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Theorem sylow3lem4 16866
Description: Lemma for sylow3 16869, 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 16865 . 2  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  =  ( # `  ( X /. ( G ~QG  N ) ) ) )
12 slwsubg 16846 . . . . . . . . . 10  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
138, 12syl 17 . . . . . . . . 9  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
14 eqid 2402 . . . . . . . . . . 11  |-  ( Gs  N )  =  ( Gs  N )
1510, 1, 5, 14nmznsg 16461 . . . . . . . . . 10  |-  ( K  e.  (SubGrp `  G
)  ->  K  e.  (NrmSGrp `  ( Gs  N ) ) )
16 nsgsubg 16449 . . . . . . . . . 10  |-  ( K  e.  (NrmSGrp `  ( Gs  N ) )  ->  K  e.  (SubGrp `  ( Gs  N ) ) )
1715, 16syl 17 . . . . . . . . 9  |-  ( K  e.  (SubGrp `  G
)  ->  K  e.  (SubGrp `  ( Gs  N ) ) )
1813, 17syl 17 . . . . . . . 8  |-  ( ph  ->  K  e.  (SubGrp `  ( Gs  N ) ) )
1910, 1, 5nmzsubg 16458 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
202, 19syl 17 . . . . . . . . . 10  |-  ( ph  ->  N  e.  (SubGrp `  G ) )
2114subgbas 16421 . . . . . . . . . 10  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  ( Gs  N
) ) )
2220, 21syl 17 . . . . . . . . 9  |-  ( ph  ->  N  =  ( Base `  ( Gs  N ) ) )
231subgss 16418 . . . . . . . . . . 11  |-  ( N  e.  (SubGrp `  G
)  ->  N  C_  X
)
2420, 23syl 17 . . . . . . . . . 10  |-  ( ph  ->  N  C_  X )
25 ssfi 7695 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  N  C_  X )  ->  N  e.  Fin )
263, 24, 25syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  N  e.  Fin )
2722, 26eqeltrrd 2491 . . . . . . . 8  |-  ( ph  ->  ( Base `  ( Gs  N ) )  e. 
Fin )
28 eqid 2402 . . . . . . . . 9  |-  ( Base `  ( Gs  N ) )  =  ( Base `  ( Gs  N ) )
2928lagsubg 16479 . . . . . . . 8  |-  ( ( K  e.  (SubGrp `  ( Gs  N ) )  /\  ( Base `  ( Gs  N
) )  e.  Fin )  ->  ( # `  K
)  ||  ( # `  ( Base `  ( Gs  N ) ) ) )
3018, 27, 29syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  ||  ( # `  ( Base `  ( Gs  N ) ) ) )
3122fveq2d 5809 . . . . . . 7  |-  ( ph  ->  ( # `  N
)  =  ( # `  ( Base `  ( Gs  N ) ) ) )
3230, 31breqtrrd 4420 . . . . . 6  |-  ( ph  ->  ( # `  K
)  ||  ( # `  N
) )
33 eqid 2402 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
3433subg0cl 16425 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  K
)
3513, 34syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  G
)  e.  K )
36 ne0i 3743 . . . . . . . . . 10  |-  ( ( 0g `  G )  e.  K  ->  K  =/=  (/) )
3735, 36syl 17 . . . . . . . . 9  |-  ( ph  ->  K  =/=  (/) )
381subgss 16418 . . . . . . . . . . . 12  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
3913, 38syl 17 . . . . . . . . . . 11  |-  ( ph  ->  K  C_  X )
40 ssfi 7695 . . . . . . . . . . 11  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
413, 39, 40syl2anc 659 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Fin )
42 hashnncl 12391 . . . . . . . . . 10  |-  ( K  e.  Fin  ->  (
( # `  K )  e.  NN  <->  K  =/=  (/) ) )
4341, 42syl 17 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  K
)  e.  NN  <->  K  =/=  (/) ) )
4437, 43mpbird 232 . . . . . . . 8  |-  ( ph  ->  ( # `  K
)  e.  NN )
4544nnzd 10927 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
46 hashcl 12382 . . . . . . . . 9  |-  ( N  e.  Fin  ->  ( # `
 N )  e. 
NN0 )
4726, 46syl 17 . . . . . . . 8  |-  ( ph  ->  ( # `  N
)  e.  NN0 )
4847nn0zd 10926 . . . . . . 7  |-  ( ph  ->  ( # `  N
)  e.  ZZ )
49 pwfi 7769 . . . . . . . . . . 11  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
503, 49sylib 196 . . . . . . . . . 10  |-  ( ph  ->  ~P X  e.  Fin )
51 eqid 2402 . . . . . . . . . . . . 13  |-  ( G ~QG  N )  =  ( G ~QG  N )
521, 51eqger 16467 . . . . . . . . . . . 12  |-  ( N  e.  (SubGrp `  G
)  ->  ( G ~QG  N
)  Er  X )
5320, 52syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( G ~QG  N )  Er  X
)
5453qsss 7329 . . . . . . . . . 10  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  C_  ~P X )
55 ssfi 7695 . . . . . . . . . 10  |-  ( ( ~P X  e.  Fin  /\  ( X /. ( G ~QG  N ) )  C_  ~P X )  ->  ( X /. ( G ~QG  N ) )  e.  Fin )
5650, 54, 55syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  e. 
Fin )
57 hashcl 12382 . . . . . . . . 9  |-  ( ( X /. ( G ~QG  N ) )  e.  Fin  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  NN0 )
5856, 57syl 17 . . . . . . . 8  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  NN0 )
5958nn0zd 10926 . . . . . . 7  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  ZZ )
60 dvdscmul 14111 . . . . . . 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 1230 . . . . . 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 12382 . . . . . . . . 9  |-  ( X  e.  Fin  ->  ( # `
 X )  e. 
NN0 )
643, 63syl 17 . . . . . . . 8  |-  ( ph  ->  ( # `  X
)  e.  NN0 )
6564nn0cnd 10815 . . . . . . 7  |-  ( ph  ->  ( # `  X
)  e.  CC )
6644nncnd 10512 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  e.  CC )
6744nnne0d 10541 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  =/=  0 )
6865, 66, 67divcan1d 10282 . . . . . 6  |-  ( ph  ->  ( ( ( # `  X )  /  ( # `
 K ) )  x.  ( # `  K
) )  =  (
# `  X )
)
691, 51, 20, 3lagsubg2 16478 . . . . . 6  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  N
) ) )
7068, 69eqtrd 2443 . . . . 5  |-  ( ph  ->  ( ( ( # `  X )  /  ( # `
 K ) )  x.  ( # `  K
) )  =  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) ) )
7162, 70breqtrrd 4420 . . . 4  |-  ( ph  ->  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 K ) ) 
||  ( ( (
# `  X )  /  ( # `  K
) )  x.  ( # `
 K ) ) )
721lagsubg 16479 . . . . . . 7  |-  ( ( K  e.  (SubGrp `  G )  /\  X  e.  Fin )  ->  ( # `
 K )  ||  ( # `  X ) )
7313, 3, 72syl2anc 659 . . . . . 6  |-  ( ph  ->  ( # `  K
)  ||  ( # `  X
) )
7464nn0zd 10926 . . . . . . 7  |-  ( ph  ->  ( # `  X
)  e.  ZZ )
75 dvdsval2 14090 . . . . . . 7  |-  ( ( ( # `  K
)  e.  ZZ  /\  ( # `  K )  =/=  0  /\  ( # `
 X )  e.  ZZ )  ->  (
( # `  K ) 
||  ( # `  X
)  <->  ( ( # `  X )  /  ( # `
 K ) )  e.  ZZ ) )
7645, 67, 74, 75syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  ||  ( # `  X
)  <->  ( ( # `  X )  /  ( # `
 K ) )  e.  ZZ ) )
7773, 76mpbid 210 . . . . 5  |-  ( ph  ->  ( ( # `  X
)  /  ( # `  K ) )  e.  ZZ )
78 dvdsmulcr 14114 . . . . 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 1234 . . . 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 16860 . . . 4  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
8281oveq2d 6250 . . 3  |-  ( ph  ->  ( ( # `  X
)  /  ( # `  K ) )  =  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
8380, 82breqtrd 4418 . 2  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) ) )
8411, 83eqbrtrd 4414 1  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   {crab 2757    C_ wss 3413   (/)c0 3737   ~Pcpw 3954   class class class wbr 4394    |-> cmpt 4452   ran crn 4943   ` cfv 5525  (class class class)co 6234    |-> cmpt2 6236    Er wer 7265   /.cqs 7267   Fincfn 7474   0cc0 9442    x. cmul 9447    / cdiv 10167   NNcn 10496   NN0cn0 10756   ZZcz 10825   ^cexp 12120   #chash 12359    || cdvds 14087   Primecprime 14318    pCnt cpc 14461   Basecbs 14733   ↾s cress 14734   +g cplusg 14801   0gc0g 14946   Grpcgrp 16269   -gcsg 16271  SubGrpcsubg 16411  NrmSGrpcnsg 16412   ~QG cqg 16413   pSyl cslw 16768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-disj 4366  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-2o 7088  df-oadd 7091  df-omul 7092  df-er 7268  df-ec 7270  df-qs 7274  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-oi 7889  df-card 8272  df-acn 8275  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-q 11146  df-rp 11184  df-fz 11644  df-fzo 11768  df-fl 11879  df-mod 11948  df-seq 12062  df-exp 12121  df-fac 12308  df-bc 12335  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-clim 13367  df-sum 13565  df-dvds 14088  df-gcd 14246  df-prm 14319  df-pc 14462  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-ress 14740  df-plusg 14814  df-0g 14948  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-submnd 16183  df-grp 16273  df-minusg 16274  df-sbg 16275  df-mulg 16276  df-subg 16414  df-nsg 16415  df-eqg 16416  df-ghm 16481  df-ga 16544  df-od 16769  df-pgp 16771  df-slw 16772
This theorem is referenced by:  sylow3  16869
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