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Theorem sylow1 15239
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 2438 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
8 eqid 2438 . . 3  |-  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  =  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }
9 oveq2 6091 . . . . . . 7  |-  ( s  =  z  ->  (
u ( +g  `  G
) s )  =  ( u ( +g  `  G ) z ) )
109cbvmptv 4302 . . . . . 6  |-  ( s  e.  v  |->  ( u ( +g  `  G
) s ) )  =  ( z  e.  v  |->  ( u ( +g  `  G ) z ) )
11 oveq1 6090 . . . . . . 7  |-  ( u  =  x  ->  (
u ( +g  `  G
) z )  =  ( x ( +g  `  G ) z ) )
1211mpteq2dv 4298 . . . . . 6  |-  ( u  =  x  ->  (
z  e.  v  |->  ( u ( +g  `  G
) z ) )  =  ( z  e.  v  |->  ( x ( +g  `  G ) z ) ) )
1310, 12syl5eq 2482 . . . . 5  |-  ( u  =  x  ->  (
s  e.  v  |->  ( u ( +g  `  G
) s ) )  =  ( z  e.  v  |->  ( x ( +g  `  G ) z ) ) )
1413rneqd 5099 . . . 4  |-  ( u  =  x  ->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) )  =  ran  (
z  e.  v  |->  ( x ( +g  `  G
) z ) ) )
15 mpteq1 4291 . . . . 5  |-  ( v  =  y  ->  (
z  e.  v  |->  ( x ( +g  `  G
) z ) )  =  ( z  e.  y  |->  ( x ( +g  `  G ) z ) ) )
1615rneqd 5099 . . . 4  |-  ( v  =  y  ->  ran  ( z  e.  v 
|->  ( x ( +g  `  G ) z ) )  =  ran  (
z  e.  y  |->  ( x ( +g  `  G
) z ) ) )
1714, 16cbvmpt2v 6154 . . 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 3887 . . . . . 6  |-  ( ( a  =  x  /\  b  =  y )  ->  { a ,  b }  =  { x ,  y } )
1918sseq1d 3377 . . . . 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 6091 . . . . . . 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 21 . . . . . . 7  |-  ( b  =  y  ->  b  =  y )
2220, 21eqeqan12d 2453 . . . . . 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 2728 . . . . 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 693 . . . 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 4279 . . 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 15236 . 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 453 . . 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
) ) )  ->  G  e.  Grp )
283adantr 453 . . 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
) ) )  ->  X  e.  Fin )
294adantr 453 . . 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
) ) )  ->  P  e.  Prime )
305adantr 453 . . 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
) ) )  ->  N  e.  NN0 )
316adantr 453 . . 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
) ) )  -> 
( P ^ N
)  ||  ( # `  X
) )
32 simprl 734 . . 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
) ) )  ->  h  e.  { s  e.  ~P X  |  (
# `  s )  =  ( P ^ N ) } )
33 eqid 2438 . . 3  |-  { 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 735 . . 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
) ) )  -> 
( 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 15238 . 2  |-  ( (
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 ) )
3626, 35rexlimddv 2836 1  |-  ( ph  ->  E. g  e.  (SubGrp `  G ) ( # `  g )  =  ( P ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   {crab 2711    C_ wss 3322   ~Pcpw 3801   {cpr 3817   class class class wbr 4214   {copab 4267    e. cmpt 4268   ran crn 4881   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   [cec 6905   Fincfn 7111    <_ cle 9123    - cmin 9293   NN0cn0 10223   ^cexp 11384   #chash 11620    || cdivides 12854   Primecprime 13081    pCnt cpc 13212   Basecbs 13471   +g cplusg 13531   Grpcgrp 14687  SubGrpcsubg 14940
This theorem is referenced by:  odcau  15240  slwhash  15260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-disj 4185  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-ec 6909  df-qs 6913  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-dvds 12855  df-gcd 13009  df-prm 13082  df-pc 13213  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-subg 14943  df-eqg 14945  df-ga 15069
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