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Theorem sylow1 14930
Description: Sylow's first theorem. If  P ^ N is a prime power that divides the cardinality of  G, then  G has a supgroup with size  P ^ N. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypotheses
Ref Expression
sylow1.x  |-  X  =  ( Base `  G
)
sylow1.g  |-  ( ph  ->  G  e.  Grp )
sylow1.f  |-  ( ph  ->  X  e.  Fin )
sylow1.p  |-  ( ph  ->  P  e.  Prime )
sylow1.n  |-  ( ph  ->  N  e.  NN0 )
sylow1.d  |-  ( ph  ->  ( P ^ N
)  ||  ( # `  X
) )
Assertion
Ref Expression
sylow1  |-  ( ph  ->  E. g  e.  (SubGrp `  G ) ( # `  g )  =  ( P ^ N ) )
Distinct variable groups:    g, N    g, X    g, G    P, g    ph, g

Proof of Theorem sylow1
Dummy variables  a 
b  s  u  x  y  z  h  k  t  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sylow1.x . . 3  |-  X  =  ( Base `  G
)
2 sylow1.g . . 3  |-  ( ph  ->  G  e.  Grp )
3 sylow1.f . . 3  |-  ( ph  ->  X  e.  Fin )
4 sylow1.p . . 3  |-  ( ph  ->  P  e.  Prime )
5 sylow1.n . . 3  |-  ( ph  ->  N  e.  NN0 )
6 sylow1.d . . 3  |-  ( ph  ->  ( P ^ N
)  ||  ( # `  X
) )
7 eqid 2296 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
8 eqid 2296 . . 3  |-  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  =  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }
9 oveq2 5882 . . . . . . 7  |-  ( s  =  z  ->  (
u ( +g  `  G
) s )  =  ( u ( +g  `  G ) z ) )
109cbvmptv 4127 . . . . . 6  |-  ( s  e.  v  |->  ( u ( +g  `  G
) s ) )  =  ( z  e.  v  |->  ( u ( +g  `  G ) z ) )
11 oveq1 5881 . . . . . . 7  |-  ( u  =  x  ->  (
u ( +g  `  G
) z )  =  ( x ( +g  `  G ) z ) )
1211mpteq2dv 4123 . . . . . 6  |-  ( u  =  x  ->  (
z  e.  v  |->  ( u ( +g  `  G
) z ) )  =  ( z  e.  v  |->  ( x ( +g  `  G ) z ) ) )
1310, 12syl5eq 2340 . . . . 5  |-  ( u  =  x  ->  (
s  e.  v  |->  ( u ( +g  `  G
) s ) )  =  ( z  e.  v  |->  ( x ( +g  `  G ) z ) ) )
1413rneqd 4922 . . . 4  |-  ( u  =  x  ->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) )  =  ran  (
z  e.  v  |->  ( x ( +g  `  G
) z ) ) )
15 mpteq1 4116 . . . . 5  |-  ( v  =  y  ->  (
z  e.  v  |->  ( x ( +g  `  G
) z ) )  =  ( z  e.  y  |->  ( x ( +g  `  G ) z ) ) )
1615rneqd 4922 . . . 4  |-  ( v  =  y  ->  ran  ( z  e.  v 
|->  ( x ( +g  `  G ) z ) )  =  ran  (
z  e.  y  |->  ( x ( +g  `  G
) z ) ) )
1714, 16cbvmpt2v 5942 . . 3  |-  ( u  e.  X ,  v  e.  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  |->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) ) )  =  ( x  e.  X , 
y  e.  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  |->  ran  ( z  e.  y 
|->  ( x ( +g  `  G ) z ) ) )
18 preq12 3721 . . . . . 6  |-  ( ( a  =  x  /\  b  =  y )  ->  { a ,  b }  =  { x ,  y } )
1918sseq1d 3218 . . . . 5  |-  ( ( a  =  x  /\  b  =  y )  ->  ( { a ,  b }  C_  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  <->  { x ,  y }  C_  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } ) )
20 oveq2 5882 . . . . . . 7  |-  ( a  =  x  ->  (
k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) x ) )
21 id 19 . . . . . . 7  |-  ( b  =  y  ->  b  =  y )
2220, 21eqeqan12d 2311 . . . . . 6  |-  ( ( a  =  x  /\  b  =  y )  ->  ( ( k ( u  e.  X , 
v  e.  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  |->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) ) ) a )  =  b  <->  ( k
( u  e.  X ,  v  e.  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  |->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) ) ) x )  =  y ) )
2322rexbidv 2577 . . . . 5  |-  ( ( a  =  x  /\  b  =  y )  ->  ( E. k  e.  X  ( k ( u  e.  X , 
v  e.  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  |->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) ) ) a )  =  b  <->  E. k  e.  X  ( k
( u  e.  X ,  v  e.  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  |->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) ) ) x )  =  y ) )
2419, 23anbi12d 691 . . . 4  |-  ( ( a  =  x  /\  b  =  y )  ->  ( ( { a ,  b }  C_  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X , 
v  e.  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  |->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) ) ) a )  =  b )  <->  ( {
x ,  y } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) x )  =  y ) ) )
2524cbvopabv 4104 . . 3  |-  { <. a ,  b >.  |  ( { a ,  b }  C_  { s  e.  ~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) }  =  { <. x ,  y >.  |  ( { x ,  y }  C_  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X , 
v  e.  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  |->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) ) ) x )  =  y ) }
261, 2, 3, 4, 5, 6, 7, 8, 17, 25sylow1lem3 14927 . 2  |-  ( ph  ->  E. h  e.  {
s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  ( P  pCnt  ( # `
 [ h ] { <. a ,  b
>.  |  ( {
a ,  b } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
) )
272adantr 451 . . . . 5  |-  ( (
ph  /\  ( h  e.  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  /\  ( P  pCnt  (
# `  [ h ] { <. a ,  b
>.  |  ( {
a ,  b } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
) ) )  ->  G  e.  Grp )
283adantr 451 . . . . 5  |-  ( (
ph  /\  ( h  e.  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  /\  ( P  pCnt  (
# `  [ h ] { <. a ,  b
>.  |  ( {
a ,  b } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
) ) )  ->  X  e.  Fin )
294adantr 451 . . . . 5  |-  ( (
ph  /\  ( h  e.  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  /\  ( P  pCnt  (
# `  [ h ] { <. a ,  b
>.  |  ( {
a ,  b } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
) ) )  ->  P  e.  Prime )
305adantr 451 . . . . 5  |-  ( (
ph  /\  ( h  e.  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  /\  ( P  pCnt  (
# `  [ h ] { <. a ,  b
>.  |  ( {
a ,  b } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
) ) )  ->  N  e.  NN0 )
316adantr 451 . . . . 5  |-  ( (
ph  /\  ( h  e.  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  /\  ( P  pCnt  (
# `  [ h ] { <. a ,  b
>.  |  ( {
a ,  b } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
) ) )  -> 
( P ^ N
)  ||  ( # `  X
) )
32 simprl 732 . . . . 5  |-  ( (
ph  /\  ( h  e.  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  /\  ( P  pCnt  (
# `  [ h ] { <. a ,  b
>.  |  ( {
a ,  b } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
) ) )  ->  h  e.  { s  e.  ~P X  |  (
# `  s )  =  ( P ^ N ) } )
33 eqid 2296 . . . . 5  |-  { t  e.  X  |  ( t ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) h )  =  h }  =  { t  e.  X  |  ( t ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) h )  =  h }
34 simprr 733 . . . . 5  |-  ( (
ph  /\  ( h  e.  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  /\  ( P  pCnt  (
# `  [ h ] { <. a ,  b
>.  |  ( {
a ,  b } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
) ) )  -> 
( P  pCnt  ( # `
 [ h ] { <. a ,  b
>.  |  ( {
a ,  b } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
) )
351, 27, 28, 29, 30, 31, 7, 8, 17, 25, 32, 33, 34sylow1lem5 14929 . . . 4  |-  ( (
ph  /\  ( h  e.  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  /\  ( P  pCnt  (
# `  [ h ] { <. a ,  b
>.  |  ( {
a ,  b } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
) ) )  ->  E. g  e.  (SubGrp `  G ) ( # `  g )  =  ( P ^ N ) )
3635expr 598 . . 3  |-  ( (
ph  /\  h  e.  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } )  ->  ( ( P  pCnt  ( # `  [
h ] { <. a ,  b >.  |  ( { a ,  b }  C_  { s  e.  ~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
)  ->  E. g  e.  (SubGrp `  G )
( # `  g )  =  ( P ^ N ) ) )
3736rexlimdva 2680 . 2  |-  ( ph  ->  ( E. h  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }  ( P  pCnt  ( # `
 [ h ] { <. a ,  b
>.  |  ( {
a ,  b } 
C_  { s  e. 
~P X  |  (
# `  s )  =  ( P ^ N ) }  /\  E. k  e.  X  ( k ( u  e.  X ,  v  e. 
{ s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) } 
|->  ran  ( s  e.  v  |->  ( u ( +g  `  G ) s ) ) ) a )  =  b ) } ) )  <_  ( ( P 
pCnt  ( # `  X
) )  -  N
)  ->  E. g  e.  (SubGrp `  G )
( # `  g )  =  ( P ^ N ) ) )
3826, 37mpd 14 1  |-  ( ph  ->  E. g  e.  (SubGrp `  G ) ( # `  g )  =  ( P ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560    C_ wss 3165   ~Pcpw 3638   {cpr 3654   class class class wbr 4039   {copab 4092    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   [cec 6674   Fincfn 6879    <_ cle 8884    - cmin 9053   NN0cn0 9981   ^cexp 11120   #chash 11353    || cdivides 12547   Primecprime 12774    pCnt cpc 12905   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378  SubGrpcsubg 14631
This theorem is referenced by:  odcau  14931  slwhash  14951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634  df-eqg 14636  df-ga 14760
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