Step |
Hyp |
Ref |
Expression |
1 |
|
sylow3.x |
|- X = ( Base ` G ) |
2 |
|
sylow3.g |
|- ( ph -> G e. Grp ) |
3 |
|
sylow3.xf |
|- ( ph -> X e. Fin ) |
4 |
|
sylow3.p |
|- ( ph -> P e. Prime ) |
5 |
|
sylow3.n |
|- N = ( # ` ( P pSyl G ) ) |
6 |
1
|
slwn0 |
|- ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) -> ( P pSyl G ) =/= (/) ) |
7 |
2 3 4 6
|
syl3anc |
|- ( ph -> ( P pSyl G ) =/= (/) ) |
8 |
|
n0 |
|- ( ( P pSyl G ) =/= (/) <-> E. k k e. ( P pSyl G ) ) |
9 |
7 8
|
sylib |
|- ( ph -> E. k k e. ( P pSyl G ) ) |
10 |
2
|
adantr |
|- ( ( ph /\ k e. ( P pSyl G ) ) -> G e. Grp ) |
11 |
3
|
adantr |
|- ( ( ph /\ k e. ( P pSyl G ) ) -> X e. Fin ) |
12 |
4
|
adantr |
|- ( ( ph /\ k e. ( P pSyl G ) ) -> P e. Prime ) |
13 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
14 |
|
eqid |
|- ( -g ` G ) = ( -g ` G ) |
15 |
|
oveq2 |
|- ( c = z -> ( a ( +g ` G ) c ) = ( a ( +g ` G ) z ) ) |
16 |
15
|
oveq1d |
|- ( c = z -> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) = ( ( a ( +g ` G ) z ) ( -g ` G ) a ) ) |
17 |
16
|
cbvmptv |
|- ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) ) |
18 |
|
oveq1 |
|- ( a = x -> ( a ( +g ` G ) z ) = ( x ( +g ` G ) z ) ) |
19 |
|
id |
|- ( a = x -> a = x ) |
20 |
18 19
|
oveq12d |
|- ( a = x -> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) = ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) |
21 |
20
|
mpteq2dv |
|- ( a = x -> ( z e. b |-> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) ) |
22 |
17 21
|
syl5eq |
|- ( a = x -> ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) ) |
23 |
22
|
rneqd |
|- ( a = x -> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ran ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) ) |
24 |
|
mpteq1 |
|- ( b = y -> ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) = ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) ) |
25 |
24
|
rneqd |
|- ( b = y -> ran ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) = ran ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) ) |
26 |
23 25
|
cbvmpov |
|- ( a e. X , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) = ( x e. X , y e. ( P pSyl G ) |-> ran ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) ) |
27 |
|
simpr |
|- ( ( ph /\ k e. ( P pSyl G ) ) -> k e. ( P pSyl G ) ) |
28 |
|
eqid |
|- { u e. X | ( u ( a e. X , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) k ) = k } = { u e. X | ( u ( a e. X , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) k ) = k } |
29 |
|
eqid |
|- { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. k <-> ( y ( +g ` G ) x ) e. k ) } = { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. k <-> ( y ( +g ` G ) x ) e. k ) } |
30 |
1 10 11 12 13 14 26 27 28 29
|
sylow3lem4 |
|- ( ( ph /\ k e. ( P pSyl G ) ) -> ( # ` ( P pSyl G ) ) || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) ) |
31 |
5 30
|
eqbrtrid |
|- ( ( ph /\ k e. ( P pSyl G ) ) -> N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) ) |
32 |
5
|
oveq1i |
|- ( N mod P ) = ( ( # ` ( P pSyl G ) ) mod P ) |
33 |
23 25
|
cbvmpov |
|- ( a e. k , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) = ( x e. k , y e. ( P pSyl G ) |-> ran ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) ) |
34 |
|
eqid |
|- { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. s <-> ( y ( +g ` G ) x ) e. s ) } = { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. s <-> ( y ( +g ` G ) x ) e. s ) } |
35 |
1 10 11 12 13 14 27 33 34
|
sylow3lem6 |
|- ( ( ph /\ k e. ( P pSyl G ) ) -> ( ( # ` ( P pSyl G ) ) mod P ) = 1 ) |
36 |
32 35
|
syl5eq |
|- ( ( ph /\ k e. ( P pSyl G ) ) -> ( N mod P ) = 1 ) |
37 |
31 36
|
jca |
|- ( ( ph /\ k e. ( P pSyl G ) ) -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) ) |
38 |
9 37
|
exlimddv |
|- ( ph -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) ) |