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Theorem sylow3lem4 16446
Description: Lemma for sylow3 16449, 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 16445 . 2  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  =  ( # `  ( X /. ( G ~QG  N ) ) ) )
12 slwsubg 16426 . . . . . . . . . 10  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
138, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
14 eqid 2467 . . . . . . . . . . 11  |-  ( Gs  N )  =  ( Gs  N )
1510, 1, 5, 14nmznsg 16040 . . . . . . . . . 10  |-  ( K  e.  (SubGrp `  G
)  ->  K  e.  (NrmSGrp `  ( Gs  N ) ) )
16 nsgsubg 16028 . . . . . . . . . 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 16037 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
202, 19syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  e.  (SubGrp `  G ) )
2114subgbas 16000 . . . . . . . . . 10  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  ( Gs  N
) ) )
2220, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  N  =  ( Base `  ( Gs  N ) ) )
231subgss 15997 . . . . . . . . . . 11  |-  ( N  e.  (SubGrp `  G
)  ->  N  C_  X
)
2420, 23syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  C_  X )
25 ssfi 7737 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  N  C_  X )  ->  N  e.  Fin )
263, 24, 25syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  N  e.  Fin )
2722, 26eqeltrrd 2556 . . . . . . . 8  |-  ( ph  ->  ( Base `  ( Gs  N ) )  e. 
Fin )
28 eqid 2467 . . . . . . . . 9  |-  ( Base `  ( Gs  N ) )  =  ( Base `  ( Gs  N ) )
2928lagsubg 16058 . . . . . . . 8  |-  ( ( K  e.  (SubGrp `  ( Gs  N ) )  /\  ( Base `  ( Gs  N
) )  e.  Fin )  ->  ( # `  K
)  ||  ( # `  ( Base `  ( Gs  N ) ) ) )
3018, 27, 29syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  ||  ( # `  ( Base `  ( Gs  N ) ) ) )
3122fveq2d 5868 . . . . . . 7  |-  ( ph  ->  ( # `  N
)  =  ( # `  ( Base `  ( Gs  N ) ) ) )
3230, 31breqtrrd 4473 . . . . . 6  |-  ( ph  ->  ( # `  K
)  ||  ( # `  N
) )
33 eqid 2467 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
3433subg0cl 16004 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  K
)
3513, 34syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  G
)  e.  K )
36 ne0i 3791 . . . . . . . . . 10  |-  ( ( 0g `  G )  e.  K  ->  K  =/=  (/) )
3735, 36syl 16 . . . . . . . . 9  |-  ( ph  ->  K  =/=  (/) )
381subgss 15997 . . . . . . . . . . . 12  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
3913, 38syl 16 . . . . . . . . . . 11  |-  ( ph  ->  K  C_  X )
40 ssfi 7737 . . . . . . . . . . 11  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
413, 39, 40syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Fin )
42 hashnncl 12400 . . . . . . . . . 10  |-  ( K  e.  Fin  ->  (
( # `  K )  e.  NN  <->  K  =/=  (/) ) )
4341, 42syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  K
)  e.  NN  <->  K  =/=  (/) ) )
4437, 43mpbird 232 . . . . . . . 8  |-  ( ph  ->  ( # `  K
)  e.  NN )
4544nnzd 10961 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
46 hashcl 12392 . . . . . . . . 9  |-  ( N  e.  Fin  ->  ( # `
 N )  e. 
NN0 )
4726, 46syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  N
)  e.  NN0 )
4847nn0zd 10960 . . . . . . 7  |-  ( ph  ->  ( # `  N
)  e.  ZZ )
49 pwfi 7811 . . . . . . . . . . 11  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
503, 49sylib 196 . . . . . . . . . 10  |-  ( ph  ->  ~P X  e.  Fin )
51 eqid 2467 . . . . . . . . . . . . 13  |-  ( G ~QG  N )  =  ( G ~QG  N )
521, 51eqger 16046 . . . . . . . . . . . 12  |-  ( N  e.  (SubGrp `  G
)  ->  ( G ~QG  N
)  Er  X )
5320, 52syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( G ~QG  N )  Er  X
)
5453qsss 7369 . . . . . . . . . 10  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  C_  ~P X )
55 ssfi 7737 . . . . . . . . . 10  |-  ( ( ~P X  e.  Fin  /\  ( X /. ( G ~QG  N ) )  C_  ~P X )  ->  ( X /. ( G ~QG  N ) )  e.  Fin )
5650, 54, 55syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  e. 
Fin )
57 hashcl 12392 . . . . . . . . 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 10960 . . . . . . 7  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  ZZ )
60 dvdscmul 13867 . . . . . . 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 1228 . . . . . 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 12392 . . . . . . . . 9  |-  ( X  e.  Fin  ->  ( # `
 X )  e. 
NN0 )
643, 63syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  X
)  e.  NN0 )
6564nn0cnd 10850 . . . . . . 7  |-  ( ph  ->  ( # `  X
)  e.  CC )
6644nncnd 10548 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  e.  CC )
6744nnne0d 10576 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  =/=  0 )
6865, 66, 67divcan1d 10317 . . . . . 6  |-  ( ph  ->  ( ( ( # `  X )  /  ( # `
 K ) )  x.  ( # `  K
) )  =  (
# `  X )
)
691, 51, 20, 3lagsubg2 16057 . . . . . 6  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  N
) ) )
7068, 69eqtrd 2508 . . . . 5  |-  ( ph  ->  ( ( ( # `  X )  /  ( # `
 K ) )  x.  ( # `  K
) )  =  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) ) )
7162, 70breqtrrd 4473 . . . 4  |-  ( ph  ->  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 K ) ) 
||  ( ( (
# `  X )  /  ( # `  K
) )  x.  ( # `
 K ) ) )
721lagsubg 16058 . . . . . . 7  |-  ( ( K  e.  (SubGrp `  G )  /\  X  e.  Fin )  ->  ( # `
 K )  ||  ( # `  X ) )
7313, 3, 72syl2anc 661 . . . . . 6  |-  ( ph  ->  ( # `  K
)  ||  ( # `  X
) )
7464nn0zd 10960 . . . . . . 7  |-  ( ph  ->  ( # `  X
)  e.  ZZ )
75 dvdsval2 13846 . . . . . . 7  |-  ( ( ( # `  K
)  e.  ZZ  /\  ( # `  K )  =/=  0  /\  ( # `
 X )  e.  ZZ )  ->  (
( # `  K ) 
||  ( # `  X
)  <->  ( ( # `  X )  /  ( # `
 K ) )  e.  ZZ ) )
7645, 67, 74, 75syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  ||  ( # `  X
)  <->  ( ( # `  X )  /  ( # `
 K ) )  e.  ZZ ) )
7773, 76mpbid 210 . . . . 5  |-  ( ph  ->  ( ( # `  X
)  /  ( # `  K ) )  e.  ZZ )
78 dvdsmulcr 13870 . . . . 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 1232 . . . 4  |-  ( ph  ->  ( ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  K
) )  ||  (
( ( # `  X
)  /  ( # `  K ) )  x.  ( # `  K
) )  <->  ( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( # `  K
) ) ) )
8071, 79mpbid 210 . . 3  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( # `  K
) ) )
811, 3, 8slwhash 16440 . . . 4  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
8281oveq2d 6298 . . 3  |-  ( ph  ->  ( ( # `  X
)  /  ( # `  K ) )  =  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
8380, 82breqtrd 4471 . 2  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) ) )
8411, 83eqbrtrd 4467 1  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505   ran crn 5000   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284    Er wer 7305   /.cqs 7307   Fincfn 7513   0cc0 9488    x. cmul 9493    / cdiv 10202   NNcn 10532   NN0cn0 10791   ZZcz 10860   ^cexp 12130   #chash 12369    || cdivides 13843   Primecprime 14072    pCnt cpc 14215   Basecbs 14486   ↾s cress 14487   +g cplusg 14551   0gc0g 14691   Grpcgrp 15723   -gcsg 15726  SubGrpcsubg 15990  NrmSGrpcnsg 15991   ~QG cqg 15992   pSyl cslw 16348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-ec 7310  df-qs 7314  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-dvds 13844  df-gcd 14000  df-prm 14073  df-pc 14216  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-0g 14693  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-nsg 15994  df-eqg 15995  df-ghm 16060  df-ga 16123  df-od 16349  df-pgp 16351  df-slw 16352
This theorem is referenced by:  sylow3  16449
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