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Theorem sylow1lem5 14929
Description: Lemma for sylow1 14930. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly  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
) )
sylow1lem.a  |-  .+  =  ( +g  `  G )
sylow1lem.s  |-  S  =  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }
sylow1lem.m  |-  .(+)  =  ( x  e.  X , 
y  e.  S  |->  ran  ( z  e.  y 
|->  ( x  .+  z
) ) )
sylow1lem3.1  |-  .~  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  S  /\  E. g  e.  X  (
g  .(+)  x )  =  y ) }
sylow1lem4.b  |-  ( ph  ->  B  e.  S )
sylow1lem4.h  |-  H  =  { u  e.  X  |  ( u  .(+)  B )  =  B }
sylow1lem5.l  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  <_  (
( P  pCnt  ( # `
 X ) )  -  N ) )
Assertion
Ref Expression
sylow1lem5  |-  ( ph  ->  E. h  e.  (SubGrp `  G ) ( # `  h )  =  ( P ^ N ) )
Distinct variable groups:    g, s, u, x, y, z, B   
g, h, H, x, y    S, g, u, x, y, z    g, N   
h, s, u, z, N, x, y    g, X, h, s, u, x, y, z    .+ , s, u, x, y, z    z,  .~   
.(+) , g, u, x, y, z    g, G, h, s, u, x, y, z    P, g, h, s, u, x, y, z    ph, u, x, y, z
Allowed substitution hints:    ph( g, h, s)    B( h)    .+ ( g, h)    .(+) (
h, s)    .~ ( x, y, u, g, h, s)    S( h, s)    H( z, u, s)

Proof of Theorem sylow1lem5
StepHypRef Expression
1 sylow1.x . . . 4  |-  X  =  ( Base `  G
)
2 sylow1.g . . . 4  |-  ( ph  ->  G  e.  Grp )
3 sylow1.f . . . 4  |-  ( ph  ->  X  e.  Fin )
4 sylow1.p . . . 4  |-  ( ph  ->  P  e.  Prime )
5 sylow1.n . . . 4  |-  ( ph  ->  N  e.  NN0 )
6 sylow1.d . . . 4  |-  ( ph  ->  ( P ^ N
)  ||  ( # `  X
) )
7 sylow1lem.a . . . 4  |-  .+  =  ( +g  `  G )
8 sylow1lem.s . . . 4  |-  S  =  { s  e.  ~P X  |  ( # `  s
)  =  ( P ^ N ) }
9 sylow1lem.m . . . 4  |-  .(+)  =  ( x  e.  X , 
y  e.  S  |->  ran  ( z  e.  y 
|->  ( x  .+  z
) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9sylow1lem2 14926 . . 3  |-  ( ph  -> 
.(+)  e.  ( G  GrpAct  S ) )
11 sylow1lem4.b . . 3  |-  ( ph  ->  B  e.  S )
12 sylow1lem4.h . . . 4  |-  H  =  { u  e.  X  |  ( u  .(+)  B )  =  B }
131, 12gastacl 14779 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  S )  /\  B  e.  S )  ->  H  e.  (SubGrp `  G )
)
1410, 11, 13syl2anc 642 . 2  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
15 sylow1lem3.1 . . . 4  |-  .~  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  S  /\  E. g  e.  X  (
g  .(+)  x )  =  y ) }
161, 2, 3, 4, 5, 6, 7, 8, 9, 15, 11, 12sylow1lem4 14928 . . 3  |-  ( ph  ->  ( # `  H
)  <_  ( P ^ N ) )
17 sylow1lem5.l . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  <_  (
( P  pCnt  ( # `
 X ) )  -  N ) )
1815, 1gaorber 14778 . . . . . . . . . . . . . . . 16  |-  (  .(+)  e.  ( G  GrpAct  S )  ->  .~  Er  S
)
1910, 18syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  .~  Er  S )
20 erdm 6686 . . . . . . . . . . . . . . 15  |-  (  .~  Er  S  ->  dom  .~  =  S )
2119, 20syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  .~  =  S )
2211, 21eleqtrrd 2373 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  dom  .~  )
23 ecdmn0 6718 . . . . . . . . . . . . 13  |-  ( B  e.  dom  .~  <->  [ B ]  .~  =/=  (/) )
2422, 23sylib 188 . . . . . . . . . . . 12  |-  ( ph  ->  [ B ]  .~  =/=  (/) )
25 pwfi 7167 . . . . . . . . . . . . . . . 16  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
263, 25sylib 188 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ~P X  e.  Fin )
27 ssrab2 3271 . . . . . . . . . . . . . . . 16  |-  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  C_  ~P X
288, 27eqsstri 3221 . . . . . . . . . . . . . . 15  |-  S  C_  ~P X
29 ssfi 7099 . . . . . . . . . . . . . . 15  |-  ( ( ~P X  e.  Fin  /\  S  C_  ~P X
)  ->  S  e.  Fin )
3026, 28, 29sylancl 643 . . . . . . . . . . . . . 14  |-  ( ph  ->  S  e.  Fin )
3119ecss 6717 . . . . . . . . . . . . . 14  |-  ( ph  ->  [ B ]  .~  C_  S )
32 ssfi 7099 . . . . . . . . . . . . . 14  |-  ( ( S  e.  Fin  /\  [ B ]  .~  C_  S
)  ->  [ B ]  .~  e.  Fin )
3330, 31, 32syl2anc 642 . . . . . . . . . . . . 13  |-  ( ph  ->  [ B ]  .~  e.  Fin )
34 hashnncl 11370 . . . . . . . . . . . . 13  |-  ( [ B ]  .~  e.  Fin  ->  ( ( # `  [ B ]  .~  )  e.  NN  <->  [ B ]  .~  =/=  (/) ) )
3533, 34syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  [ B ]  .~  )  e.  NN  <->  [ B ]  .~  =/=  (/) ) )
3624, 35mpbird 223 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  [ B ]  .~  )  e.  NN )
374, 36pccld 12919 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  e.  NN0 )
3837nn0red 10035 . . . . . . . . 9  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  e.  RR )
395nn0red 10035 . . . . . . . . 9  |-  ( ph  ->  N  e.  RR )
401grpbn0 14527 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  X  =/=  (/) )
412, 40syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  X  =/=  (/) )
42 hashnncl 11370 . . . . . . . . . . . . 13  |-  ( X  e.  Fin  ->  (
( # `  X )  e.  NN  <->  X  =/=  (/) ) )
433, 42syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  X
)  e.  NN  <->  X  =/=  (/) ) )
4441, 43mpbird 223 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  X
)  e.  NN )
454, 44pccld 12919 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  NN0 )
4645nn0red 10035 . . . . . . . . 9  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  RR )
47 leaddsub 9266 . . . . . . . . 9  |-  ( ( ( P  pCnt  ( # `
 [ B ]  .~  ) )  e.  RR  /\  N  e.  RR  /\  ( P  pCnt  ( # `  X ) )  e.  RR )  ->  (
( ( P  pCnt  (
# `  [ B ]  .~  ) )  +  N )  <_  ( P  pCnt  ( # `  X
) )  <->  ( P  pCnt  ( # `  [ B ]  .~  )
)  <_  ( ( P  pCnt  ( # `  X
) )  -  N
) ) )
4838, 39, 46, 47syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
pCnt  ( # `  [ B ]  .~  )
)  +  N )  <_  ( P  pCnt  (
# `  X )
)  <->  ( P  pCnt  (
# `  [ B ]  .~  ) )  <_ 
( ( P  pCnt  (
# `  X )
)  -  N ) ) )
4917, 48mpbird 223 . . . . . . 7  |-  ( ph  ->  ( ( P  pCnt  (
# `  [ B ]  .~  ) )  +  N )  <_  ( P  pCnt  ( # `  X
) ) )
50 eqid 2296 . . . . . . . . . . 11  |-  ( G ~QG  H )  =  ( G ~QG  H )
511, 12, 50, 15orbsta2 14784 . . . . . . . . . 10  |-  ( ( (  .(+)  e.  ( G  GrpAct  S )  /\  B  e.  S )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  [ B ]  .~  )  x.  ( # `
 H ) ) )
5210, 11, 3, 51syl21anc 1181 . . . . . . . . 9  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  [ B ]  .~  )  x.  ( # `
 H ) ) )
5352oveq2d 5890 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  =  ( P  pCnt  ( ( # `  [ B ]  .~  )  x.  ( # `  H
) ) ) )
5436nnzd 10132 . . . . . . . . 9  |-  ( ph  ->  ( # `  [ B ]  .~  )  e.  ZZ )
5536nnne0d 9806 . . . . . . . . 9  |-  ( ph  ->  ( # `  [ B ]  .~  )  =/=  0 )
56 eqid 2296 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  =  ( 0g `  G
)
5756subg0cl 14645 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  H
)
5814, 57syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0g `  G
)  e.  H )
59 ne0i 3474 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  H  ->  H  =/=  (/) )
6058, 59syl 15 . . . . . . . . . . 11  |-  ( ph  ->  H  =/=  (/) )
61 ssrab2 3271 . . . . . . . . . . . . . 14  |-  { u  e.  X  |  (
u  .(+)  B )  =  B }  C_  X
6212, 61eqsstri 3221 . . . . . . . . . . . . 13  |-  H  C_  X
63 ssfi 7099 . . . . . . . . . . . . 13  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
643, 62, 63sylancl 643 . . . . . . . . . . . 12  |-  ( ph  ->  H  e.  Fin )
65 hashnncl 11370 . . . . . . . . . . . 12  |-  ( H  e.  Fin  ->  (
( # `  H )  e.  NN  <->  H  =/=  (/) ) )
6664, 65syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  H
)  e.  NN  <->  H  =/=  (/) ) )
6760, 66mpbird 223 . . . . . . . . . 10  |-  ( ph  ->  ( # `  H
)  e.  NN )
6867nnzd 10132 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
6967nnne0d 9806 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  =/=  0 )
70 pcmul 12920 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  (
( # `  [ B ]  .~  )  e.  ZZ  /\  ( # `  [ B ]  .~  )  =/=  0 )  /\  (
( # `  H )  e.  ZZ  /\  ( # `
 H )  =/=  0 ) )  -> 
( P  pCnt  (
( # `  [ B ]  .~  )  x.  ( # `
 H ) ) )  =  ( ( P  pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) )
714, 54, 55, 68, 69, 70syl122anc 1191 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  (
( # `  [ B ]  .~  )  x.  ( # `
 H ) ) )  =  ( ( P  pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) )
7253, 71eqtrd 2328 . . . . . . 7  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  =  ( ( P 
pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) )
7349, 72breqtrd 4063 . . . . . 6  |-  ( ph  ->  ( ( P  pCnt  (
# `  [ B ]  .~  ) )  +  N )  <_  (
( P  pCnt  ( # `
 [ B ]  .~  ) )  +  ( P  pCnt  ( # `  H
) ) ) )
744, 67pccld 12919 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 H ) )  e.  NN0 )
7574nn0red 10035 . . . . . . 7  |-  ( ph  ->  ( P  pCnt  ( # `
 H ) )  e.  RR )
7639, 75, 38leadd2d 9383 . . . . . 6  |-  ( ph  ->  ( N  <_  ( P  pCnt  ( # `  H
) )  <->  ( ( P  pCnt  ( # `  [ B ]  .~  )
)  +  N )  <_  ( ( P 
pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) ) )
7773, 76mpbird 223 . . . . 5  |-  ( ph  ->  N  <_  ( P  pCnt  ( # `  H
) ) )
78 pcdvdsb 12937 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( # `
 H )  e.  ZZ  /\  N  e. 
NN0 )  ->  ( N  <_  ( P  pCnt  (
# `  H )
)  <->  ( P ^ N )  ||  ( # `
 H ) ) )
794, 68, 5, 78syl3anc 1182 . . . . 5  |-  ( ph  ->  ( N  <_  ( P  pCnt  ( # `  H
) )  <->  ( P ^ N )  ||  ( # `
 H ) ) )
8077, 79mpbid 201 . . . 4  |-  ( ph  ->  ( P ^ N
)  ||  ( # `  H
) )
81 prmnn 12777 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
824, 81syl 15 . . . . . . 7  |-  ( ph  ->  P  e.  NN )
8382, 5nnexpcld 11282 . . . . . 6  |-  ( ph  ->  ( P ^ N
)  e.  NN )
8483nnzd 10132 . . . . 5  |-  ( ph  ->  ( P ^ N
)  e.  ZZ )
85 dvdsle 12590 . . . . 5  |-  ( ( ( P ^ N
)  e.  ZZ  /\  ( # `  H )  e.  NN )  -> 
( ( P ^ N )  ||  ( # `
 H )  -> 
( P ^ N
)  <_  ( # `  H
) ) )
8684, 67, 85syl2anc 642 . . . 4  |-  ( ph  ->  ( ( P ^ N )  ||  ( # `
 H )  -> 
( P ^ N
)  <_  ( # `  H
) ) )
8780, 86mpd 14 . . 3  |-  ( ph  ->  ( P ^ N
)  <_  ( # `  H
) )
88 hashcl 11366 . . . . . 6  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
8964, 88syl 15 . . . . 5  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
9089nn0red 10035 . . . 4  |-  ( ph  ->  ( # `  H
)  e.  RR )
9183nnred 9777 . . . 4  |-  ( ph  ->  ( P ^ N
)  e.  RR )
9290, 91letri3d 8977 . . 3  |-  ( ph  ->  ( ( # `  H
)  =  ( P ^ N )  <->  ( ( # `
 H )  <_ 
( P ^ N
)  /\  ( P ^ N )  <_  ( # `
 H ) ) ) )
9316, 87, 92mpbir2and 888 . 2  |-  ( ph  ->  ( # `  H
)  =  ( P ^ N ) )
94 fveq2 5541 . . . 4  |-  ( h  =  H  ->  ( # `
 h )  =  ( # `  H
) )
9594eqeq1d 2304 . . 3  |-  ( h  =  H  ->  (
( # `  h )  =  ( P ^ N )  <->  ( # `  H
)  =  ( P ^ N ) ) )
9695rspcev 2897 . 2  |-  ( ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ N
) )  ->  E. h  e.  (SubGrp `  G )
( # `  h )  =  ( P ^ N ) )
9714, 93, 96syl2anc 642 1  |-  ( ph  ->  E. h  e.  (SubGrp `  G ) ( # `  h )  =  ( P ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {cpr 3654   class class class wbr 4039   {copab 4092    e. cmpt 4093   dom cdm 4705   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    Er wer 6673   [cec 6674   Fincfn 6879   RRcr 8752   0cc0 8753    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120   #chash 11353    || cdivides 12547   Primecprime 12774    pCnt cpc 12905   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378  SubGrpcsubg 14631   ~QG cqg 14633    GrpAct cga 14759
This theorem is referenced by:  sylow1  14930
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-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-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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