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Theorem sylow3lem6 16448
Description: Lemma for sylow3 16449, second part. Using the lemma sylow2a 16435, show that the number of sylow subgroups is equivalent  mod  P to the number of fixed points under the group action. But  K is the unique element of the set of Sylow subgroups that is fixed under the group action, so there is exactly one fixed point and so  ( ( # `  ( P pSyl  G ) )  mod  P )  =  1. (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypotheses
Ref Expression
sylow3.x  |-  X  =  ( Base `  G
)
sylow3.g  |-  ( ph  ->  G  e.  Grp )
sylow3.xf  |-  ( ph  ->  X  e.  Fin )
sylow3.p  |-  ( ph  ->  P  e.  Prime )
sylow3lem5.a  |-  .+  =  ( +g  `  G )
sylow3lem5.d  |-  .-  =  ( -g `  G )
sylow3lem5.k  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
sylow3lem5.m  |-  .(+)  =  ( x  e.  K , 
y  e.  ( P pSyl 
G )  |->  ran  (
z  e.  y  |->  ( ( x  .+  z
)  .-  x )
) )
sylow3lem6.n  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  s  <-> 
( y  .+  x
)  e.  s ) }
Assertion
Ref Expression
sylow3lem6  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  mod 
P )  =  1 )
Distinct variable groups:    x, y,
z,  .-    x, s, y, z,  .(+)    K, s, x, y, z    z, N   
x, X, y, z    G, s, x, y, z    ph, s, x, y, z   
x,  .+ , y, z    P, s, x, y, z
Allowed substitution hints:    .+ ( s)    .- ( s)    N( x, y, s)    X( s)

Proof of Theorem sylow3lem6
Dummy variables  w  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . 5  |-  ( Base `  ( Gs  K ) )  =  ( Base `  ( Gs  K ) )
2 sylow3.x . . . . . 6  |-  X  =  ( Base `  G
)
3 sylow3.g . . . . . 6  |-  ( ph  ->  G  e.  Grp )
4 sylow3.xf . . . . . 6  |-  ( ph  ->  X  e.  Fin )
5 sylow3.p . . . . . 6  |-  ( ph  ->  P  e.  Prime )
6 sylow3lem5.a . . . . . 6  |-  .+  =  ( +g  `  G )
7 sylow3lem5.d . . . . . 6  |-  .-  =  ( -g `  G )
8 sylow3lem5.k . . . . . 6  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
9 sylow3lem5.m . . . . . 6  |-  .(+)  =  ( x  e.  K , 
y  e.  ( P pSyl 
G )  |->  ran  (
z  e.  y  |->  ( ( x  .+  z
)  .-  x )
) )
102, 3, 4, 5, 6, 7, 8, 9sylow3lem5 16447 . . . . 5  |-  ( ph  -> 
.(+)  e.  ( ( Gs  K )  GrpAct  ( P pSyl 
G ) ) )
11 eqid 2467 . . . . . . 7  |-  ( Gs  K )  =  ( Gs  K )
1211slwpgp 16429 . . . . . 6  |-  ( K  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  K ) )
138, 12syl 16 . . . . 5  |-  ( ph  ->  P pGrp  ( Gs  K ) )
14 slwsubg 16426 . . . . . . . 8  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
158, 14syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
1611subgbas 16000 . . . . . . 7  |-  ( K  e.  (SubGrp `  G
)  ->  K  =  ( Base `  ( Gs  K
) ) )
1715, 16syl 16 . . . . . 6  |-  ( ph  ->  K  =  ( Base `  ( Gs  K ) ) )
182subgss 15997 . . . . . . . 8  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
1915, 18syl 16 . . . . . . 7  |-  ( ph  ->  K  C_  X )
20 ssfi 7737 . . . . . . 7  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
214, 19, 20syl2anc 661 . . . . . 6  |-  ( ph  ->  K  e.  Fin )
2217, 21eqeltrrd 2556 . . . . 5  |-  ( ph  ->  ( Base `  ( Gs  K ) )  e. 
Fin )
23 pwfi 7811 . . . . . . 7  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
244, 23sylib 196 . . . . . 6  |-  ( ph  ->  ~P X  e.  Fin )
25 slwsubg 16426 . . . . . . . . 9  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  (SubGrp `  G ) )
262subgss 15997 . . . . . . . . 9  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  X
)
2725, 26syl 16 . . . . . . . 8  |-  ( x  e.  ( P pSyl  G
)  ->  x  C_  X
)
28 selpw 4017 . . . . . . . 8  |-  ( x  e.  ~P X  <->  x  C_  X
)
2927, 28sylibr 212 . . . . . . 7  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  ~P X )
3029ssriv 3508 . . . . . 6  |-  ( P pSyl 
G )  C_  ~P X
31 ssfi 7737 . . . . . 6  |-  ( ( ~P X  e.  Fin  /\  ( P pSyl  G ) 
C_  ~P X )  -> 
( P pSyl  G )  e.  Fin )
3224, 30, 31sylancl 662 . . . . 5  |-  ( ph  ->  ( P pSyl  G )  e.  Fin )
33 eqid 2467 . . . . 5  |-  { s  e.  ( P pSyl  G
)  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s }  =  { s  e.  ( P pSyl  G )  | 
A. g  e.  (
Base `  ( Gs  K
) ) ( g 
.(+)  s )  =  s }
34 eqid 2467 . . . . 5  |-  { <. z ,  w >.  |  ( { z ,  w }  C_  ( P pSyl  G
)  /\  E. h  e.  ( Base `  ( Gs  K ) ) ( h  .(+)  z )  =  w ) }  =  { <. z ,  w >.  |  ( { z ,  w }  C_  ( P pSyl  G )  /\  E. h  e.  (
Base `  ( Gs  K
) ) ( h 
.(+)  z )  =  w ) }
351, 10, 13, 22, 32, 33, 34sylow2a 16435 . . . 4  |-  ( ph  ->  P  ||  ( (
# `  ( P pSyl  G ) )  -  ( # `
 { s  e.  ( P pSyl  G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } ) ) )
36 eqcom 2476 . . . . . . . . . . . . . 14  |-  ( ran  ( z  e.  s 
|->  ( ( g  .+  z )  .-  g
) )  =  s  <-> 
s  =  ran  (
z  e.  s  |->  ( ( g  .+  z
)  .-  g )
) )
3719adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  K  C_  X
)
3837sselda 3504 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  g  e.  X )
3938biantrurd 508 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
s  =  ran  (
z  e.  s  |->  ( ( g  .+  z
)  .-  g )
)  <->  ( g  e.  X  /\  s  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) ) ) )
4036, 39syl5bb 257 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  ( ran  ( z  e.  s 
|->  ( ( g  .+  z )  .-  g
) )  =  s  <-> 
( g  e.  X  /\  s  =  ran  ( z  e.  s 
|->  ( ( g  .+  z )  .-  g
) ) ) ) )
41 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  g  e.  K )
42 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  s  e.  ( P pSyl  G ) )
43 simpr 461 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  g  /\  y  =  s )  ->  y  =  s )
44 simpl 457 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  g  /\  y  =  s )  ->  x  =  g )
4544oveq1d 6297 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  g  /\  y  =  s )  ->  ( x  .+  z
)  =  ( g 
.+  z ) )
4645, 44oveq12d 6300 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  g  /\  y  =  s )  ->  ( ( x  .+  z )  .-  x
)  =  ( ( g  .+  z ) 
.-  g ) )
4743, 46mpteq12dv 4525 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  g  /\  y  =  s )  ->  ( z  e.  y 
|->  ( ( x  .+  z )  .-  x
) )  =  ( z  e.  s  |->  ( ( g  .+  z
)  .-  g )
) )
4847rneqd 5228 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  g  /\  y  =  s )  ->  ran  ( z  e.  y  |->  ( ( x 
.+  z )  .-  x ) )  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) )
49 vex 3116 . . . . . . . . . . . . . . . . . 18  |-  s  e. 
_V
5049mptex 6129 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  s  |->  ( ( g  .+  z ) 
.-  g ) )  e.  _V
5150rnex 6715 . . . . . . . . . . . . . . . 16  |-  ran  (
z  e.  s  |->  ( ( g  .+  z
)  .-  g )
)  e.  _V
5248, 9, 51ovmpt2a 6415 . . . . . . . . . . . . . . 15  |-  ( ( g  e.  K  /\  s  e.  ( P pSyl  G ) )  ->  (
g  .(+)  s )  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) )
5341, 42, 52syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
g  .(+)  s )  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) )
5453eqeq1d 2469 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
( g  .(+)  s )  =  s  <->  ran  ( z  e.  s  |->  ( ( g  .+  z ) 
.-  g ) )  =  s ) )
55 slwsubg 16426 . . . . . . . . . . . . . . 15  |-  ( s  e.  ( P pSyl  G
)  ->  s  e.  (SubGrp `  G ) )
5655ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  s  e.  (SubGrp `  G )
)
57 eqid 2467 . . . . . . . . . . . . . . 15  |-  ( z  e.  s  |->  ( ( g  .+  z ) 
.-  g ) )  =  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) )
58 sylow3lem6.n . . . . . . . . . . . . . . 15  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  s  <-> 
( y  .+  x
)  e.  s ) }
592, 6, 7, 57, 58conjnmzb 16096 . . . . . . . . . . . . . 14  |-  ( s  e.  (SubGrp `  G
)  ->  ( g  e.  N  <->  ( g  e.  X  /\  s  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) ) ) )
6056, 59syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
g  e.  N  <->  ( g  e.  X  /\  s  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) ) ) )
6140, 54, 603bitr4d 285 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
( g  .(+)  s )  =  s  <->  g  e.  N ) )
6261ralbidva 2900 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  K  (
g  .(+)  s )  =  s  <->  A. g  e.  K  g  e.  N )
)
63 dfss3 3494 . . . . . . . . . . 11  |-  ( K 
C_  N  <->  A. g  e.  K  g  e.  N )
6462, 63syl6bbr 263 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  K  (
g  .(+)  s )  =  s  <->  K  C_  N ) )
6517adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  K  =  ( Base `  ( Gs  K
) ) )
6665raleqdv 3064 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  K  (
g  .(+)  s )  =  s  <->  A. g  e.  (
Base `  ( Gs  K
) ) ( g 
.(+)  s )  =  s ) )
67 eqid 2467 . . . . . . . . . . . . 13  |-  ( Base `  ( Gs  N ) )  =  ( Base `  ( Gs  N ) )
683ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  G  e.  Grp )
6958, 2, 6nmzsubg 16037 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
7068, 69syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  e.  (SubGrp `  G )
)
71 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( Gs  N )  =  ( Gs  N )
7271subgbas 16000 . . . . . . . . . . . . . . 15  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  ( Gs  N
) ) )
7370, 72syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  =  ( Base `  ( Gs  N ) ) )
744ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  X  e.  Fin )
752subgss 15997 . . . . . . . . . . . . . . . 16  |-  ( N  e.  (SubGrp `  G
)  ->  N  C_  X
)
7670, 75syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  C_  X )
77 ssfi 7737 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  Fin  /\  N  C_  X )  ->  N  e.  Fin )
7874, 76, 77syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  e.  Fin )
7973, 78eqeltrrd 2556 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  ( Base `  ( Gs  N ) )  e.  Fin )
808ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  K  e.  ( P pSyl  G ) )
81 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  K  C_  N )
8271subgslw 16432 . . . . . . . . . . . . . 14  |-  ( ( N  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  N
)  ->  K  e.  ( P pSyl  ( Gs  N
) ) )
8370, 80, 81, 82syl3anc 1228 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  K  e.  ( P pSyl  ( Gs  N ) ) )
84 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  ( P pSyl  G ) )
8555ad2antlr 726 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  (SubGrp `  G )
)
8658, 2, 6ssnmz 16038 . . . . . . . . . . . . . . 15  |-  ( s  e.  (SubGrp `  G
)  ->  s  C_  N )
8785, 86syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  C_  N )
8871subgslw 16432 . . . . . . . . . . . . . 14  |-  ( ( N  e.  (SubGrp `  G )  /\  s  e.  ( P pSyl  G )  /\  s  C_  N
)  ->  s  e.  ( P pSyl  ( Gs  N
) ) )
8970, 84, 87, 88syl3anc 1228 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  ( P pSyl  ( Gs  N ) ) )
90 fvex 5874 . . . . . . . . . . . . . . . . 17  |-  ( Base `  G )  e.  _V
912, 90eqeltri 2551 . . . . . . . . . . . . . . . 16  |-  X  e. 
_V
9291rabex 4598 . . . . . . . . . . . . . . 15  |-  { x  e.  X  |  A. y  e.  X  (
( x  .+  y
)  e.  s  <->  ( y  .+  x )  e.  s ) }  e.  _V
9358, 92eqeltri 2551 . . . . . . . . . . . . . 14  |-  N  e. 
_V
9471, 6ressplusg 14593 . . . . . . . . . . . . . 14  |-  ( N  e.  _V  ->  .+  =  ( +g  `  ( Gs  N ) ) )
9593, 94ax-mp 5 . . . . . . . . . . . . 13  |-  .+  =  ( +g  `  ( Gs  N ) )
96 eqid 2467 . . . . . . . . . . . . 13  |-  ( -g `  ( Gs  N ) )  =  ( -g `  ( Gs  N ) )
9767, 79, 83, 89, 95, 96sylow2 16442 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  E. g  e.  ( Base `  ( Gs  N ) ) K  =  ran  ( z  e.  s  |->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) ) )
9858, 2, 6, 71nmznsg 16040 . . . . . . . . . . . . . . . 16  |-  ( s  e.  (SubGrp `  G
)  ->  s  e.  (NrmSGrp `  ( Gs  N ) ) )
9985, 98syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  (NrmSGrp `  ( Gs  N
) ) )
100 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( z  e.  s  |->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) )  =  ( z  e.  s  |->  ( ( g  .+  z
) ( -g `  ( Gs  N ) ) g ) )
10167, 95, 96, 100conjnsg 16097 . . . . . . . . . . . . . . 15  |-  ( ( s  e.  (NrmSGrp `  ( Gs  N ) )  /\  g  e.  ( Base `  ( Gs  N ) ) )  ->  s  =  ran  ( z  e.  s 
|->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) ) )
10299, 101sylan 471 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  /\  g  e.  ( Base `  ( Gs  N ) ) )  ->  s  =  ran  ( z  e.  s 
|->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) ) )
103 eqeq2 2482 . . . . . . . . . . . . . 14  |-  ( K  =  ran  ( z  e.  s  |->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) )  ->  (
s  =  K  <->  s  =  ran  ( z  e.  s 
|->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) ) ) )
104102, 103syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  /\  g  e.  ( Base `  ( Gs  N ) ) )  ->  ( K  =  ran  ( z  e.  s  |->  ( ( g 
.+  z ) (
-g `  ( Gs  N
) ) g ) )  ->  s  =  K ) )
105104rexlimdva 2955 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  ( E. g  e.  ( Base `  ( Gs  N ) ) K  =  ran  ( z  e.  s 
|->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) )  ->  s  =  K ) )
10697, 105mpd 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  =  K )
107 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  s  =  K )
10815ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  K  e.  (SubGrp `  G )
)
109107, 108eqeltrd 2555 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  s  e.  (SubGrp `  G )
)
110109, 86syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  s  C_  N )
111107, 110eqsstr3d 3539 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  K  C_  N )
112106, 111impbida 830 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( K  C_  N  <->  s  =  K ) )
11364, 66, 1123bitr3d 283 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s  <->  s  =  K ) )
114113rabbidva 3104 . . . . . . . 8  |-  ( ph  ->  { s  e.  ( P pSyl  G )  | 
A. g  e.  (
Base `  ( Gs  K
) ) ( g 
.(+)  s )  =  s }  =  {
s  e.  ( P pSyl 
G )  |  s  =  K } )
115 rabsn 4094 . . . . . . . . 9  |-  ( K  e.  ( P pSyl  G
)  ->  { s  e.  ( P pSyl  G )  |  s  =  K }  =  { K } )
1168, 115syl 16 . . . . . . . 8  |-  ( ph  ->  { s  e.  ( P pSyl  G )  |  s  =  K }  =  { K } )
117114, 116eqtrd 2508 . . . . . . 7  |-  ( ph  ->  { s  e.  ( P pSyl  G )  | 
A. g  e.  (
Base `  ( Gs  K
) ) ( g 
.(+)  s )  =  s }  =  { K } )
118117fveq2d 5868 . . . . . 6  |-  ( ph  ->  ( # `  {
s  e.  ( P pSyl 
G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } )  =  ( # `  { K } ) )
119 hashsng 12402 . . . . . . 7  |-  ( K  e.  ( P pSyl  G
)  ->  ( # `  { K } )  =  1 )
1208, 119syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { K } )  =  1 )
121118, 120eqtrd 2508 . . . . 5  |-  ( ph  ->  ( # `  {
s  e.  ( P pSyl 
G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } )  =  1 )
122121oveq2d 6298 . . . 4  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  -  ( # `  { s  e.  ( P pSyl  G
)  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } ) )  =  ( (
# `  ( P pSyl  G ) )  -  1 ) )
12335, 122breqtrd 4471 . . 3  |-  ( ph  ->  P  ||  ( (
# `  ( P pSyl  G ) )  -  1 ) )
124 prmnn 14075 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
1255, 124syl 16 . . . 4  |-  ( ph  ->  P  e.  NN )
126 hashcl 12392 . . . . . 6  |-  ( ( P pSyl  G )  e. 
Fin  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
12732, 126syl 16 . . . . 5  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
128127nn0zd 10960 . . . 4  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e.  ZZ )
129 1zzd 10891 . . . 4  |-  ( ph  ->  1  e.  ZZ )
130 moddvds 13850 . . . 4  |-  ( ( P  e.  NN  /\  ( # `  ( P pSyl 
G ) )  e.  ZZ  /\  1  e.  ZZ )  ->  (
( ( # `  ( P pSyl  G ) )  mod 
P )  =  ( 1  mod  P )  <-> 
P  ||  ( ( # `
 ( P pSyl  G
) )  -  1 ) ) )
131125, 128, 129, 130syl3anc 1228 . . 3  |-  ( ph  ->  ( ( ( # `  ( P pSyl  G ) )  mod  P )  =  ( 1  mod 
P )  <->  P  ||  (
( # `  ( P pSyl 
G ) )  - 
1 ) ) )
132123, 131mpbird 232 . 2  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  mod 
P )  =  ( 1  mod  P ) )
133 prmuz2 14090 . . 3  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
134 eluz2b2 11150 . . . 4  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
135 nnre 10539 . . . . 5  |-  ( P  e.  NN  ->  P  e.  RR )
136 1mod 11992 . . . . 5  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
137135, 136sylan 471 . . . 4  |-  ( ( P  e.  NN  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
138134, 137sylbi 195 . . 3  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( 1  mod  P )  =  1 )
1395, 133, 1383syl 20 . 2  |-  ( ph  ->  ( 1  mod  P
)  =  1 )
140132, 139eqtrd 2508 1  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  mod 
P )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   {csn 4027   {cpr 4029   class class class wbr 4447   {copab 4504    |-> cmpt 4505   ran crn 5000   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   Fincfn 7513   RRcr 9487   1c1 9489    < clt 9624    - cmin 9801   NNcn 10532   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078    mod cmo 11960   #chash 12369    || cdivides 13843   Primecprime 14072   Basecbs 14486   ↾s cress 14487   +g cplusg 14551   Grpcgrp 15723   -gcsg 15726  SubGrpcsubg 15990  NrmSGrpcnsg 15991   pGrp cpgp 16347   pSyl cslw 16348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-ec 7310  df-qs 7314  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-dvds 13844  df-gcd 14000  df-prm 14073  df-pc 14216  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-0g 14693  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-nsg 15994  df-eqg 15995  df-ghm 16060  df-ga 16123  df-od 16349  df-pgp 16351  df-slw 16352
This theorem is referenced by:  sylow3  16449
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