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Theorem sylow3lem6 14959
Description: Lemma for sylow3 14960, second part. Using the lemma sylow2a 14946, 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 2296 . . . . 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 14958 . . . . 5  |-  ( ph  -> 
.(+)  e.  ( ( Gs  K )  GrpAct  ( P pSyl 
G ) ) )
11 eqid 2296 . . . . . . 7  |-  ( Gs  K )  =  ( Gs  K )
1211slwpgp 14940 . . . . . 6  |-  ( K  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  K ) )
138, 12syl 15 . . . . 5  |-  ( ph  ->  P pGrp  ( Gs  K ) )
14 slwsubg 14937 . . . . . . . 8  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
158, 14syl 15 . . . . . . 7  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
1611subgbas 14641 . . . . . . 7  |-  ( K  e.  (SubGrp `  G
)  ->  K  =  ( Base `  ( Gs  K
) ) )
1715, 16syl 15 . . . . . 6  |-  ( ph  ->  K  =  ( Base `  ( Gs  K ) ) )
182subgss 14638 . . . . . . . 8  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
1915, 18syl 15 . . . . . . 7  |-  ( ph  ->  K  C_  X )
20 ssfi 7099 . . . . . . 7  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
214, 19, 20syl2anc 642 . . . . . 6  |-  ( ph  ->  K  e.  Fin )
2217, 21eqeltrrd 2371 . . . . 5  |-  ( ph  ->  ( Base `  ( Gs  K ) )  e. 
Fin )
23 pwfi 7167 . . . . . . 7  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
244, 23sylib 188 . . . . . 6  |-  ( ph  ->  ~P X  e.  Fin )
25 slwsubg 14937 . . . . . . . . 9  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  (SubGrp `  G ) )
262subgss 14638 . . . . . . . . 9  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  X
)
2725, 26syl 15 . . . . . . . 8  |-  ( x  e.  ( P pSyl  G
)  ->  x  C_  X
)
28 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
2928elpw 3644 . . . . . . . 8  |-  ( x  e.  ~P X  <->  x  C_  X
)
3027, 29sylibr 203 . . . . . . 7  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  ~P X )
3130ssriv 3197 . . . . . 6  |-  ( P pSyl 
G )  C_  ~P X
32 ssfi 7099 . . . . . 6  |-  ( ( ~P X  e.  Fin  /\  ( P pSyl  G ) 
C_  ~P X )  -> 
( P pSyl  G )  e.  Fin )
3324, 31, 32sylancl 643 . . . . 5  |-  ( ph  ->  ( P pSyl  G )  e.  Fin )
34 eqid 2296 . . . . 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 }
35 eqid 2296 . . . . 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 ) }
361, 10, 13, 22, 33, 34, 35sylow2a 14946 . . . 4  |-  ( ph  ->  P  ||  ( (
# `  ( P pSyl  G ) )  -  ( # `
 { s  e.  ( P pSyl  G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } ) ) )
37 eqcom 2298 . . . . . . . . . . . . . 14  |-  ( ran  ( z  e.  s 
|->  ( ( g  .+  z )  .-  g
) )  =  s  <-> 
s  =  ran  (
z  e.  s  |->  ( ( g  .+  z
)  .-  g )
) )
3819adantr 451 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  K  C_  X
)
3938sselda 3193 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  g  e.  X )
4039biantrurd 494 . . . . . . . . . . . . . 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 ) ) ) ) )
4137, 40syl5bb 248 . . . . . . . . . . . . 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
) ) ) ) )
42 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  g  e.  K )
43 simplr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  s  e.  ( P pSyl  G ) )
44 simpr 447 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  g  /\  y  =  s )  ->  y  =  s )
45 simpl 443 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  g  /\  y  =  s )  ->  x  =  g )
4645oveq1d 5889 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  g  /\  y  =  s )  ->  ( x  .+  z
)  =  ( g 
.+  z ) )
4746, 45oveq12d 5892 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  g  /\  y  =  s )  ->  ( ( x  .+  z )  .-  x
)  =  ( ( g  .+  z ) 
.-  g ) )
4844, 47mpteq12dv 4114 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  g  /\  y  =  s )  ->  ( z  e.  y 
|->  ( ( x  .+  z )  .-  x
) )  =  ( z  e.  s  |->  ( ( g  .+  z
)  .-  g )
) )
4948rneqd 4922 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  g  /\  y  =  s )  ->  ran  ( z  e.  y  |->  ( ( x 
.+  z )  .-  x ) )  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) )
50 vex 2804 . . . . . . . . . . . . . . . . . 18  |-  s  e. 
_V
5150mptex 5762 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  s  |->  ( ( g  .+  z ) 
.-  g ) )  e.  _V
5251rnex 4958 . . . . . . . . . . . . . . . 16  |-  ran  (
z  e.  s  |->  ( ( g  .+  z
)  .-  g )
)  e.  _V
5349, 9, 52ovmpt2a 5994 . . . . . . . . . . . . . . 15  |-  ( ( g  e.  K  /\  s  e.  ( P pSyl  G ) )  ->  (
g  .(+)  s )  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) )
5442, 43, 53syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
g  .(+)  s )  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) )
5554eqeq1d 2304 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
( g  .(+)  s )  =  s  <->  ran  ( z  e.  s  |->  ( ( g  .+  z ) 
.-  g ) )  =  s ) )
56 slwsubg 14937 . . . . . . . . . . . . . . 15  |-  ( s  e.  ( P pSyl  G
)  ->  s  e.  (SubGrp `  G ) )
5756ad2antlr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  s  e.  (SubGrp `  G )
)
58 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( z  e.  s  |->  ( ( g  .+  z ) 
.-  g ) )  =  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) )
59 sylow3lem6.n . . . . . . . . . . . . . . 15  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  s  <-> 
( y  .+  x
)  e.  s ) }
602, 6, 7, 58, 59conjnmzb 14733 . . . . . . . . . . . . . 14  |-  ( s  e.  (SubGrp `  G
)  ->  ( g  e.  N  <->  ( g  e.  X  /\  s  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) ) ) )
6157, 60syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
g  e.  N  <->  ( g  e.  X  /\  s  =  ran  ( z  e.  s  |->  ( ( g 
.+  z )  .-  g ) ) ) ) )
6241, 55, 613bitr4d 276 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  g  e.  K )  ->  (
( g  .(+)  s )  =  s  <->  g  e.  N ) )
6362ralbidva 2572 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  K  (
g  .(+)  s )  =  s  <->  A. g  e.  K  g  e.  N )
)
64 dfss3 3183 . . . . . . . . . . 11  |-  ( K 
C_  N  <->  A. g  e.  K  g  e.  N )
6563, 64syl6bbr 254 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  K  (
g  .(+)  s )  =  s  <->  K  C_  N ) )
6617adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  K  =  ( Base `  ( Gs  K
) ) )
6766raleqdv 2755 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  K  (
g  .(+)  s )  =  s  <->  A. g  e.  (
Base `  ( Gs  K
) ) ( g 
.(+)  s )  =  s ) )
68 eqid 2296 . . . . . . . . . . . . 13  |-  ( Base `  ( Gs  N ) )  =  ( Base `  ( Gs  N ) )
693ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  G  e.  Grp )
7059, 2, 6nmzsubg 14674 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
7169, 70syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  e.  (SubGrp `  G )
)
72 eqid 2296 . . . . . . . . . . . . . . . 16  |-  ( Gs  N )  =  ( Gs  N )
7372subgbas 14641 . . . . . . . . . . . . . . 15  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  ( Gs  N
) ) )
7471, 73syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  =  ( Base `  ( Gs  N ) ) )
754ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  X  e.  Fin )
762subgss 14638 . . . . . . . . . . . . . . . 16  |-  ( N  e.  (SubGrp `  G
)  ->  N  C_  X
)
7771, 76syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  C_  X )
78 ssfi 7099 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  Fin  /\  N  C_  X )  ->  N  e.  Fin )
7975, 77, 78syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  N  e.  Fin )
8074, 79eqeltrrd 2371 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  ( Base `  ( Gs  N ) )  e.  Fin )
818ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  K  e.  ( P pSyl  G ) )
82 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  K  C_  N )
8372subgslw 14943 . . . . . . . . . . . . . 14  |-  ( ( N  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  N
)  ->  K  e.  ( P pSyl  ( Gs  N
) ) )
8471, 81, 82, 83syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  K  e.  ( P pSyl  ( Gs  N ) ) )
85 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  ( P pSyl  G ) )
8656ad2antlr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  (SubGrp `  G )
)
8759, 2, 6ssnmz 14675 . . . . . . . . . . . . . . 15  |-  ( s  e.  (SubGrp `  G
)  ->  s  C_  N )
8886, 87syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  C_  N )
8972subgslw 14943 . . . . . . . . . . . . . 14  |-  ( ( N  e.  (SubGrp `  G )  /\  s  e.  ( P pSyl  G )  /\  s  C_  N
)  ->  s  e.  ( P pSyl  ( Gs  N
) ) )
9071, 85, 88, 89syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  ( P pSyl  ( Gs  N ) ) )
91 fvex 5555 . . . . . . . . . . . . . . . . 17  |-  ( Base `  G )  e.  _V
922, 91eqeltri 2366 . . . . . . . . . . . . . . . 16  |-  X  e. 
_V
9392rabex 4181 . . . . . . . . . . . . . . 15  |-  { x  e.  X  |  A. y  e.  X  (
( x  .+  y
)  e.  s  <->  ( y  .+  x )  e.  s ) }  e.  _V
9459, 93eqeltri 2366 . . . . . . . . . . . . . 14  |-  N  e. 
_V
9572, 6ressplusg 13266 . . . . . . . . . . . . . 14  |-  ( N  e.  _V  ->  .+  =  ( +g  `  ( Gs  N ) ) )
9694, 95ax-mp 8 . . . . . . . . . . . . 13  |-  .+  =  ( +g  `  ( Gs  N ) )
97 eqid 2296 . . . . . . . . . . . . 13  |-  ( -g `  ( Gs  N ) )  =  ( -g `  ( Gs  N ) )
9868, 80, 84, 90, 96, 97sylow2 14953 . . . . . . . . . . . 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 ) ) )
9959, 2, 6, 72nmznsg 14677 . . . . . . . . . . . . . . . 16  |-  ( s  e.  (SubGrp `  G
)  ->  s  e.  (NrmSGrp `  ( Gs  N ) ) )
10086, 99syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  e.  (NrmSGrp `  ( Gs  N
) ) )
101 eqid 2296 . . . . . . . . . . . . . . . 16  |-  ( z  e.  s  |->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) )  =  ( z  e.  s  |->  ( ( g  .+  z
) ( -g `  ( Gs  N ) ) g ) )
10268, 96, 97, 101conjnsg 14734 . . . . . . . . . . . . . . 15  |-  ( ( s  e.  (NrmSGrp `  ( Gs  N ) )  /\  g  e.  ( Base `  ( Gs  N ) ) )  ->  s  =  ran  ( z  e.  s 
|->  ( ( g  .+  z ) ( -g `  ( Gs  N ) ) g ) ) )
103100, 102sylan 457 . . . . . . . . . . . . . 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 ) ) )
104 eqeq2 2305 . . . . . . . . . . . . . 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 ) ) ) )
105103, 104syl5ibrcom 213 . . . . . . . . . . . . 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 ) )
106105rexlimdva 2680 . . . . . . . . . . . 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 ) )
10798, 106mpd 14 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  K  C_  N )  ->  s  =  K )
108 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  s  =  K )
10915ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  K  e.  (SubGrp `  G )
)
110108, 109eqeltrd 2370 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  s  e.  (SubGrp `  G )
)
111110, 87syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  s  C_  N )
112108, 111eqsstr3d 3226 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  ( P pSyl  G ) )  /\  s  =  K )  ->  K  C_  N )
113107, 112impbida 805 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( K  C_  N  <->  s  =  K ) )
11465, 67, 1133bitr3d 274 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( P pSyl  G )
)  ->  ( A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s  <->  s  =  K ) )
115114rabbidva 2792 . . . . . . . 8  |-  ( ph  ->  { s  e.  ( P pSyl  G )  | 
A. g  e.  (
Base `  ( Gs  K
) ) ( g 
.(+)  s )  =  s }  =  {
s  e.  ( P pSyl 
G )  |  s  =  K } )
116 rabsn 3710 . . . . . . . . 9  |-  ( K  e.  ( P pSyl  G
)  ->  { s  e.  ( P pSyl  G )  |  s  =  K }  =  { K } )
1178, 116syl 15 . . . . . . . 8  |-  ( ph  ->  { s  e.  ( P pSyl  G )  |  s  =  K }  =  { K } )
118115, 117eqtrd 2328 . . . . . . 7  |-  ( ph  ->  { s  e.  ( P pSyl  G )  | 
A. g  e.  (
Base `  ( Gs  K
) ) ( g 
.(+)  s )  =  s }  =  { K } )
119118fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( # `  {
s  e.  ( P pSyl 
G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } )  =  ( # `  { K } ) )
120 hashsng 11372 . . . . . . 7  |-  ( K  e.  ( P pSyl  G
)  ->  ( # `  { K } )  =  1 )
1218, 120syl 15 . . . . . 6  |-  ( ph  ->  ( # `  { K } )  =  1 )
122119, 121eqtrd 2328 . . . . 5  |-  ( ph  ->  ( # `  {
s  e.  ( P pSyl 
G )  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } )  =  1 )
123122oveq2d 5890 . . . 4  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  -  ( # `  { s  e.  ( P pSyl  G
)  |  A. g  e.  ( Base `  ( Gs  K ) ) ( g  .(+)  s )  =  s } ) )  =  ( (
# `  ( P pSyl  G ) )  -  1 ) )
12436, 123breqtrd 4063 . . 3  |-  ( ph  ->  P  ||  ( (
# `  ( P pSyl  G ) )  -  1 ) )
125 prmnn 12777 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
1265, 125syl 15 . . . 4  |-  ( ph  ->  P  e.  NN )
127 hashcl 11366 . . . . . 6  |-  ( ( P pSyl  G )  e. 
Fin  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
12833, 127syl 15 . . . . 5  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
129128nn0zd 10131 . . . 4  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e.  ZZ )
130 1z 10069 . . . . 5  |-  1  e.  ZZ
131130a1i 10 . . . 4  |-  ( ph  ->  1  e.  ZZ )
132 moddvds 12554 . . . 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 ) ) )
133126, 129, 131, 132syl3anc 1182 . . 3  |-  ( ph  ->  ( ( ( # `  ( P pSyl  G ) )  mod  P )  =  ( 1  mod 
P )  <->  P  ||  (
( # `  ( P pSyl 
G ) )  - 
1 ) ) )
134124, 133mpbird 223 . 2  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  mod 
P )  =  ( 1  mod  P ) )
135 prmuz2 12792 . . 3  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
136 eluz2b2 10306 . . . 4  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
137 nnre 9769 . . . . 5  |-  ( P  e.  NN  ->  P  e.  RR )
138 1mod 11012 . . . . 5  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
139137, 138sylan 457 . . . 4  |-  ( ( P  e.  NN  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
140136, 139sylbi 187 . . 3  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( 1  mod  P )  =  1 )
1415, 135, 1403syl 18 . 2  |-  ( ph  ->  ( 1  mod  P
)  =  1 )
142134, 141eqtrd 2328 1  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  mod 
P )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   {csn 3653   {cpr 3654   class class class wbr 4039   {copab 4092    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Fincfn 6879   RRcr 8752   1c1 8754    < clt 8883    - cmin 9053   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246    mod cmo 10989   #chash 11353    || cdivides 12547   Primecprime 12774   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   Grpcgrp 14378   -gcsg 14381  SubGrpcsubg 14631  NrmSGrpcnsg 14632   pGrp cpgp 14858   pSyl cslw 14859
This theorem is referenced by:  sylow3  14960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-nsg 14635  df-eqg 14636  df-ghm 14697  df-ga 14760  df-od 14860  df-pgp 14862  df-slw 14863
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