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Theorem sylow1lem5 15238
Description: Lemma for sylow1 15239. 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 15235 . . 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 15088 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  S )  /\  B  e.  S )  ->  H  e.  (SubGrp `  G )
)
1410, 11, 13syl2anc 644 . 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 15237 . . 3  |-  ( ph  ->  ( # `  H
)  <_  ( P ^ N ) )
17 sylow1lem5.l . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  <_  (
( P  pCnt  ( # `
 X ) )  -  N ) )
1815, 1gaorber 15087 . . . . . . . . . . . . . . . 16  |-  (  .(+)  e.  ( G  GrpAct  S )  ->  .~  Er  S
)
1910, 18syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  .~  Er  S )
20 erdm 6917 . . . . . . . . . . . . . . 15  |-  (  .~  Er  S  ->  dom  .~  =  S )
2119, 20syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  .~  =  S )
2211, 21eleqtrrd 2515 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  dom  .~  )
23 ecdmn0 6949 . . . . . . . . . . . . 13  |-  ( B  e.  dom  .~  <->  [ B ]  .~  =/=  (/) )
2422, 23sylib 190 . . . . . . . . . . . 12  |-  ( ph  ->  [ B ]  .~  =/=  (/) )
25 pwfi 7404 . . . . . . . . . . . . . . . 16  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
263, 25sylib 190 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ~P X  e.  Fin )
27 ssrab2 3430 . . . . . . . . . . . . . . . 16  |-  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  C_  ~P X
288, 27eqsstri 3380 . . . . . . . . . . . . . . 15  |-  S  C_  ~P X
29 ssfi 7331 . . . . . . . . . . . . . . 15  |-  ( ( ~P X  e.  Fin  /\  S  C_  ~P X
)  ->  S  e.  Fin )
3026, 28, 29sylancl 645 . . . . . . . . . . . . . 14  |-  ( ph  ->  S  e.  Fin )
3119ecss 6948 . . . . . . . . . . . . . 14  |-  ( ph  ->  [ B ]  .~  C_  S )
32 ssfi 7331 . . . . . . . . . . . . . 14  |-  ( ( S  e.  Fin  /\  [ B ]  .~  C_  S
)  ->  [ B ]  .~  e.  Fin )
3330, 31, 32syl2anc 644 . . . . . . . . . . . . 13  |-  ( ph  ->  [ B ]  .~  e.  Fin )
34 hashnncl 11647 . . . . . . . . . . . . 13  |-  ( [ B ]  .~  e.  Fin  ->  ( ( # `  [ B ]  .~  )  e.  NN  <->  [ B ]  .~  =/=  (/) ) )
3533, 34syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  [ B ]  .~  )  e.  NN  <->  [ B ]  .~  =/=  (/) ) )
3624, 35mpbird 225 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  [ B ]  .~  )  e.  NN )
374, 36pccld 13226 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  e.  NN0 )
3837nn0red 10277 . . . . . . . . 9  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  e.  RR )
395nn0red 10277 . . . . . . . . 9  |-  ( ph  ->  N  e.  RR )
401grpbn0 14836 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  X  =/=  (/) )
412, 40syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  X  =/=  (/) )
42 hashnncl 11647 . . . . . . . . . . . . 13  |-  ( X  e.  Fin  ->  (
( # `  X )  e.  NN  <->  X  =/=  (/) ) )
433, 42syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  X
)  e.  NN  <->  X  =/=  (/) ) )
4441, 43mpbird 225 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  X
)  e.  NN )
454, 44pccld 13226 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  NN0 )
4645nn0red 10277 . . . . . . . . 9  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  RR )
47 leaddsub 9506 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
pCnt  ( # `  [ B ]  .~  )
)  +  N )  <_  ( P  pCnt  (
# `  X )
)  <->  ( P  pCnt  (
# `  [ B ]  .~  ) )  <_ 
( ( P  pCnt  (
# `  X )
)  -  N ) ) )
4917, 48mpbird 225 . . . . . . 7  |-  ( ph  ->  ( ( P  pCnt  (
# `  [ B ]  .~  ) )  +  N )  <_  ( P  pCnt  ( # `  X
) ) )
50 eqid 2438 . . . . . . . . . . 11  |-  ( G ~QG  H )  =  ( G ~QG  H )
511, 12, 50, 15orbsta2 15093 . . . . . . . . . 10  |-  ( ( (  .(+)  e.  ( G  GrpAct  S )  /\  B  e.  S )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  [ B ]  .~  )  x.  ( # `
 H ) ) )
5210, 11, 3, 51syl21anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  [ B ]  .~  )  x.  ( # `
 H ) ) )
5352oveq2d 6099 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  =  ( P  pCnt  ( ( # `  [ B ]  .~  )  x.  ( # `  H
) ) ) )
5436nnzd 10376 . . . . . . . . 9  |-  ( ph  ->  ( # `  [ B ]  .~  )  e.  ZZ )
5536nnne0d 10046 . . . . . . . . 9  |-  ( ph  ->  ( # `  [ B ]  .~  )  =/=  0 )
56 eqid 2438 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  =  ( 0g `  G
)
5756subg0cl 14954 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  H
)
5814, 57syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0g `  G
)  e.  H )
59 ne0i 3636 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  H  ->  H  =/=  (/) )
6058, 59syl 16 . . . . . . . . . . 11  |-  ( ph  ->  H  =/=  (/) )
61 ssrab2 3430 . . . . . . . . . . . . . 14  |-  { u  e.  X  |  (
u  .(+)  B )  =  B }  C_  X
6212, 61eqsstri 3380 . . . . . . . . . . . . 13  |-  H  C_  X
63 ssfi 7331 . . . . . . . . . . . . 13  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
643, 62, 63sylancl 645 . . . . . . . . . . . 12  |-  ( ph  ->  H  e.  Fin )
65 hashnncl 11647 . . . . . . . . . . . 12  |-  ( H  e.  Fin  ->  (
( # `  H )  e.  NN  <->  H  =/=  (/) ) )
6664, 65syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  H
)  e.  NN  <->  H  =/=  (/) ) )
6760, 66mpbird 225 . . . . . . . . . 10  |-  ( ph  ->  ( # `  H
)  e.  NN )
6867nnzd 10376 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
6967nnne0d 10046 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  =/=  0 )
70 pcmul 13227 . . . . . . . . 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 1194 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  (
( # `  [ B ]  .~  )  x.  ( # `
 H ) ) )  =  ( ( P  pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) )
7253, 71eqtrd 2470 . . . . . . 7  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  =  ( ( P 
pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) )
7349, 72breqtrd 4238 . . . . . 6  |-  ( ph  ->  ( ( P  pCnt  (
# `  [ B ]  .~  ) )  +  N )  <_  (
( P  pCnt  ( # `
 [ B ]  .~  ) )  +  ( P  pCnt  ( # `  H
) ) ) )
744, 67pccld 13226 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 H ) )  e.  NN0 )
7574nn0red 10277 . . . . . . 7  |-  ( ph  ->  ( P  pCnt  ( # `
 H ) )  e.  RR )
7639, 75, 38leadd2d 9623 . . . . . 6  |-  ( ph  ->  ( N  <_  ( P  pCnt  ( # `  H
) )  <->  ( ( P  pCnt  ( # `  [ B ]  .~  )
)  +  N )  <_  ( ( P 
pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) ) )
7773, 76mpbird 225 . . . . 5  |-  ( ph  ->  N  <_  ( P  pCnt  ( # `  H
) ) )
78 pcdvdsb 13244 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( # `
 H )  e.  ZZ  /\  N  e. 
NN0 )  ->  ( N  <_  ( P  pCnt  (
# `  H )
)  <->  ( P ^ N )  ||  ( # `
 H ) ) )
794, 68, 5, 78syl3anc 1185 . . . . 5  |-  ( ph  ->  ( N  <_  ( P  pCnt  ( # `  H
) )  <->  ( P ^ N )  ||  ( # `
 H ) ) )
8077, 79mpbid 203 . . . 4  |-  ( ph  ->  ( P ^ N
)  ||  ( # `  H
) )
81 prmnn 13084 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
824, 81syl 16 . . . . . . 7  |-  ( ph  ->  P  e.  NN )
8382, 5nnexpcld 11546 . . . . . 6  |-  ( ph  ->  ( P ^ N
)  e.  NN )
8483nnzd 10376 . . . . 5  |-  ( ph  ->  ( P ^ N
)  e.  ZZ )
85 dvdsle 12897 . . . . 5  |-  ( ( ( P ^ N
)  e.  ZZ  /\  ( # `  H )  e.  NN )  -> 
( ( P ^ N )  ||  ( # `
 H )  -> 
( P ^ N
)  <_  ( # `  H
) ) )
8684, 67, 85syl2anc 644 . . . 4  |-  ( ph  ->  ( ( P ^ N )  ||  ( # `
 H )  -> 
( P ^ N
)  <_  ( # `  H
) ) )
8780, 86mpd 15 . . 3  |-  ( ph  ->  ( P ^ N
)  <_  ( # `  H
) )
88 hashcl 11641 . . . . . 6  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
8964, 88syl 16 . . . . 5  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
9089nn0red 10277 . . . 4  |-  ( ph  ->  ( # `  H
)  e.  RR )
9183nnred 10017 . . . 4  |-  ( ph  ->  ( P ^ N
)  e.  RR )
9290, 91letri3d 9217 . . 3  |-  ( ph  ->  ( ( # `  H
)  =  ( P ^ N )  <->  ( ( # `
 H )  <_ 
( P ^ N
)  /\  ( P ^ N )  <_  ( # `
 H ) ) ) )
9316, 87, 92mpbir2and 890 . 2  |-  ( ph  ->  ( # `  H
)  =  ( P ^ N ) )
94 fveq2 5730 . . . 4  |-  ( h  =  H  ->  ( # `
 h )  =  ( # `  H
) )
9594eqeq1d 2446 . . 3  |-  ( h  =  H  ->  (
( # `  h )  =  ( P ^ N )  <->  ( # `  H
)  =  ( P ^ N ) ) )
9695rspcev 3054 . 2  |-  ( ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ N
) )  ->  E. h  e.  (SubGrp `  G )
( # `  h )  =  ( P ^ N ) )
9714, 93, 96syl2anc 644 1  |-  ( ph  ->  E. h  e.  (SubGrp `  G ) ( # `  h )  =  ( P ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {cpr 3817   class class class wbr 4214   {copab 4267    e. cmpt 4268   dom cdm 4880   ran crn 4881   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085    Er wer 6904   [cec 6905   Fincfn 7111   RRcr 8991   0cc0 8992    + caddc 8995    x. cmul 8997    <_ cle 9123    - cmin 9293   NNcn 10002   NN0cn0 10223   ZZcz 10284   ^cexp 11384   #chash 11620    || cdivides 12854   Primecprime 13081    pCnt cpc 13212   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Grpcgrp 14687  SubGrpcsubg 14940   ~QG cqg 14942    GrpAct cga 15068
This theorem is referenced by:  sylow1  15239
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-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-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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