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Theorem sylow1lem5 16413
Description: Lemma for sylow1 16414. 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 16410 . . 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 16137 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  S )  /\  B  e.  S )  ->  H  e.  (SubGrp `  G )
)
1410, 11, 13syl2anc 661 . 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 16412 . . 3  |-  ( ph  ->  ( # `  H
)  <_  ( P ^ N ) )
17 sylow1lem5.l . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  <_  (
( P  pCnt  ( # `
 X ) )  -  N ) )
1815, 1gaorber 16136 . . . . . . . . . . . . . . . 16  |-  (  .(+)  e.  ( G  GrpAct  S )  ->  .~  Er  S
)
1910, 18syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  .~  Er  S )
20 erdm 7313 . . . . . . . . . . . . . . 15  |-  (  .~  Er  S  ->  dom  .~  =  S )
2119, 20syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  .~  =  S )
2211, 21eleqtrrd 2553 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  dom  .~  )
23 ecdmn0 7346 . . . . . . . . . . . . 13  |-  ( B  e.  dom  .~  <->  [ B ]  .~  =/=  (/) )
2422, 23sylib 196 . . . . . . . . . . . 12  |-  ( ph  ->  [ B ]  .~  =/=  (/) )
25 pwfi 7806 . . . . . . . . . . . . . . . 16  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
263, 25sylib 196 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ~P X  e.  Fin )
27 ssrab2 3580 . . . . . . . . . . . . . . . 16  |-  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  C_  ~P X
288, 27eqsstri 3529 . . . . . . . . . . . . . . 15  |-  S  C_  ~P X
29 ssfi 7732 . . . . . . . . . . . . . . 15  |-  ( ( ~P X  e.  Fin  /\  S  C_  ~P X
)  ->  S  e.  Fin )
3026, 28, 29sylancl 662 . . . . . . . . . . . . . 14  |-  ( ph  ->  S  e.  Fin )
3119ecss 7345 . . . . . . . . . . . . . 14  |-  ( ph  ->  [ B ]  .~  C_  S )
32 ssfi 7732 . . . . . . . . . . . . . 14  |-  ( ( S  e.  Fin  /\  [ B ]  .~  C_  S
)  ->  [ B ]  .~  e.  Fin )
3330, 31, 32syl2anc 661 . . . . . . . . . . . . 13  |-  ( ph  ->  [ B ]  .~  e.  Fin )
34 hashnncl 12393 . . . . . . . . . . . . 13  |-  ( [ B ]  .~  e.  Fin  ->  ( ( # `  [ B ]  .~  )  e.  NN  <->  [ B ]  .~  =/=  (/) ) )
3533, 34syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  [ B ]  .~  )  e.  NN  <->  [ B ]  .~  =/=  (/) ) )
3624, 35mpbird 232 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  [ B ]  .~  )  e.  NN )
374, 36pccld 14224 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  e.  NN0 )
3837nn0red 10844 . . . . . . . . 9  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  e.  RR )
395nn0red 10844 . . . . . . . . 9  |-  ( ph  ->  N  e.  RR )
401grpbn0 15875 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  X  =/=  (/) )
412, 40syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  X  =/=  (/) )
42 hashnncl 12393 . . . . . . . . . . . . 13  |-  ( X  e.  Fin  ->  (
( # `  X )  e.  NN  <->  X  =/=  (/) ) )
433, 42syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  X
)  e.  NN  <->  X  =/=  (/) ) )
4441, 43mpbird 232 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  X
)  e.  NN )
454, 44pccld 14224 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  NN0 )
4645nn0red 10844 . . . . . . . . 9  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  RR )
47 leaddsub 10019 . . . . . . . . 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 1223 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
pCnt  ( # `  [ B ]  .~  )
)  +  N )  <_  ( P  pCnt  (
# `  X )
)  <->  ( P  pCnt  (
# `  [ B ]  .~  ) )  <_ 
( ( P  pCnt  (
# `  X )
)  -  N ) ) )
4917, 48mpbird 232 . . . . . . 7  |-  ( ph  ->  ( ( P  pCnt  (
# `  [ B ]  .~  ) )  +  N )  <_  ( P  pCnt  ( # `  X
) ) )
50 eqid 2462 . . . . . . . . . . 11  |-  ( G ~QG  H )  =  ( G ~QG  H )
511, 12, 50, 15orbsta2 16142 . . . . . . . . . 10  |-  ( ( (  .(+)  e.  ( G  GrpAct  S )  /\  B  e.  S )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  [ B ]  .~  )  x.  ( # `
 H ) ) )
5210, 11, 3, 51syl21anc 1222 . . . . . . . . 9  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  [ B ]  .~  )  x.  ( # `
 H ) ) )
5352oveq2d 6293 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  =  ( P  pCnt  ( ( # `  [ B ]  .~  )  x.  ( # `  H
) ) ) )
5436nnzd 10956 . . . . . . . . 9  |-  ( ph  ->  ( # `  [ B ]  .~  )  e.  ZZ )
5536nnne0d 10571 . . . . . . . . 9  |-  ( ph  ->  ( # `  [ B ]  .~  )  =/=  0 )
56 eqid 2462 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  =  ( 0g `  G
)
5756subg0cl 15999 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  H
)
5814, 57syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0g `  G
)  e.  H )
59 ne0i 3786 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  H  ->  H  =/=  (/) )
6058, 59syl 16 . . . . . . . . . . 11  |-  ( ph  ->  H  =/=  (/) )
61 ssrab2 3580 . . . . . . . . . . . . . 14  |-  { u  e.  X  |  (
u  .(+)  B )  =  B }  C_  X
6212, 61eqsstri 3529 . . . . . . . . . . . . 13  |-  H  C_  X
63 ssfi 7732 . . . . . . . . . . . . 13  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
643, 62, 63sylancl 662 . . . . . . . . . . . 12  |-  ( ph  ->  H  e.  Fin )
65 hashnncl 12393 . . . . . . . . . . . 12  |-  ( H  e.  Fin  ->  (
( # `  H )  e.  NN  <->  H  =/=  (/) ) )
6664, 65syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  H
)  e.  NN  <->  H  =/=  (/) ) )
6760, 66mpbird 232 . . . . . . . . . 10  |-  ( ph  ->  ( # `  H
)  e.  NN )
6867nnzd 10956 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
6967nnne0d 10571 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  =/=  0 )
70 pcmul 14225 . . . . . . . . 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 1232 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  (
( # `  [ B ]  .~  )  x.  ( # `
 H ) ) )  =  ( ( P  pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) )
7253, 71eqtrd 2503 . . . . . . 7  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  =  ( ( P 
pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) )
7349, 72breqtrd 4466 . . . . . 6  |-  ( ph  ->  ( ( P  pCnt  (
# `  [ B ]  .~  ) )  +  N )  <_  (
( P  pCnt  ( # `
 [ B ]  .~  ) )  +  ( P  pCnt  ( # `  H
) ) ) )
744, 67pccld 14224 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 H ) )  e.  NN0 )
7574nn0red 10844 . . . . . . 7  |-  ( ph  ->  ( P  pCnt  ( # `
 H ) )  e.  RR )
7639, 75, 38leadd2d 10138 . . . . . 6  |-  ( ph  ->  ( N  <_  ( P  pCnt  ( # `  H
) )  <->  ( ( P  pCnt  ( # `  [ B ]  .~  )
)  +  N )  <_  ( ( P 
pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) ) )
7773, 76mpbird 232 . . . . 5  |-  ( ph  ->  N  <_  ( P  pCnt  ( # `  H
) ) )
78 pcdvdsb 14242 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( # `
 H )  e.  ZZ  /\  N  e. 
NN0 )  ->  ( N  <_  ( P  pCnt  (
# `  H )
)  <->  ( P ^ N )  ||  ( # `
 H ) ) )
794, 68, 5, 78syl3anc 1223 . . . . 5  |-  ( ph  ->  ( N  <_  ( P  pCnt  ( # `  H
) )  <->  ( P ^ N )  ||  ( # `
 H ) ) )
8077, 79mpbid 210 . . . 4  |-  ( ph  ->  ( P ^ N
)  ||  ( # `  H
) )
81 prmnn 14070 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
824, 81syl 16 . . . . . . 7  |-  ( ph  ->  P  e.  NN )
8382, 5nnexpcld 12288 . . . . . 6  |-  ( ph  ->  ( P ^ N
)  e.  NN )
8483nnzd 10956 . . . . 5  |-  ( ph  ->  ( P ^ N
)  e.  ZZ )
85 dvdsle 13881 . . . . 5  |-  ( ( ( P ^ N
)  e.  ZZ  /\  ( # `  H )  e.  NN )  -> 
( ( P ^ N )  ||  ( # `
 H )  -> 
( P ^ N
)  <_  ( # `  H
) ) )
8684, 67, 85syl2anc 661 . . . 4  |-  ( ph  ->  ( ( P ^ N )  ||  ( # `
 H )  -> 
( P ^ N
)  <_  ( # `  H
) ) )
8780, 86mpd 15 . . 3  |-  ( ph  ->  ( P ^ N
)  <_  ( # `  H
) )
88 hashcl 12385 . . . . . 6  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
8964, 88syl 16 . . . . 5  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
9089nn0red 10844 . . . 4  |-  ( ph  ->  ( # `  H
)  e.  RR )
9183nnred 10542 . . . 4  |-  ( ph  ->  ( P ^ N
)  e.  RR )
9290, 91letri3d 9717 . . 3  |-  ( ph  ->  ( ( # `  H
)  =  ( P ^ N )  <->  ( ( # `
 H )  <_ 
( P ^ N
)  /\  ( P ^ N )  <_  ( # `
 H ) ) ) )
9316, 87, 92mpbir2and 915 . 2  |-  ( ph  ->  ( # `  H
)  =  ( P ^ N ) )
94 fveq2 5859 . . . 4  |-  ( h  =  H  ->  ( # `
 h )  =  ( # `  H
) )
9594eqeq1d 2464 . . 3  |-  ( h  =  H  ->  (
( # `  h )  =  ( P ^ N )  <->  ( # `  H
)  =  ( P ^ N ) ) )
9695rspcev 3209 . 2  |-  ( ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ N
) )  ->  E. h  e.  (SubGrp `  G )
( # `  h )  =  ( P ^ N ) )
9714, 93, 96syl2anc 661 1  |-  ( ph  ->  E. h  e.  (SubGrp `  G ) ( # `  h )  =  ( P ^ N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657   E.wrex 2810   {crab 2813    C_ wss 3471   (/)c0 3780   ~Pcpw 4005   {cpr 4024   class class class wbr 4442   {copab 4499    |-> cmpt 4500   dom cdm 4994   ran crn 4995   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279    Er wer 7300   [cec 7301   Fincfn 7508   RRcr 9482   0cc0 9483    + caddc 9486    x. cmul 9488    <_ cle 9620    - cmin 9796   NNcn 10527   NN0cn0 10786   ZZcz 10855   ^cexp 12124   #chash 12362    || cdivides 13838   Primecprime 14067    pCnt cpc 14210   Basecbs 14481   +g cplusg 14546   0gc0g 14686   Grpcgrp 15718  SubGrpcsubg 15985   ~QG cqg 15987    GrpAct cga 16117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-disj 4413  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-ec 7305  df-qs 7309  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-q 11174  df-rp 11212  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-clim 13262  df-sum 13460  df-dvds 13839  df-gcd 13995  df-prm 14068  df-pc 14211  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-0g 14688  df-mnd 15723  df-grp 15853  df-minusg 15854  df-subg 15988  df-eqg 15990  df-ga 16118
This theorem is referenced by:  sylow1  16414
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