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Theorem sylow3 15259
Description: Sylow's third theorem. The number of Sylow subgroups is a divisor of  |  G  |  /  d, where  d is the common order of a Sylow subgroup, and is equivalent to  1  mod  P. (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 )
sylow3.n  |-  N  =  ( # `  ( P pSyl  G ) )
Assertion
Ref Expression
sylow3  |-  ( ph  ->  ( N  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) )  /\  ( N  mod  P )  =  1 ) )

Proof of Theorem sylow3
Dummy variables  a 
b  c  u  x  y  z  s  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sylow3.g . . . 4  |-  ( ph  ->  G  e.  Grp )
2 sylow3.xf . . . 4  |-  ( ph  ->  X  e.  Fin )
3 sylow3.p . . . 4  |-  ( ph  ->  P  e.  Prime )
4 sylow3.x . . . . 5  |-  X  =  ( Base `  G
)
54slwn0 15241 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  ->  ( P pSyl  G )  =/=  (/) )
61, 2, 3, 5syl3anc 1184 . . 3  |-  ( ph  ->  ( P pSyl  G )  =/=  (/) )
7 n0 3629 . . 3  |-  ( ( P pSyl  G )  =/=  (/) 
<->  E. k  k  e.  ( P pSyl  G ) )
86, 7sylib 189 . 2  |-  ( ph  ->  E. k  k  e.  ( P pSyl  G ) )
9 sylow3.n . . . 4  |-  N  =  ( # `  ( P pSyl  G ) )
101adantr 452 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  G  e.  Grp )
112adantr 452 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  X  e.  Fin )
123adantr 452 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  P  e.  Prime )
13 eqid 2435 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
14 eqid 2435 . . . . 5  |-  ( -g `  G )  =  (
-g `  G )
15 oveq2 6081 . . . . . . . . . 10  |-  ( c  =  z  ->  (
a ( +g  `  G
) c )  =  ( a ( +g  `  G ) z ) )
1615oveq1d 6088 . . . . . . . . 9  |-  ( c  =  z  ->  (
( a ( +g  `  G ) c ) ( -g `  G
) a )  =  ( ( a ( +g  `  G ) z ) ( -g `  G ) a ) )
1716cbvmptv 4292 . . . . . . . 8  |-  ( c  e.  b  |->  ( ( a ( +g  `  G
) c ) (
-g `  G )
a ) )  =  ( z  e.  b 
|->  ( ( a ( +g  `  G ) z ) ( -g `  G ) a ) )
18 oveq1 6080 . . . . . . . . . 10  |-  ( a  =  x  ->  (
a ( +g  `  G
) z )  =  ( x ( +g  `  G ) z ) )
19 id 20 . . . . . . . . . 10  |-  ( a  =  x  ->  a  =  x )
2018, 19oveq12d 6091 . . . . . . . . 9  |-  ( a  =  x  ->  (
( a ( +g  `  G ) z ) ( -g `  G
) a )  =  ( ( x ( +g  `  G ) z ) ( -g `  G ) x ) )
2120mpteq2dv 4288 . . . . . . . 8  |-  ( a  =  x  ->  (
z  e.  b  |->  ( ( a ( +g  `  G ) z ) ( -g `  G
) a ) )  =  ( z  e.  b  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2217, 21syl5eq 2479 . . . . . . 7  |-  ( a  =  x  ->  (
c  e.  b  |->  ( ( a ( +g  `  G ) c ) ( -g `  G
) a ) )  =  ( z  e.  b  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2322rneqd 5089 . . . . . 6  |-  ( a  =  x  ->  ran  ( c  e.  b 
|->  ( ( a ( +g  `  G ) c ) ( -g `  G ) a ) )  =  ran  (
z  e.  b  |->  ( ( x ( +g  `  G ) z ) ( -g `  G
) x ) ) )
24 mpteq1 4281 . . . . . . 7  |-  ( b  =  y  ->  (
z  e.  b  |->  ( ( x ( +g  `  G ) z ) ( -g `  G
) x ) )  =  ( z  e.  y  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2524rneqd 5089 . . . . . 6  |-  ( b  =  y  ->  ran  ( z  e.  b 
|->  ( ( x ( +g  `  G ) z ) ( -g `  G ) x ) )  =  ran  (
z  e.  y  |->  ( ( x ( +g  `  G ) z ) ( -g `  G
) x ) ) )
2623, 25cbvmpt2v 6144 . . . . 5  |-  ( a  e.  X ,  b  e.  ( P pSyl  G
)  |->  ran  ( c  e.  b  |->  ( ( a ( +g  `  G
) c ) (
-g `  G )
a ) ) )  =  ( x  e.  X ,  y  e.  ( P pSyl  G ) 
|->  ran  ( z  e.  y  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
27 simpr 448 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  k  e.  ( P pSyl  G )
)
28 eqid 2435 . . . . 5  |-  { u  e.  X  |  (
u ( a  e.  X ,  b  e.  ( P pSyl  G ) 
|->  ran  ( c  e.  b  |->  ( ( a ( +g  `  G
) c ) (
-g `  G )
a ) ) ) k )  =  k }  =  { u  e.  X  |  (
u ( a  e.  X ,  b  e.  ( P pSyl  G ) 
|->  ran  ( c  e.  b  |->  ( ( a ( +g  `  G
) c ) (
-g `  G )
a ) ) ) k )  =  k }
29 eqid 2435 . . . . 5  |-  { x  e.  X  |  A. y  e.  X  (
( x ( +g  `  G ) y )  e.  k  <->  ( y
( +g  `  G ) x )  e.  k ) }  =  {
x  e.  X  |  A. y  e.  X  ( ( x ( +g  `  G ) y )  e.  k  <-> 
( y ( +g  `  G ) x )  e.  k ) }
304, 10, 11, 12, 13, 14, 26, 27, 28, 29sylow3lem4 15256 . . . 4  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( # `  ( P pSyl  G ) )  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
319, 30syl5eqbr 4237 . . 3  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  N  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) ) )
329oveq1i 6083 . . . 4  |-  ( N  mod  P )  =  ( ( # `  ( P pSyl  G ) )  mod 
P )
3323, 25cbvmpt2v 6144 . . . . 5  |-  ( a  e.  k ,  b  e.  ( P pSyl  G
)  |->  ran  ( c  e.  b  |->  ( ( a ( +g  `  G
) c ) (
-g `  G )
a ) ) )  =  ( x  e.  k ,  y  e.  ( P pSyl  G ) 
|->  ran  ( z  e.  y  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
34 eqid 2435 . . . . 5  |-  { x  e.  X  |  A. y  e.  X  (
( x ( +g  `  G ) y )  e.  s  <->  ( y
( +g  `  G ) x )  e.  s ) }  =  {
x  e.  X  |  A. y  e.  X  ( ( x ( +g  `  G ) y )  e.  s  <-> 
( y ( +g  `  G ) x )  e.  s ) }
354, 10, 11, 12, 13, 14, 27, 33, 34sylow3lem6 15258 . . . 4  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( ( # `
 ( P pSyl  G
) )  mod  P
)  =  1 )
3632, 35syl5eq 2479 . . 3  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( N  mod  P )  =  1 )
3731, 36jca 519 . 2  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( N  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) )  /\  ( N  mod  P )  =  1 ) )
388, 37exlimddv 1648 1  |-  ( ph  ->  ( N  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) )  /\  ( N  mod  P )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   {crab 2701   (/)c0 3620   class class class wbr 4204    e. cmpt 4258   ran crn 4871   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   Fincfn 7101   1c1 8983    / cdiv 9669    mod cmo 11242   ^cexp 11374   #chash 11610    || cdivides 12844   Primecprime 13071    pCnt cpc 13202   Basecbs 13461   +g cplusg 13521   Grpcgrp 14677   -gcsg 14680   pSyl cslw 15158
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-er 6897  df-ec 6899  df-qs 6903  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-acn 7821  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-nsg 14934  df-eqg 14935  df-ghm 14996  df-ga 15059  df-od 15159  df-pgp 15161  df-slw 15162
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