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Theorem sylow3 15154
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 15136 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  ->  ( P pSyl  G )  =/=  (/) )
61, 2, 3, 5syl3anc 1183 . . 3  |-  ( ph  ->  ( P pSyl  G )  =/=  (/) )
7 n0 3552 . . 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 2366 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
14 eqid 2366 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
15 oveq2 5989 . . . . . . . . . . . 12  |-  ( c  =  z  ->  (
a ( +g  `  G
) c )  =  ( a ( +g  `  G ) z ) )
1615oveq1d 5996 . . . . . . . . . . 11  |-  ( c  =  z  ->  (
( a ( +g  `  G ) c ) ( -g `  G
) a )  =  ( ( a ( +g  `  G ) z ) ( -g `  G ) a ) )
1716cbvmptv 4213 . . . . . . . . . 10  |-  ( c  e.  b  |->  ( ( a ( +g  `  G
) c ) (
-g `  G )
a ) )  =  ( z  e.  b 
|->  ( ( a ( +g  `  G ) z ) ( -g `  G ) a ) )
18 oveq1 5988 . . . . . . . . . . . 12  |-  ( a  =  x  ->  (
a ( +g  `  G
) z )  =  ( x ( +g  `  G ) z ) )
19 id 19 . . . . . . . . . . . 12  |-  ( a  =  x  ->  a  =  x )
2018, 19oveq12d 5999 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
( a ( +g  `  G ) z ) ( -g `  G
) a )  =  ( ( x ( +g  `  G ) z ) ( -g `  G ) x ) )
2120mpteq2dv 4209 . . . . . . . . . 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 2410 . . . . . . . . 9  |-  ( a  =  x  ->  (
c  e.  b  |->  ( ( a ( +g  `  G ) c ) ( -g `  G
) a ) )  =  ( z  e.  b  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2322rneqd 5009 . . . . . . . 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 4202 . . . . . . . . 9  |-  ( b  =  y  ->  (
z  e.  b  |->  ( ( x ( +g  `  G ) z ) ( -g `  G
) x ) )  =  ( z  e.  y  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2524rneqd 5009 . . . . . . . 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 6052 . . . . . . 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 2366 . . . . . . 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 2366 . . . . . . 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 15151 . . . . . 6  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( # `  ( P pSyl  G ) )  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
319, 30syl5eqbr 4158 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  N  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) ) )
329oveq1i 5991 . . . . . 6  |-  ( N  mod  P )  =  ( ( # `  ( P pSyl  G ) )  mod 
P )
3323, 25cbvmpt2v 6052 . . . . . . 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 2366 . . . . . . 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 15153 . . . . . 6  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( ( # `
 ( P pSyl  G
) )  mod  P
)  =  1 )
3632, 35syl5eq 2410 . . . . 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 1641 . 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 1546    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   {crab 2632   (/)c0 3543   class class class wbr 4125    e. cmpt 4179   ran crn 4793   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   Fincfn 7006   1c1 8885    / cdiv 9570    mod cmo 11137   ^cexp 11269   #chash 11505    || cdivides 12739   Primecprime 12966    pCnt cpc 13097   Basecbs 13356   +g cplusg 13416   Grpcgrp 14572   -gcsg 14575   pSyl cslw 15053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-disj 4096  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-omul 6626  df-er 6802  df-ec 6804  df-qs 6808  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-oi 7372  df-card 7719  df-acn 7722  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-q 10468  df-rp 10506  df-fz 10936  df-fzo 11026  df-fl 11089  df-mod 11138  df-seq 11211  df-exp 11270  df-fac 11454  df-bc 11481  df-hash 11506  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-clim 12169  df-sum 12367  df-dvds 12740  df-gcd 12894  df-prm 12967  df-pc 13098  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-0g 13614  df-mnd 14577  df-submnd 14626  df-grp 14699  df-minusg 14700  df-sbg 14701  df-mulg 14702  df-subg 14828  df-nsg 14829  df-eqg 14830  df-ghm 14891  df-ga 14954  df-od 15054  df-pgp 15056  df-slw 15057
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