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Theorem sylow3 16252
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. This is part of Metamath 100 proof #72. (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 16234 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  ->  ( P pSyl  G )  =/=  (/) )
61, 2, 3, 5syl3anc 1219 . . 3  |-  ( ph  ->  ( P pSyl  G )  =/=  (/) )
7 n0 3753 . . 3  |-  ( ( P pSyl  G )  =/=  (/) 
<->  E. k  k  e.  ( P pSyl  G ) )
86, 7sylib 196 . 2  |-  ( ph  ->  E. k  k  e.  ( P pSyl  G ) )
9 sylow3.n . . . 4  |-  N  =  ( # `  ( P pSyl  G ) )
101adantr 465 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  G  e.  Grp )
112adantr 465 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  X  e.  Fin )
123adantr 465 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  P  e.  Prime )
13 eqid 2454 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
14 eqid 2454 . . . . 5  |-  ( -g `  G )  =  (
-g `  G )
15 oveq2 6207 . . . . . . . . . 10  |-  ( c  =  z  ->  (
a ( +g  `  G
) c )  =  ( a ( +g  `  G ) z ) )
1615oveq1d 6214 . . . . . . . . 9  |-  ( c  =  z  ->  (
( a ( +g  `  G ) c ) ( -g `  G
) a )  =  ( ( a ( +g  `  G ) z ) ( -g `  G ) a ) )
1716cbvmptv 4490 . . . . . . . 8  |-  ( c  e.  b  |->  ( ( a ( +g  `  G
) c ) (
-g `  G )
a ) )  =  ( z  e.  b 
|->  ( ( a ( +g  `  G ) z ) ( -g `  G ) a ) )
18 oveq1 6206 . . . . . . . . . 10  |-  ( a  =  x  ->  (
a ( +g  `  G
) z )  =  ( x ( +g  `  G ) z ) )
19 id 22 . . . . . . . . . 10  |-  ( a  =  x  ->  a  =  x )
2018, 19oveq12d 6217 . . . . . . . . 9  |-  ( a  =  x  ->  (
( a ( +g  `  G ) z ) ( -g `  G
) a )  =  ( ( x ( +g  `  G ) z ) ( -g `  G ) x ) )
2120mpteq2dv 4486 . . . . . . . 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 2507 . . . . . . 7  |-  ( a  =  x  ->  (
c  e.  b  |->  ( ( a ( +g  `  G ) c ) ( -g `  G
) a ) )  =  ( z  e.  b  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2322rneqd 5174 . . . . . 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 4479 . . . . . . 7  |-  ( b  =  y  ->  (
z  e.  b  |->  ( ( x ( +g  `  G ) z ) ( -g `  G
) x ) )  =  ( z  e.  y  |->  ( ( x ( +g  `  G
) z ) (
-g `  G )
x ) ) )
2524rneqd 5174 . . . . . 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 6274 . . . . 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 461 . . . . 5  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  k  e.  ( P pSyl  G )
)
28 eqid 2454 . . . . 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 2454 . . . . 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 16249 . . . 4  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( # `  ( P pSyl  G ) )  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
319, 30syl5eqbr 4432 . . 3  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  N  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) ) )
329oveq1i 6209 . . . 4  |-  ( N  mod  P )  =  ( ( # `  ( P pSyl  G ) )  mod 
P )
3323, 25cbvmpt2v 6274 . . . . 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 2454 . . . . 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 16251 . . . 4  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( ( # `
 ( P pSyl  G
) )  mod  P
)  =  1 )
3632, 35syl5eq 2507 . . 3  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( N  mod  P )  =  1 )
3731, 36jca 532 . 2  |-  ( (
ph  /\  k  e.  ( P pSyl  G )
)  ->  ( N  ||  ( ( # `  X
)  /  ( P ^ ( P  pCnt  (
# `  X )
) ) )  /\  ( N  mod  P )  =  1 ) )
388, 37exlimddv 1693 1  |-  ( ph  ->  ( N  ||  (
( # `  X )  /  ( P ^
( P  pCnt  ( # `
 X ) ) ) )  /\  ( N  mod  P )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2647   A.wral 2798   {crab 2802   (/)c0 3744   class class class wbr 4399    |-> cmpt 4457   ran crn 4948   ` cfv 5525  (class class class)co 6199    |-> cmpt2 6201   Fincfn 7419   1c1 9393    / cdiv 10103    mod cmo 11824   ^cexp 11981   #chash 12219    || cdivides 13652   Primecprime 13880    pCnt cpc 14020   Basecbs 14291   +g cplusg 14356   Grpcgrp 15528   -gcsg 15531   pSyl cslw 16151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-disj 4370  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-omul 7034  df-er 7210  df-ec 7212  df-qs 7216  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-oi 7834  df-card 8219  df-acn 8222  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-q 11064  df-rp 11102  df-fz 11554  df-fzo 11665  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-fac 12168  df-bc 12195  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-sum 13281  df-dvds 13653  df-gcd 13808  df-prm 13881  df-pc 14021  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-0g 14498  df-mnd 15533  df-submnd 15583  df-grp 15663  df-minusg 15664  df-sbg 15665  df-mulg 15666  df-subg 15796  df-nsg 15797  df-eqg 15798  df-ghm 15863  df-ga 15926  df-od 16152  df-pgp 16154  df-slw 16155
This theorem is referenced by: (None)
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