373 . . . . . . 7  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  ZZ )
60 dvdscmul 12876 . . . . . . 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 1184 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  ||  ( # `  N
)  ->  ( ( # `
 ( X /. ( G ~QG  N ) ) )  x.  ( # `  K
) )  ||  (
( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) ) ) )
6232, 61mpd 15 . . . . 5  |-  ( ph  ->  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 K ) ) 
||  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  N
) ) )
63 hashcl 11639 . . . . . . . . 9  |-  ( X  e.  Fin  ->  ( # `
 X )  e. 
NN0 )
643, 63syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  X
)  e.  NN0 )
6564nn0cnd 10276 . . . . . . 7  |-  ( ph  ->  ( # `  X
)  e.  CC )
6644nncnd 10016 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  e.  CC )
6744nnne0d 10044 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  =/=  0 )
6865, 66, 67divcan1d 9791 . . . . . 6  |-  ( ph  ->  ( ( ( # `  X )  /  ( # `
 K ) )  x.  ( # `  K
) )  =  (
# `  X )
)
691, 51, 20, 3lagsubg2 15001 . . . . . 6  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  N
) ) )
7068, 69eqtrd 2468 . . . . 5  |-  ( ph  ->  ( ( ( # `  X )  /  ( # `
 K ) )  x.  ( # `  K
) )  =  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) ) )
7162, 70breqtrrd 4238 . . . 4  |-  ( ph  ->  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 K ) ) 
||  ( ( (
# `  X )  /  ( # `  K
) )  x.  ( # `
 K ) ) )
721lagsubg 15002 . . . . . . 7  |-  ( ( K  e.  (SubGrp `  G )  /\  X  e.  Fin )  ->  ( # `
 K )  ||  ( # `  X ) )
7313, 3, 72syl2anc 643 . . . . . 6  |-  ( ph  ->  ( # `  K
)  ||  ( # `  X
) )
7464nn0zd 10373 . . . . . . 7  |-  ( ph  ->  ( # `  X
)  e.  ZZ )
75 dvdsval2 12855 . . . . . . 7  |-  ( ( ( # `  K
)  e.  ZZ  /\  ( # `  K )  =/=  0  /\  ( # `
 X )  e.  ZZ )  ->  (
( # `  K ) 
||  ( # `  X
)  <->  ( ( # `  X )  /  ( # `
 K ) )  e.  ZZ ) )
7645, 67, 74, 75syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  ||  ( # `  X
)  <->  ( ( # `  X )  /  ( # `
 K ) )  e.  ZZ ) )
7773, 76mpbid 202 . . . . 5  |-  ( ph  ->  ( ( # `  X
)  /  ( # `  K ) )  e.  ZZ )
78 dvdsmulcr 12879 . . . . 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 1188 . . . 4  |-  ( ph  ->  ( ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  K
) )  ||  (
( ( # `  X
)  /  ( # `  K ) )  x.  ( # `  K
) )  <->  ( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( # `  K
) ) ) )
8071, 79mpbid 202 . . 3  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( # `  K
) ) )
811, 3, 8slwhash 15258 . . . 4  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
8281oveq2d 6097 . . 3  |-  ( ph  ->  ( ( # `  X
)  /  ( # `  K ) )  =  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
8380, 82breqtrd 4236 . 2  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) ) )
8411, 83eqbrtrd 4232 1  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   {crab 2709    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   class class class wbr 4212    e. cmpt 4266   ran crn 4879   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083    Er wer 6902   /.cqs 6904   Fincfn 7109   0cc0 8990    x. cmul 8995    / cdiv 9677   NNcn 10000   NN0cn0 10221   ZZcz 10282   ^cexp 11382   #chash 11618    || cdivides 12852   Primecprime 13079    pCnt cpc 13210   Basecbs 13469   ↾s cress 13470   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685   -gcsg 14688  SubGrpcsubg 14938  NrmSGrpcnsg 14939   ~QG cqg 14940   pSyl cslw 15166
This theorem is referenced by:  sylow3  15267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-disj 4183  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-acn 7829  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-nsg 14942  df-eqg 14943  df-ghm 15004  df-ga 15067  df-od 15167  df-pgp 15169  df-slw 15170
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