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Theorem sylow1 14909
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 2283 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
8 eqid 2283 . . 3  |-  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  =  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }
9 oveq2 5827 . . . . . . 7  |-  ( s  =  z  ->  (
u ( +g  `  G
) s )  =  ( u ( +g  `  G ) z ) )
109cbvmptv 4111 . . . . . 6  |-  ( s  e.  v  |->  ( u ( +g  `  G
) s ) )  =  ( z  e.  v  |->  ( u ( +g  `  G ) z ) )
11 oveq1 5826 . . . . . . 7  |-  ( u  =  x  ->  (
u ( +g  `  G
) z )  =  ( x ( +g  `  G ) z ) )
1211mpteq2dv 4107 . . . . . 6  |-  ( u  =  x  ->  (
z  e.  v  |->  ( u ( +g  `  G
) z ) )  =  ( z  e.  v  |->  ( x ( +g  `  G ) z ) ) )
1310, 12syl5eq 2327 . . . . 5  |-  ( u  =  x  ->  (
s  e.  v  |->  ( u ( +g  `  G
) s ) )  =  ( z  e.  v  |->  ( x ( +g  `  G ) z ) ) )
1413rneqd 4904 . . . 4  |-  ( u  =  x  ->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) )  =  ran  (
z  e.  v  |->  ( x ( +g  `  G
) z ) ) )
15 mpteq1 4100 . . . . 5  |-  ( v  =  y  ->  (
z  e.  v  |->  ( x ( +g  `  G
) z ) )  =  ( z  e.  y  |->  ( x ( +g  `  G ) z ) ) )
1615rneqd 4904 . . . 4  |-  ( v  =  y  ->  ran  ( z  e.  v 
|->  ( x ( +g  `  G ) z ) )  =  ran  (
z  e.  y  |->  ( x ( +g  `  G
) z ) ) )
1714, 16cbvmpt2v 5887 . . 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 3708 . . . . . 6  |-  ( ( a  =  x  /\  b  =  y )  ->  { a ,  b }  =  { x ,  y } )
1918sseq1d 3205 . . . . 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 5827 . . . . . . 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 2298 . . . . . 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 2564 . . . . 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 4088 . . 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 14906 . 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 2283 . . . . 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 14908 . . . 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 2667 . 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 1623    e. wcel 1684   E.wrex 2544   {crab 2547    C_ wss 3152   ~Pcpw 3625   {cpr 3641   class class class wbr 4023   {copab 4076    e. cmpt 4077   ran crn 4688   ` cfv 5220  (class class class)co 5819    e. cmpt2 5821   [cec 6653   Fincfn 6858    <_ cle 8863    - cmin 9032   NN0cn0 9960   ^cexp 11099   #chash 11332    || cdivides 12526   Primecprime 12753    pCnt cpc 12884   Basecbs 13143   +g cplusg 13203   Grpcgrp 14357  SubGrpcsubg 14610
This theorem is referenced by:  odcau  14910  slwhash  14930
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4186  ax-pr 4212  ax-un 4510  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4303  df-id 4307  df-po 4312  df-so 4313  df-fr 4350  df-se 4351  df-we 4352  df-ord 4393  df-on 4394  df-lim 4395  df-suc 4396  df-om 4655  df-xp 4693  df-rel 4694  df-cnv 4695  df-co 4696  df-dm 4697  df-rn 4698  df-res 4699  df-ima 4700  df-fun 5222  df-fn 5223  df-f 5224  df-f1 5225  df-fo 5226  df-f1o 5227  df-fv 5228  df-isom 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-ec 6657  df-qs 6661  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-q 10312  df-rp 10350  df-fz 10778  df-fzo 10866  df-fl 10920  df-mod 10969  df-seq 11042  df-exp 11100  df-fac 11284  df-bc 11311  df-hash 11333  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-clim 11957  df-sum 12154  df-dvds 12527  df-gcd 12681  df-prm 12754  df-pc 12885  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-0g 13399  df-mnd 14362  df-grp 14484  df-minusg 14485  df-subg 14613  df-eqg 14615  df-ga 14739
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