MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sylow3 Unicode version

Theorem sylow3 14946
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 14928 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  ->  ( P pSyl  G )  =/=  (/) )
61, 2, 3, 5syl3anc 1182 . . 3  |-  ( ph  ->  ( P pSyl  G )  =/=  (/) )
7 n0 3466 . . 3  |-  ( ( P pSyl  G )  =/=  (/) 
<->  E. k  k  e.  ( P pSyl  G ) )
86, 7sylib 188 . 2  |-  ( ph  ->  E. k  k  e.  ( P pSyl  G ) )
9 sylow3.n . . . . . 6  |-  N  =  ( # `  ( P pSyl  G ) )
101adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  G  e.  Grp )
112adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  X  e.  Fin )
123adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  P  e.  Prime )
13 eqid 2285 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
14 eqid 2285 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
15 oveq2 5868 . . . . . . . . . . . 12  |-  ( c  =  z  ->  (
a ( +g  `  G
) c )  =  ( a ( +g  `  G ) z ) )
1615oveq1d 5875 . . . . . . . . . . 11  |-  ( c  =  z  ->  (
( a ( +g  `  G ) c ) ( -g `  G
) a )  =  ( ( a ( +g  `  G ) z ) ( -g `  G ) a ) )
1716cbvmptv 4113 . . . . . . . . . 10  |-  ( c  e.  b  |->  ( ( a ( +g  `  G
) c ) (
-g `  G )
a ) )  =  ( z  e.  b 
|->  ( ( a ( +g  `  G ) z ) ( -g `  G ) a ) )
18 oveq1 5867 . . . . . . . . . . . 12  |-  ( a  =  x  ->  (
a ( +g  `  G
) z )  =  ( x ( +g  `  G ) z ) )
19 id 19 . . . . . . . . . . . 12  |-  ( a  =  x  ->  a  =  x )
2018, 19oveq12d 5878 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
( a ( +g  `  G ) z ) ( -g `  G
) a )  =  ( ( x ( +g  `  G ) z ) ( -g `  G ) x ) )
2120mpteq2dv 4109 . . . . . . . . . 10  |-  ( a  =  x  ->  (
z  e.  b  |->  ( ( a ( +g  `  G ) z ) ( -g `  G
) a ) )  =  ( z  e.  b  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2217, 21syl5eq 2329 . . . . . . . . 9  |-  ( a  =  x  ->  (
c  e.  b  |->  ( ( a ( +g  `  G ) c ) ( -g `  G
) a ) )  =  ( z  e.  b  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2322rneqd 4908 . . . . . . . 8  |-  ( 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 4102 . . . . . . . . 9  |-  ( b  =  y  ->  (
z  e.  b  |->  ( ( x ( +g  `  G ) z ) ( -g `  G
) x ) )  =  ( z  e.  y  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2524rneqd 4908 . . . . . . . 8  |-  ( 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 5928 . . . . . . 7  |-  ( 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 447 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  k  e.  ( P pSyl  G )
)
28 eqid 2285 . . . . . . 7  |-  { 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 2285 . . . . . . 7  |-  { 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 14943 . . . . . 6  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( # `  ( P pSyl  G ) )  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
319, 30syl5eqbr 4058 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  N  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) ) )
329oveq1i 5870 . . . . . 6  |-  ( N  mod  P )  =  ( ( # `  ( P pSyl  G ) )  mod 
P )
3323, 25cbvmpt2v 5928 . . . . . . 7  |-  ( 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 2285 . . . . . . 7  |-  { 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 14945 . . . . . 6  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( ( # `
 ( P pSyl  G
) )  mod  P
)  =  1 )
3632, 35syl5eq 2329 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( N  mod  P )  =  1 )
3731, 36jca 518 . . . 4  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( N  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) )  /\  ( N  mod  P )  =  1 ) )
3837ex 423 . . 3  |-  ( ph  ->  ( k  e.  ( P pSyl  G )  -> 
( N  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) )  /\  ( N  mod  P )  =  1 ) ) )
3938exlimdv 1666 . 2  |-  ( ph  ->  ( E. k  k  e.  ( P pSyl  G
)  ->  ( N  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) )  /\  ( N  mod  P )  =  1 ) ) )
408, 39mpd 14 1  |-  ( ph  ->  ( N  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) )  /\  ( N  mod  P )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1530    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   {crab 2549   (/)c0 3457   class class class wbr 4025    e. cmpt 4079   ran crn 4692   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862   Fincfn 6865   1c1 8740    / cdiv 9425    mod cmo 10975   ^cexp 11106   #chash 11339    || cdivides 12533   Primecprime 12760    pCnt cpc 12891   Basecbs 13150   +g cplusg 13210   Grpcgrp 14364   -gcsg 14367   pSyl cslw 14845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-disj 3996  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-omul 6486  df-er 6662  df-ec 6664  df-qs 6668  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-acn 7577  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-q 10319  df-rp 10357  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-sum 12161  df-dvds 12534  df-gcd 12688  df-prm 12761  df-pc 12892  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-0g 13406  df-mnd 14369  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-mulg 14494  df-subg 14620  df-nsg 14621  df-eqg 14622  df-ghm 14683  df-ga 14746  df-od 14846  df-pgp 14848  df-slw 14849
  Copyright terms: Public domain W3C validator