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Theorem sylow1 16105
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 2443 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
8 eqid 2443 . . 3  |-  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  =  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }
9 oveq2 6102 . . . . . . 7  |-  ( s  =  z  ->  (
u ( +g  `  G
) s )  =  ( u ( +g  `  G ) z ) )
109cbvmptv 4386 . . . . . 6  |-  ( s  e.  v  |->  ( u ( +g  `  G
) s ) )  =  ( z  e.  v  |->  ( u ( +g  `  G ) z ) )
11 oveq1 6101 . . . . . . 7  |-  ( u  =  x  ->  (
u ( +g  `  G
) z )  =  ( x ( +g  `  G ) z ) )
1211mpteq2dv 4382 . . . . . 6  |-  ( u  =  x  ->  (
z  e.  v  |->  ( u ( +g  `  G
) z ) )  =  ( z  e.  v  |->  ( x ( +g  `  G ) z ) ) )
1310, 12syl5eq 2487 . . . . 5  |-  ( u  =  x  ->  (
s  e.  v  |->  ( u ( +g  `  G
) s ) )  =  ( z  e.  v  |->  ( x ( +g  `  G ) z ) ) )
1413rneqd 5070 . . . 4  |-  ( u  =  x  ->  ran  ( s  e.  v 
|->  ( u ( +g  `  G ) s ) )  =  ran  (
z  e.  v  |->  ( x ( +g  `  G
) z ) ) )
15 mpteq1 4375 . . . . 5  |-  ( v  =  y  ->  (
z  e.  v  |->  ( x ( +g  `  G
) z ) )  =  ( z  e.  y  |->  ( x ( +g  `  G ) z ) ) )
1615rneqd 5070 . . . 4  |-  ( v  =  y  ->  ran  ( z  e.  v 
|->  ( x ( +g  `  G ) z ) )  =  ran  (
z  e.  y  |->  ( x ( +g  `  G
) z ) ) )
1714, 16cbvmpt2v 6169 . . 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 3959 . . . . . 6  |-  ( ( a  =  x  /\  b  =  y )  ->  { a ,  b }  =  { x ,  y } )
1918sseq1d 3386 . . . . 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 6102 . . . . . . 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 2458 . . . . . 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 2739 . . . . 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 4364 . . 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 16102 . 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 2443 . . 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 16104 . 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 2848 1  |-  ( ph  ->  E. g  e.  (SubGrp `  G ) ( # `  g )  =  ( P ^ N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2719   {crab 2722    C_ wss 3331   ~Pcpw 3863   {cpr 3882   class class class wbr 4295   {copab 4352    e. cmpt 4353   ran crn 4844   ` cfv 5421  (class class class)co 6094    e. cmpt2 6096   [cec 7102   Fincfn 7313    <_ cle 9422    - cmin 9598   NN0cn0 10582   ^cexp 11868   #chash 12106    || cdivides 13538   Primecprime 13766    pCnt cpc 13906   Basecbs 14177   +g cplusg 14241   Grpcgrp 15413  SubGrpcsubg 15678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-disj 4266  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-ec 7106  df-qs 7110  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-oi 7727  df-card 8112  df-cda 8340  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-q 10957  df-rp 10995  df-fz 11441  df-fzo 11552  df-fl 11645  df-mod 11712  df-seq 11810  df-exp 11869  df-fac 12055  df-bc 12082  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-sum 13167  df-dvds 13539  df-gcd 13694  df-prm 13767  df-pc 13907  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-0g 14383  df-mnd 15418  df-grp 15548  df-minusg 15549  df-subg 15681  df-eqg 15683  df-ga 15811
This theorem is referenced by:  odcau  16106  slwhash  16126
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