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Theorem sylow1 16496
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. This is part of Metamath 100 proof #72. (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 2467 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
8 eqid 2467 . . 3  |-  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  =  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }
9 oveq2 6303 . . . . . . 7  |-  ( s  =  z  ->  (
u ( +g  `  G
) s )  =  ( u ( +g  `  G ) z ) )
109cbvmptv 4544 . . . . . 6  |-  ( s  e.  v  |->  ( u ( +g  `  G
) s ) )  =  ( z  e.  v  |->  ( u ( +g  `  G ) z ) )
11 oveq1 6302 . . . . . . 7  |-  ( u  =  x  ->  (
u ( +g  `  G
) z )  =  ( x ( +g  `  G ) z ) )
1211mpteq2dv 4540 . . . . . 6  |-  ( u  =  x  ->  (
z  e.  v  |->  ( u ( +g  `  G
) z ) )  =  ( z  e.  v  |->  ( x ( +g  `  G ) z ) ) )
1310, 12syl5eq 2520 . . . . 5  |-  ( u  =  x  ->  (
s  e.  v  |->  ( u ( +g  `  G
) s ) )  =  ( z  e.  v  |->  ( x ( +g  `  G ) z ) ) )
1413rneqd 5236 . . . 4  |-  ( u  =  x  ->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) )  =  ran  (
z  e.  v  |->  ( x ( +g  `  G
) z ) ) )
15 mpteq1 4533 . . . . 5  |-  ( v  =  y  ->  (
z  e.  v  |->  ( x ( +g  `  G
) z ) )  =  ( z  e.  y  |->  ( x ( +g  `  G ) z ) ) )
1615rneqd 5236 . . . 4  |-  ( v  =  y  ->  ran  ( z  e.  v 
|->  ( x ( +g  `  G ) z ) )  =  ran  (
z  e.  y  |->  ( x ( +g  `  G
) z ) ) )
1714, 16cbvmpt2v 6372 . . 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 4114 . . . . . 6  |-  ( ( a  =  x  /\  b  =  y )  ->  { a ,  b }  =  { x ,  y } )
1918sseq1d 3536 . . . . 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 6303 . . . . . . 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 22 . . . . . . 7  |-  ( b  =  y  ->  b  =  y )
2220, 21eqeqan12d 2490 . . . . . 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 2978 . . . . 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 710 . . . 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 4522 . . 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 16493 . 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 465 . . 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 465 . . 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 465 . . 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 465 . . 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 465 . . 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 755 . . 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 2467 . . 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 756 . . 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 16495 . 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 2963 1  |-  ( ph  ->  E. g  e.  (SubGrp `  G ) ( # `  g )  =  ( P ^ N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   {crab 2821    C_ wss 3481   ~Pcpw 4016   {cpr 4035   class class class wbr 4453   {copab 4510    |-> cmpt 4511   ran crn 5006   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   [cec 7321   Fincfn 7528    <_ cle 9641    - cmin 9817   NN0cn0 10807   ^cexp 12146   #chash 12385    || cdivides 13864   Primecprime 14093    pCnt cpc 14236   Basecbs 14507   +g cplusg 14572   Grpcgrp 15925  SubGrpcsubg 16067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-ec 7325  df-qs 7329  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-dvds 13865  df-gcd 14021  df-prm 14094  df-pc 14237  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-subg 16070  df-eqg 16072  df-ga 16200
This theorem is referenced by:  odcau  16497  slwhash  16517
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