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Theorem sylow3lem6 16136
Description: Lemma for sylow3 16137, second part. Using the lemma sylow2a 16123, 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 2443 . . . . 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 16135 . . . . 5  |-  ( ph  -> 
.(+)  e.  ( ( Gs  K )  GrpAct  ( P pSyl 
G ) ) )
11 eqid 2443 . . . . . . 7  |-  ( Gs  K )  =  ( Gs  K )
1211slwpgp 16117 . . . . . 6  |-  ( K  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  K ) )
138, 12syl 16 . . . . 5  |-  ( ph  ->  P pGrp  ( Gs  K ) )
14 slwsubg 16114 . . . . . . . 8  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
158, 14syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
1611subgbas 15690 . . . . . . 7  |-  ( K  e.  (SubGrp `  G
)  ->  K  =  ( Base `  ( Gs  K
) ) )
1715, 16syl 16 . . . . . 6  |-  ( ph  ->  K  =  ( Base `  ( Gs  K ) ) )
182subgss 15687 . . . . . . . 8  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
1915, 18syl 16 . . . . . . 7  |-  ( ph  ->  K  C_  X )
20 ssfi 7538 . . . . . . 7  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
214, 19, 20syl2anc 661 . . . . . 6  |-  ( ph  ->  K  e.  Fin )
2217, 21eqeltrrd 2518 . . . . 5  |-  ( ph  ->  ( Base `  ( Gs  K ) )  e. 
Fin )
23 pwfi 7611 . . . . . . 7  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
244, 23sylib 196 . . . . . 6  |-  ( ph  ->  ~P X  e.  Fin )
25 slwsubg 16114 . . . . . . . . 9  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  (SubGrp `  G ) )
262subgss 15687 . . . . . . . . 9  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  X
)
2725, 26syl 16 . . . . . . . 8  |-  ( x  e.  ( P pSyl  G
)  ->  x  C_  X
)
28 selpw 3872 . . . . . . . 8  |-  ( x  e.  ~P X  <->  x  C_  X
)
2927, 28sylibr 212 . . . . . . 7  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  ~P X )
3029ssriv 3365 . . . . . 6  |-  ( P pSyl 
G )  C_  ~P X
31 ssfi 7538 . . . . . 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 2443 . . . . 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 2443 . . . . 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 16123 . . . 4  |-  ( ph  ->  P  ||  ( (
# `  ( P pSyl  G ) )  -  ( # `
 { s  e.  ( P pSyl  G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } ) ) )
36 eqcom 2445 . . . . . . . . . . . . . 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 3361 . . . . . . . . . . . . . . 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 6111 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  g  /\  y  =  s )  ->  ( x  .+  z
)  =  ( g 
.+  z ) )
4645, 44oveq12d 6114 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  g  /\  y  =  s )  ->  ( ( x  .+  z )  .-  x
)  =  ( ( g  .+  z ) 
.-  g ) )
4743, 46mpteq12dv 4375 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  g  /\  y  =  s )  ->  ( z  e.  y 
|->  ( ( x  .+  z )  .-  x
) )  =  ( z  e.  s  |->  ( ( g  .+  z
)  .-  g )
) )
4847rneqd 5072 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  g  /\  y  =  s )  ->  ran  ( z  e.  y  |->  ( ( x 
.+  z )  .-  x ) )  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) )
49 vex 2980 . . . . . . . . . . . . . . . . . 18  |-  s  e. 
_V
5049mptex 5953 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  s  |->  ( ( g  .+  z ) 
.-  g ) )  e.  _V
5150rnex 6517 . . . . . . . . . . . . . . . 16  |-  ran  (
z  e.  s  |->  ( ( g  .+  z
)  .-  g )
)  e.  _V
5248, 9, 51ovmpt2a 6226 . . . . . . . . . . . . . . 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 2451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
( g  .(+)  s )  =  s  <->  ran  ( z  e.  s  |->  ( ( g  .+  z ) 
.-  g ) )  =  s ) )
55 slwsubg 16114 . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . 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 15786 . . . . . . . . . . . . . 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 2736 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  K  (
g  .(+)  s )  =  s  <->  A. g  e.  K  g  e.  N )
)
63 dfss3 3351 . . . . . . . . . . 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 2928 . . . . . . . . . 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 2443 . . . . . . . . . . . . 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 15727 . . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . 16  |-  ( Gs  N )  =  ( Gs  N )
7271subgbas 15690 . . . . . . . . . . . . . . 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 15687 . . . . . . . . . . . . . . . 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 7538 . . . . . . . . . . . . . . 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 2518 . . . . . . . . . . . . 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 16120 . . . . . . . . . . . . . 14  |-  ( ( N  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  N
)  ->  K  e.  ( P pSyl  ( Gs  N
) ) )
8370, 80, 81, 82syl3anc 1218 . . . . . . . . . . . . 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 15728 . . . . . . . . . . . . . . 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 16120 . . . . . . . . . . . . . 14  |-  ( ( N  e.  (SubGrp `  G )  /\  s  e.  ( P pSyl  G )  /\  s  C_  N
)  ->  s  e.  ( P pSyl  ( Gs  N
) ) )
8970, 84, 87, 88syl3anc 1218 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  ( P pSyl  ( Gs  N ) ) )
90 fvex 5706 . . . . . . . . . . . . . . . . 17  |-  ( Base `  G )  e.  _V
912, 90eqeltri 2513 . . . . . . . . . . . . . . . 16  |-  X  e. 
_V
9291rabex 4448 . . . . . . . . . . . . . . 15  |-  { x  e.  X  |  A. y  e.  X  (
( x  .+  y
)  e.  s  <->  ( y  .+  x )  e.  s ) }  e.  _V
9358, 92eqeltri 2513 . . . . . . . . . . . . . 14  |-  N  e. 
_V
9471, 6ressplusg 14285 . . . . . . . . . . . . . 14  |-  ( N  e.  _V  ->  .+  =  ( +g  `  ( Gs  N ) ) )
9593, 94ax-mp 5 . . . . . . . . . . . . 13  |-  .+  =  ( +g  `  ( Gs  N ) )
96 eqid 2443 . . . . . . . . . . . . 13  |-  ( -g `  ( Gs  N ) )  =  ( -g `  ( Gs  N ) )
9767, 79, 83, 89, 95, 96sylow2 16130 . . . . . . . . . . . 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 15730 . . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . 16  |-  ( z  e.  s  |->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) )  =  ( z  e.  s  |->  ( ( g  .+  z
) ( -g `  ( Gs  N ) ) g ) )
10167, 95, 96, 100conjnsg 15787 . . . . . . . . . . . . . . 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 2452 . . . . . . . . . . . . . 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 2846 . . . . . . . . . . . 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 2517 . . . . . . . . . . . . 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 3396 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  K  C_  N )
112106, 111impbida 828 . . . . . . . . . 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 2968 . . . . . . . 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 3947 . . . . . . . . 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 2475 . . . . . . 7  |-  ( ph  ->  { s  e.  ( P pSyl  G )  | 
A. g  e.  (
Base `  ( Gs  K
) ) ( g 
.(+)  s )  =  s }  =  { K } )
118117fveq2d 5700 . . . . . 6  |-  ( ph  ->  ( # `  {
s  e.  ( P pSyl 
G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } )  =  ( # `  { K } ) )
119 hashsng 12141 . . . . . . 7  |-  ( K  e.  ( P pSyl  G
)  ->  ( # `  { K } )  =  1 )
1208, 119syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { K } )  =  1 )
121118, 120eqtrd 2475 . . . . 5  |-  ( ph  ->  ( # `  {
s  e.  ( P pSyl 
G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } )  =  1 )
122121oveq2d 6112 . . . 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 4321 . . 3  |-  ( ph  ->  P  ||  ( (
# `  ( P pSyl  G ) )  -  1 ) )
124 prmnn 13771 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
1255, 124syl 16 . . . 4  |-  ( ph  ->  P  e.  NN )
126 hashcl 12131 . . . . . 6  |-  ( ( P pSyl  G )  e. 
Fin  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
12732, 126syl 16 . . . . 5  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
128127nn0zd 10750 . . . 4  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e.  ZZ )
129 1zzd 10682 . . . 4  |-  ( ph  ->  1  e.  ZZ )
130 moddvds 13547 . . . 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 1218 . . 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 13786 . . 3  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
134 eluz2b2 10932 . . . 4  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
135 nnre 10334 . . . . 5  |-  ( P  e.  NN  ->  P  e.  RR )
136 1mod 11745 . . . . 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 2475 1  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  mod 
P )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   {crab 2724   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865   {csn 3882   {cpr 3884   class class class wbr 4297   {copab 4354    e. cmpt 4355   ran crn 4846   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   Fincfn 7315   RRcr 9286   1c1 9288    < clt 9423    - cmin 9600   NNcn 10327   2c2 10376   NN0cn0 10584   ZZcz 10651   ZZ>=cuz 10866    mod cmo 11713   #chash 12108    || cdivides 13540   Primecprime 13768   Basecbs 14179   ↾s cress 14180   +g cplusg 14243   Grpcgrp 15415   -gcsg 15418  SubGrpcsubg 15680  NrmSGrpcnsg 15681   pGrp cpgp 16035   pSyl cslw 16036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-disj 4268  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-omul 6930  df-er 7106  df-ec 7108  df-qs 7112  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-acn 8117  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-bc 12084  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-dvds 13541  df-gcd 13696  df-prm 13769  df-pc 13909  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-subg 15683  df-nsg 15684  df-eqg 15685  df-ghm 15750  df-ga 15813  df-od 16037  df-pgp 16039  df-slw 16040
This theorem is referenced by:  sylow3  16137
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