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Theorem sylow1lem5 16600
Description: Lemma for sylow1 16601. 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 16597 . . 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 16325 . . 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 16599 . . 3  |-  ( ph  ->  ( # `  H
)  <_  ( P ^ N ) )
17 sylow1lem5.l . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  <_  (
( P  pCnt  ( # `
 X ) )  -  N ) )
1815, 1gaorber 16324 . . . . . . . . . . . . . . . 16  |-  (  .(+)  e.  ( G  GrpAct  S )  ->  .~  Er  S
)
1910, 18syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  .~  Er  S )
20 erdm 7323 . . . . . . . . . . . . . . 15  |-  (  .~  Er  S  ->  dom  .~  =  S )
2119, 20syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  .~  =  S )
2211, 21eleqtrrd 2534 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  dom  .~  )
23 ecdmn0 7356 . . . . . . . . . . . . 13  |-  ( B  e.  dom  .~  <->  [ B ]  .~  =/=  (/) )
2422, 23sylib 196 . . . . . . . . . . . 12  |-  ( ph  ->  [ B ]  .~  =/=  (/) )
25 pwfi 7817 . . . . . . . . . . . . . . . 16  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
263, 25sylib 196 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ~P X  e.  Fin )
27 ssrab2 3570 . . . . . . . . . . . . . . . 16  |-  { s  e.  ~P X  | 
( # `  s )  =  ( P ^ N ) }  C_  ~P X
288, 27eqsstri 3519 . . . . . . . . . . . . . . 15  |-  S  C_  ~P X
29 ssfi 7742 . . . . . . . . . . . . . . 15  |-  ( ( ~P X  e.  Fin  /\  S  C_  ~P X
)  ->  S  e.  Fin )
3026, 28, 29sylancl 662 . . . . . . . . . . . . . 14  |-  ( ph  ->  S  e.  Fin )
3119ecss 7355 . . . . . . . . . . . . . 14  |-  ( ph  ->  [ B ]  .~  C_  S )
32 ssfi 7742 . . . . . . . . . . . . . 14  |-  ( ( S  e.  Fin  /\  [ B ]  .~  C_  S
)  ->  [ B ]  .~  e.  Fin )
3330, 31, 32syl2anc 661 . . . . . . . . . . . . 13  |-  ( ph  ->  [ B ]  .~  e.  Fin )
34 hashnncl 12417 . . . . . . . . . . . . 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 14355 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  e.  NN0 )
3837nn0red 10860 . . . . . . . . 9  |-  ( ph  ->  ( P  pCnt  ( # `
 [ B ]  .~  ) )  e.  RR )
395nn0red 10860 . . . . . . . . 9  |-  ( ph  ->  N  e.  RR )
401grpbn0 16057 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  X  =/=  (/) )
412, 40syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  X  =/=  (/) )
42 hashnncl 12417 . . . . . . . . . . . . 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 14355 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  NN0 )
4645nn0red 10860 . . . . . . . . 9  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  RR )
47 leaddsub 10035 . . . . . . . . 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 1229 . . . . . . . 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 2443 . . . . . . . . . . 11  |-  ( G ~QG  H )  =  ( G ~QG  H )
511, 12, 50, 15orbsta2 16330 . . . . . . . . . 10  |-  ( ( (  .(+)  e.  ( G  GrpAct  S )  /\  B  e.  S )  /\  X  e.  Fin )  ->  ( # `  X
)  =  ( (
# `  [ B ]  .~  )  x.  ( # `
 H ) ) )
5210, 11, 3, 51syl21anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  [ B ]  .~  )  x.  ( # `
 H ) ) )
5352oveq2d 6297 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  =  ( P  pCnt  ( ( # `  [ B ]  .~  )  x.  ( # `  H
) ) ) )
5436nnzd 10974 . . . . . . . . 9  |-  ( ph  ->  ( # `  [ B ]  .~  )  e.  ZZ )
5536nnne0d 10587 . . . . . . . . 9  |-  ( ph  ->  ( # `  [ B ]  .~  )  =/=  0 )
56 eqid 2443 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  =  ( 0g `  G
)
5756subg0cl 16187 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  H
)
5814, 57syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0g `  G
)  e.  H )
59 ne0i 3776 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  H  ->  H  =/=  (/) )
6058, 59syl 16 . . . . . . . . . . 11  |-  ( ph  ->  H  =/=  (/) )
61 ssrab2 3570 . . . . . . . . . . . . . 14  |-  { u  e.  X  |  (
u  .(+)  B )  =  B }  C_  X
6212, 61eqsstri 3519 . . . . . . . . . . . . 13  |-  H  C_  X
63 ssfi 7742 . . . . . . . . . . . . 13  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
643, 62, 63sylancl 662 . . . . . . . . . . . 12  |-  ( ph  ->  H  e.  Fin )
65 hashnncl 12417 . . . . . . . . . . . 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 10974 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
6967nnne0d 10587 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  =/=  0 )
70 pcmul 14356 . . . . . . . . 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 1238 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  (
( # `  [ B ]  .~  )  x.  ( # `
 H ) ) )  =  ( ( P  pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) )
7253, 71eqtrd 2484 . . . . . . 7  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  =  ( ( P 
pCnt  ( # `  [ B ]  .~  )
)  +  ( P 
pCnt  ( # `  H
) ) ) )
7349, 72breqtrd 4461 . . . . . 6  |-  ( ph  ->  ( ( P  pCnt  (
# `  [ B ]  .~  ) )  +  N )  <_  (
( P  pCnt  ( # `
 [ B ]  .~  ) )  +  ( P  pCnt  ( # `  H
) ) ) )
744, 67pccld 14355 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( # `
 H ) )  e.  NN0 )
7574nn0red 10860 . . . . . . 7  |-  ( ph  ->  ( P  pCnt  ( # `
 H ) )  e.  RR )
7639, 75, 38leadd2d 10154 . . . . . 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 14373 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( # `
 H )  e.  ZZ  /\  N  e. 
NN0 )  ->  ( N  <_  ( P  pCnt  (
# `  H )
)  <->  ( P ^ N )  ||  ( # `
 H ) ) )
794, 68, 5, 78syl3anc 1229 . . . . 5  |-  ( ph  ->  ( N  <_  ( P  pCnt  ( # `  H
) )  <->  ( P ^ N )  ||  ( # `
 H ) ) )
8077, 79mpbid 210 . . . 4  |-  ( ph  ->  ( P ^ N
)  ||  ( # `  H
) )
81 prmnn 14201 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
824, 81syl 16 . . . . . . 7  |-  ( ph  ->  P  e.  NN )
8382, 5nnexpcld 12312 . . . . . 6  |-  ( ph  ->  ( P ^ N
)  e.  NN )
8483nnzd 10974 . . . . 5  |-  ( ph  ->  ( P ^ N
)  e.  ZZ )
85 dvdsle 14012 . . . . 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 12409 . . . . . 6  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
8964, 88syl 16 . . . . 5  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
9089nn0red 10860 . . . 4  |-  ( ph  ->  ( # `  H
)  e.  RR )
9183nnred 10558 . . . 4  |-  ( ph  ->  ( P ^ N
)  e.  RR )
9290, 91letri3d 9730 . . 3  |-  ( ph  ->  ( ( # `  H
)  =  ( P ^ N )  <->  ( ( # `
 H )  <_ 
( P ^ N
)  /\  ( P ^ N )  <_  ( # `
 H ) ) ) )
9316, 87, 92mpbir2and 922 . 2  |-  ( ph  ->  ( # `  H
)  =  ( P ^ N ) )
94 fveq2 5856 . . . 4  |-  ( h  =  H  ->  ( # `
 h )  =  ( # `  H
) )
9594eqeq1d 2445 . . 3  |-  ( h  =  H  ->  (
( # `  h )  =  ( P ^ N )  <->  ( # `  H
)  =  ( P ^ N ) ) )
9695rspcev 3196 . 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 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794   {crab 2797    C_ wss 3461   (/)c0 3770   ~Pcpw 3997   {cpr 4016   class class class wbr 4437   {copab 4494    |-> cmpt 4495   dom cdm 4989   ran crn 4990   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283    Er wer 7310   [cec 7311   Fincfn 7518   RRcr 9494   0cc0 9495    + caddc 9498    x. cmul 9500    <_ cle 9632    - cmin 9810   NNcn 10543   NN0cn0 10802   ZZcz 10871   ^cexp 12147   #chash 12386    || cdvds 13967   Primecprime 14198    pCnt cpc 14341   Basecbs 14613   +g cplusg 14678   0gc0g 14818   Grpcgrp 16031  SubGrpcsubg 16173   ~QG cqg 16175    GrpAct cga 16305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-ec 7315  df-qs 7319  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11092  df-q 11193  df-rp 11231  df-fz 11683  df-fzo 11806  df-fl 11910  df-mod 11978  df-seq 12089  df-exp 12148  df-hash 12387  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050  df-clim 13292  df-sum 13490  df-dvds 13968  df-gcd 14126  df-prm 14199  df-pc 14342  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-0g 14820  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-grp 16035  df-minusg 16036  df-subg 16176  df-eqg 16178  df-ga 16306
This theorem is referenced by:  sylow1  16601
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