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Theorem sylow3lem4 16134
Description: Lemma for sylow3 16137, 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 16133 . 2  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  =  ( # `  ( X /. ( G ~QG  N ) ) ) )
12 slwsubg 16114 . . . . . . . . . 10  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
138, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
14 eqid 2443 . . . . . . . . . . 11  |-  ( Gs  N )  =  ( Gs  N )
1510, 1, 5, 14nmznsg 15730 . . . . . . . . . 10  |-  ( K  e.  (SubGrp `  G
)  ->  K  e.  (NrmSGrp `  ( Gs  N ) ) )
16 nsgsubg 15718 . . . . . . . . . 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 15727 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
202, 19syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  e.  (SubGrp `  G ) )
2114subgbas 15690 . . . . . . . . . 10  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  ( Gs  N
) ) )
2220, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  N  =  ( Base `  ( Gs  N ) ) )
231subgss 15687 . . . . . . . . . . 11  |-  ( N  e.  (SubGrp `  G
)  ->  N  C_  X
)
2420, 23syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  C_  X )
25 ssfi 7538 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  N  C_  X )  ->  N  e.  Fin )
263, 24, 25syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  N  e.  Fin )
2722, 26eqeltrrd 2518 . . . . . . . 8  |-  ( ph  ->  ( Base `  ( Gs  N ) )  e. 
Fin )
28 eqid 2443 . . . . . . . . 9  |-  ( Base `  ( Gs  N ) )  =  ( Base `  ( Gs  N ) )
2928lagsubg 15748 . . . . . . . 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 5700 . . . . . . 7  |-  ( ph  ->  ( # `  N
)  =  ( # `  ( Base `  ( Gs  N ) ) ) )
3230, 31breqtrrd 4323 . . . . . 6  |-  ( ph  ->  ( # `  K
)  ||  ( # `  N
) )
33 eqid 2443 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
3433subg0cl 15694 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  K
)
3513, 34syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  G
)  e.  K )
36 ne0i 3648 . . . . . . . . . 10  |-  ( ( 0g `  G )  e.  K  ->  K  =/=  (/) )
3735, 36syl 16 . . . . . . . . 9  |-  ( ph  ->  K  =/=  (/) )
381subgss 15687 . . . . . . . . . . . 12  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
3913, 38syl 16 . . . . . . . . . . 11  |-  ( ph  ->  K  C_  X )
40 ssfi 7538 . . . . . . . . . . 11  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
413, 39, 40syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Fin )
42 hashnncl 12139 . . . . . . . . . 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 10751 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
46 hashcl 12131 . . . . . . . . 9  |-  ( N  e.  Fin  ->  ( # `
 N )  e. 
NN0 )
4726, 46syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  N
)  e.  NN0 )
4847nn0zd 10750 . . . . . . 7  |-  ( ph  ->  ( # `  N
)  e.  ZZ )
49 pwfi 7611 . . . . . . . . . . 11  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
503, 49sylib 196 . . . . . . . . . 10  |-  ( ph  ->  ~P X  e.  Fin )
51 eqid 2443 . . . . . . . . . . . . 13  |-  ( G ~QG  N )  =  ( G ~QG  N )
521, 51eqger 15736 . . . . . . . . . . . 12  |-  ( N  e.  (SubGrp `  G
)  ->  ( G ~QG  N
)  Er  X )
5320, 52syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( G ~QG  N )  Er  X
)
5453qsss 7166 . . . . . . . . . 10  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  C_  ~P X )
55 ssfi 7538 . . . . . . . . . 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 12131 . . . . . . . . 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 10750 . . . . . . 7  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  ZZ )
60 dvdscmul 13564 . . . . . . 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 1218 . . . . . 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 12131 . . . . . . . . 9  |-  ( X  e.  Fin  ->  ( # `
 X )  e. 
NN0 )
643, 63syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  X
)  e.  NN0 )
6564nn0cnd 10643 . . . . . . 7  |-  ( ph  ->  ( # `  X
)  e.  CC )
6644nncnd 10343 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  e.  CC )
6744nnne0d 10371 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  =/=  0 )
6865, 66, 67divcan1d 10113 . . . . . 6  |-  ( ph  ->  ( ( ( # `  X )  /  ( # `
 K ) )  x.  ( # `  K
) )  =  (
# `  X )
)
691, 51, 20, 3lagsubg2 15747 . . . . . 6  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  N
) ) )
7068, 69eqtrd 2475 . . . . 5  |-  ( ph  ->  ( ( ( # `  X )  /  ( # `
 K ) )  x.  ( # `  K
) )  =  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) ) )
7162, 70breqtrrd 4323 . . . 4  |-  ( ph  ->  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 K ) ) 
||  ( ( (
# `  X )  /  ( # `  K
) )  x.  ( # `
 K ) ) )
721lagsubg 15748 . . . . . . 7  |-  ( ( K  e.  (SubGrp `  G )  /\  X  e.  Fin )  ->  ( # `
 K )  ||  ( # `  X ) )
7313, 3, 72syl2anc 661 . . . . . 6  |-  ( ph  ->  ( # `  K
)  ||  ( # `  X
) )
7464nn0zd 10750 . . . . . . 7  |-  ( ph  ->  ( # `  X
)  e.  ZZ )
75 dvdsval2 13543 . . . . . . 7  |-  ( ( ( # `  K
)  e.  ZZ  /\  ( # `  K )  =/=  0  /\  ( # `
 X )  e.  ZZ )  ->  (
( # `  K ) 
||  ( # `  X
)  <->  ( ( # `  X )  /  ( # `
 K ) )  e.  ZZ ) )
7645, 67, 74, 75syl3anc 1218 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  ||  ( # `  X
)  <->  ( ( # `  X )  /  ( # `
 K ) )  e.  ZZ ) )
7773, 76mpbid 210 . . . . 5  |-  ( ph  ->  ( ( # `  X
)  /  ( # `  K ) )  e.  ZZ )
78 dvdsmulcr 13567 . . . . 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 1222 . . . 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 16128 . . . 4  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
8281oveq2d 6112 . . 3  |-  ( ph  ->  ( ( # `  X
)  /  ( # `  K ) )  =  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
8380, 82breqtrd 4321 . 2  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) ) )
8411, 83eqbrtrd 4317 1  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   {crab 2724    C_ wss 3333   (/)c0 3642   ~Pcpw 3865   class class class wbr 4297    e. cmpt 4355   ran crn 4846   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098    Er wer 7103   /.cqs 7105   Fincfn 7315   0cc0 9287    x. cmul 9292    / cdiv 9998   NNcn 10327   NN0cn0 10584   ZZcz 10651   ^cexp 11870   #chash 12108    || cdivides 13540   Primecprime 13768    pCnt cpc 13908   Basecbs 14179   ↾s cress 14180   +g cplusg 14243   0gc0g 14383   Grpcgrp 15415   -gcsg 15418  SubGrpcsubg 15680  NrmSGrpcnsg 15681   ~QG cqg 15682   pSyl cslw 16036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-disj 4268  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-omul 6930  df-er 7106  df-ec 7108  df-qs 7112  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-acn 8117  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-bc 12084  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-dvds 13541  df-gcd 13696  df-prm 13769  df-pc 13909  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-subg 15683  df-nsg 15684  df-eqg 15685  df-ghm 15750  df-ga 15813  df-od 16037  df-pgp 16039  df-slw 16040
This theorem is referenced by:  sylow3  16137
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