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Theorem sylow3 14940
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 14922 . . . 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 3465 . . 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 2284 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
14 eqid 2284 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
15 oveq2 5828 . . . . . . . . . . . 12  |-  ( c  =  z  ->  (
a ( +g  `  G
) c )  =  ( a ( +g  `  G ) z ) )
1615oveq1d 5835 . . . . . . . . . . 11  |-  ( c  =  z  ->  (
( a ( +g  `  G ) c ) ( -g `  G
) a )  =  ( ( a ( +g  `  G ) z ) ( -g `  G ) a ) )
1716cbvmptv 4112 . . . . . . . . . 10  |-  ( c  e.  b  |->  ( ( a ( +g  `  G
) c ) (
-g `  G )
a ) )  =  ( z  e.  b 
|->  ( ( a ( +g  `  G ) z ) ( -g `  G ) a ) )
18 oveq1 5827 . . . . . . . . . . . 12  |-  ( a  =  x  ->  (
a ( +g  `  G
) z )  =  ( x ( +g  `  G ) z ) )
19 id 19 . . . . . . . . . . . 12  |-  ( a  =  x  ->  a  =  x )
2018, 19oveq12d 5838 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
( a ( +g  `  G ) z ) ( -g `  G
) a )  =  ( ( x ( +g  `  G ) z ) ( -g `  G ) x ) )
2120mpteq2dv 4108 . . . . . . . . . 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 2328 . . . . . . . . 9  |-  ( a  =  x  ->  (
c  e.  b  |->  ( ( a ( +g  `  G ) c ) ( -g `  G
) a ) )  =  ( z  e.  b  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2322rneqd 4905 . . . . . . . 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 4101 . . . . . . . . 9  |-  ( b  =  y  ->  (
z  e.  b  |->  ( ( x ( +g  `  G ) z ) ( -g `  G
) x ) )  =  ( z  e.  y  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2524rneqd 4905 . . . . . . . 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 5888 . . . . . . 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 2284 . . . . . . 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 2284 . . . . . . 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 14937 . . . . . 6  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( # `  ( P pSyl  G ) )  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
319, 30syl5eqbr 4057 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  N  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) ) )
329oveq1i 5830 . . . . . 6  |-  ( N  mod  P )  =  ( ( # `  ( P pSyl  G ) )  mod 
P )
3323, 25cbvmpt2v 5888 . . . . . . 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 2284 . . . . . . 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 14939 . . . . . 6  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( ( # `
 ( P pSyl  G
) )  mod  P
)  =  1 )
3632, 35syl5eq 2328 . . . . 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 1665 . 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 1528    = wceq 1623    e. wcel 1685    =/= wne 2447   A.wral 2544   {crab 2548   (/)c0 3456   class class class wbr 4024    e. cmpt 4078   ran crn 4689   ` cfv 5221  (class class class)co 5820    e. cmpt2 5822   Fincfn 6859   1c1 8734    / cdiv 9419    mod cmo 10969   ^cexp 11100   #chash 11333    || cdivides 12527   Primecprime 12754    pCnt cpc 12885   Basecbs 13144   +g cplusg 13204   Grpcgrp 14358   -gcsg 14361   pSyl cslw 14839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-disj 3995  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-omul 6480  df-er 6656  df-ec 6658  df-qs 6662  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-acn 7571  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-q 10313  df-rp 10351  df-fz 10779  df-fzo 10867  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-fac 11285  df-bc 11312  df-hash 11334  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958  df-sum 12155  df-dvds 12528  df-gcd 12682  df-prm 12755  df-pc 12886  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-0g 13400  df-mnd 14363  df-submnd 14412  df-grp 14485  df-minusg 14486  df-sbg 14487  df-mulg 14488  df-subg 14614  df-nsg 14615  df-eqg 14616  df-ghm 14677  df-ga 14740  df-od 14840  df-pgp 14842  df-slw 14843
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