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Theorem sylow3lem6 17362
Description: Lemma for sylow3 17363, second part. Using the lemma sylow2a 17349, 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 2471 . . . . 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 17361 . . . . 5  |-  ( ph  -> 
.(+)  e.  ( ( Gs  K )  GrpAct  ( P pSyl 
G ) ) )
11 eqid 2471 . . . . . . 7  |-  ( Gs  K )  =  ( Gs  K )
1211slwpgp 17343 . . . . . 6  |-  ( K  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  K ) )
138, 12syl 17 . . . . 5  |-  ( ph  ->  P pGrp  ( Gs  K ) )
14 slwsubg 17340 . . . . . . . 8  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
158, 14syl 17 . . . . . . 7  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
1611subgbas 16899 . . . . . . 7  |-  ( K  e.  (SubGrp `  G
)  ->  K  =  ( Base `  ( Gs  K
) ) )
1715, 16syl 17 . . . . . 6  |-  ( ph  ->  K  =  ( Base `  ( Gs  K ) ) )
182subgss 16896 . . . . . . . 8  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
1915, 18syl 17 . . . . . . 7  |-  ( ph  ->  K  C_  X )
20 ssfi 7810 . . . . . . 7  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
214, 19, 20syl2anc 673 . . . . . 6  |-  ( ph  ->  K  e.  Fin )
2217, 21eqeltrrd 2550 . . . . 5  |-  ( ph  ->  ( Base `  ( Gs  K ) )  e. 
Fin )
23 pwfi 7887 . . . . . . 7  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
244, 23sylib 201 . . . . . 6  |-  ( ph  ->  ~P X  e.  Fin )
25 slwsubg 17340 . . . . . . . . 9  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  (SubGrp `  G ) )
262subgss 16896 . . . . . . . . 9  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  X
)
2725, 26syl 17 . . . . . . . 8  |-  ( x  e.  ( P pSyl  G
)  ->  x  C_  X
)
28 selpw 3949 . . . . . . . 8  |-  ( x  e.  ~P X  <->  x  C_  X
)
2927, 28sylibr 217 . . . . . . 7  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  ~P X )
3029ssriv 3422 . . . . . 6  |-  ( P pSyl 
G )  C_  ~P X
31 ssfi 7810 . . . . . 6  |-  ( ( ~P X  e.  Fin  /\  ( P pSyl  G ) 
C_  ~P X )  -> 
( P pSyl  G )  e.  Fin )
3224, 30, 31sylancl 675 . . . . 5  |-  ( ph  ->  ( P pSyl  G )  e.  Fin )
33 eqid 2471 . . . . 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 2471 . . . . 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 17349 . . . 4  |-  ( ph  ->  P  ||  ( (
# `  ( P pSyl  G ) )  -  ( # `
 { s  e.  ( P pSyl  G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } ) ) )
36 eqcom 2478 . . . . . . . . . . . . . 14  |-  ( ran  ( z  e.  s 
|->  ( ( g  .+  z )  .-  g
) )  =  s  <-> 
s  =  ran  (
z  e.  s  |->  ( ( g  .+  z
)  .-  g )
) )
3719adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  K  C_  X
)
3837sselda 3418 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  g  e.  X )
3938biantrurd 516 . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  g  e.  K )
42 simplr 770 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  s  e.  ( P pSyl  G ) )
43 simpr 468 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  g  /\  y  =  s )  ->  y  =  s )
44 simpl 464 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  g  /\  y  =  s )  ->  x  =  g )
4544oveq1d 6323 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  g  /\  y  =  s )  ->  ( x  .+  z
)  =  ( g 
.+  z ) )
4645, 44oveq12d 6326 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  g  /\  y  =  s )  ->  ( ( x  .+  z )  .-  x
)  =  ( ( g  .+  z ) 
.-  g ) )
4743, 46mpteq12dv 4474 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  g  /\  y  =  s )  ->  ( z  e.  y 
|->  ( ( x  .+  z )  .-  x
) )  =  ( z  e.  s  |->  ( ( g  .+  z
)  .-  g )
) )
4847rneqd 5068 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  g  /\  y  =  s )  ->  ran  ( z  e.  y  |->  ( ( x 
.+  z )  .-  x ) )  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) )
49 vex 3034 . . . . . . . . . . . . . . . . . 18  |-  s  e. 
_V
5049mptex 6152 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  s  |->  ( ( g  .+  z ) 
.-  g ) )  e.  _V
5150rnex 6746 . . . . . . . . . . . . . . . 16  |-  ran  (
z  e.  s  |->  ( ( g  .+  z
)  .-  g )
)  e.  _V
5248, 9, 51ovmpt2a 6446 . . . . . . . . . . . . . . 15  |-  ( ( g  e.  K  /\  s  e.  ( P pSyl  G ) )  ->  (
g  .(+)  s )  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) )
5341, 42, 52syl2anc 673 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
g  .(+)  s )  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) )
5453eqeq1d 2473 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
( g  .(+)  s )  =  s  <->  ran  ( z  e.  s  |->  ( ( g  .+  z ) 
.-  g ) )  =  s ) )
55 slwsubg 17340 . . . . . . . . . . . . . . 15  |-  ( s  e.  ( P pSyl  G
)  ->  s  e.  (SubGrp `  G ) )
5655ad2antlr 741 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  s  e.  (SubGrp `  G )
)
57 eqid 2471 . . . . . . . . . . . . . . 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 16995 . . . . . . . . . . . . . 14  |-  ( s  e.  (SubGrp `  G
)  ->  ( g  e.  N  <->  ( g  e.  X  /\  s  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) ) ) )
6056, 59syl 17 . . . . . . . . . . . . 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 293 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
( g  .(+)  s )  =  s  <->  g  e.  N ) )
6261ralbidva 2828 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  K  (
g  .(+)  s )  =  s  <->  A. g  e.  K  g  e.  N )
)
63 dfss3 3408 . . . . . . . . . . 11  |-  ( K 
C_  N  <->  A. g  e.  K  g  e.  N )
6462, 63syl6bbr 271 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  K  (
g  .(+)  s )  =  s  <->  K  C_  N ) )
6517adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  K  =  ( Base `  ( Gs  K
) ) )
6665raleqdv 2979 . . . . . . . . . 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 2471 . . . . . . . . . . . . 13  |-  ( Base `  ( Gs  N ) )  =  ( Base `  ( Gs  N ) )
683ad2antrr 740 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  G  e.  Grp )
6958, 2, 6nmzsubg 16936 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
7068, 69syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  e.  (SubGrp `  G )
)
71 eqid 2471 . . . . . . . . . . . . . . . 16  |-  ( Gs  N )  =  ( Gs  N )
7271subgbas 16899 . . . . . . . . . . . . . . 15  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  ( Gs  N
) ) )
7370, 72syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  =  ( Base `  ( Gs  N ) ) )
744ad2antrr 740 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  X  e.  Fin )
752subgss 16896 . . . . . . . . . . . . . . . 16  |-  ( N  e.  (SubGrp `  G
)  ->  N  C_  X
)
7670, 75syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  C_  X )
77 ssfi 7810 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  Fin  /\  N  C_  X )  ->  N  e.  Fin )
7874, 76, 77syl2anc 673 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  e.  Fin )
7973, 78eqeltrrd 2550 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  ( Base `  ( Gs  N ) )  e.  Fin )
808ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  K  e.  ( P pSyl  G ) )
81 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  K  C_  N )
8271subgslw 17346 . . . . . . . . . . . . . 14  |-  ( ( N  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  N
)  ->  K  e.  ( P pSyl  ( Gs  N
) ) )
8370, 80, 81, 82syl3anc 1292 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  K  e.  ( P pSyl  ( Gs  N ) ) )
84 simplr 770 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  ( P pSyl  G ) )
8555ad2antlr 741 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  (SubGrp `  G )
)
8658, 2, 6ssnmz 16937 . . . . . . . . . . . . . . 15  |-  ( s  e.  (SubGrp `  G
)  ->  s  C_  N )
8785, 86syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  C_  N )
8871subgslw 17346 . . . . . . . . . . . . . 14  |-  ( ( N  e.  (SubGrp `  G )  /\  s  e.  ( P pSyl  G )  /\  s  C_  N
)  ->  s  e.  ( P pSyl  ( Gs  N
) ) )
8970, 84, 87, 88syl3anc 1292 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  ( P pSyl  ( Gs  N ) ) )
90 fvex 5889 . . . . . . . . . . . . . . . 16  |-  ( Base `  G )  e.  _V
912, 90eqeltri 2545 . . . . . . . . . . . . . . 15  |-  X  e. 
_V
9258, 91rabex2 4552 . . . . . . . . . . . . . 14  |-  N  e. 
_V
9371, 6ressplusg 15317 . . . . . . . . . . . . . 14  |-  ( N  e.  _V  ->  .+  =  ( +g  `  ( Gs  N ) ) )
9492, 93ax-mp 5 . . . . . . . . . . . . 13  |-  .+  =  ( +g  `  ( Gs  N ) )
95 eqid 2471 . . . . . . . . . . . . 13  |-  ( -g `  ( Gs  N ) )  =  ( -g `  ( Gs  N ) )
9667, 79, 83, 89, 94, 95sylow2 17356 . . . . . . . . . . . 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 ) ) )
9758, 2, 6, 71nmznsg 16939 . . . . . . . . . . . . . . . 16  |-  ( s  e.  (SubGrp `  G
)  ->  s  e.  (NrmSGrp `  ( Gs  N ) ) )
9885, 97syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  (NrmSGrp `  ( Gs  N
) ) )
99 eqid 2471 . . . . . . . . . . . . . . . 16  |-  ( z  e.  s  |->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) )  =  ( z  e.  s  |->  ( ( g  .+  z
) ( -g `  ( Gs  N ) ) g ) )
10067, 94, 95, 99conjnsg 16996 . . . . . . . . . . . . . . 15  |-  ( ( s  e.  (NrmSGrp `  ( Gs  N ) )  /\  g  e.  ( Base `  ( Gs  N ) ) )  ->  s  =  ran  ( z  e.  s 
|->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) ) )
10198, 100sylan 479 . . . . . . . . . . . . . 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 ) ) )
102 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 ) ) ) )
103101, 102syl5ibrcom 230 . . . . . . . . . . . . 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 ) )
104103rexlimdva 2871 . . . . . . . . . . . 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 ) )
10596, 104mpd 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  =  K )
106 simpr 468 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  s  =  K )
10715ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  K  e.  (SubGrp `  G )
)
108106, 107eqeltrd 2549 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  s  e.  (SubGrp `  G )
)
109108, 86syl 17 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  s  C_  N )
110106, 109eqsstr3d 3453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  K  C_  N )
111105, 110impbida 850 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( K  C_  N  <->  s  =  K ) )
11264, 66, 1113bitr3d 291 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s  <->  s  =  K ) )
113112rabbidva 3021 . . . . . . . 8  |-  ( ph  ->  { s  e.  ( P pSyl  G )  | 
A. g  e.  (
Base `  ( Gs  K
) ) ( g 
.(+)  s )  =  s }  =  {
s  e.  ( P pSyl 
G )  |  s  =  K } )
114 rabsn 4030 . . . . . . . . 9  |-  ( K  e.  ( P pSyl  G
)  ->  { s  e.  ( P pSyl  G )  |  s  =  K }  =  { K } )
1158, 114syl 17 . . . . . . . 8  |-  ( ph  ->  { s  e.  ( P pSyl  G )  |  s  =  K }  =  { K } )
116113, 115eqtrd 2505 . . . . . . 7  |-  ( ph  ->  { s  e.  ( P pSyl  G )  | 
A. g  e.  (
Base `  ( Gs  K
) ) ( g 
.(+)  s )  =  s }  =  { K } )
117116fveq2d 5883 . . . . . 6  |-  ( ph  ->  ( # `  {
s  e.  ( P pSyl 
G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } )  =  ( # `  { K } ) )
118 hashsng 12587 . . . . . . 7  |-  ( K  e.  ( P pSyl  G
)  ->  ( # `  { K } )  =  1 )
1198, 118syl 17 . . . . . 6  |-  ( ph  ->  ( # `  { K } )  =  1 )
120117, 119eqtrd 2505 . . . . 5  |-  ( ph  ->  ( # `  {
s  e.  ( P pSyl 
G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } )  =  1 )
121120oveq2d 6324 . . . 4  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  -  ( # `  { s  e.  ( P pSyl  G
)  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } ) )  =  ( (
# `  ( P pSyl  G ) )  -  1 ) )
12235, 121breqtrd 4420 . . 3  |-  ( ph  ->  P  ||  ( (
# `  ( P pSyl  G ) )  -  1 ) )
123 prmnn 14704 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
1245, 123syl 17 . . . 4  |-  ( ph  ->  P  e.  NN )
125 hashcl 12576 . . . . . 6  |-  ( ( P pSyl  G )  e. 
Fin  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
12632, 125syl 17 . . . . 5  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
127126nn0zd 11061 . . . 4  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e.  ZZ )
128 1zzd 10992 . . . 4  |-  ( ph  ->  1  e.  ZZ )
129 moddvds 14389 . . . 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 ) ) )
130124, 127, 128, 129syl3anc 1292 . . 3  |-  ( ph  ->  ( ( ( # `  ( P pSyl  G ) )  mod  P )  =  ( 1  mod 
P )  <->  P  ||  (
( # `  ( P pSyl 
G ) )  - 
1 ) ) )
131122, 130mpbird 240 . 2  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  mod 
P )  =  ( 1  mod  P ) )
132 prmuz2 14721 . . 3  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
133 eluz2b2 11254 . . . 4  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
134 nnre 10638 . . . . 5  |-  ( P  e.  NN  ->  P  e.  RR )
135 1mod 12162 . . . . 5  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
136134, 135sylan 479 . . . 4  |-  ( ( P  e.  NN  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
137133, 136sylbi 200 . . 3  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( 1  mod  P )  =  1 )
1385, 132, 1373syl 18 . 2  |-  ( ph  ->  ( 1  mod  P
)  =  1 )
139131, 138eqtrd 2505 1  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  mod 
P )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    C_ wss 3390   ~Pcpw 3942   {csn 3959   {cpr 3961   class class class wbr 4395   {copab 4453    |-> cmpt 4454   ran crn 4840   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   Fincfn 7587   RRcr 9556   1c1 9558    < clt 9693    - cmin 9880   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182    mod cmo 12129   #chash 12553    || cdvds 14382   Primecprime 14701   Basecbs 15199   ↾s cress 15200   +g cplusg 15268   Grpcgrp 16747   -gcsg 16749  SubGrpcsubg 16889  NrmSGrpcnsg 16890   pGrp cpgp 17247   pSyl cslw 17249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-nsg 16893  df-eqg 16894  df-ghm 16959  df-ga 17022  df-od 17250  df-pgp 17254  df-slw 17256
This theorem is referenced by:  sylow3  17363
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