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 |
|
sylow3lem1.a |
|- .+ = ( +g ` G ) |
6 |
|
sylow3lem1.d |
|- .- = ( -g ` G ) |
7 |
|
sylow3lem1.m |
|- .(+) = ( x e. X , y e. ( P pSyl G ) |-> ran ( z e. y |-> ( ( x .+ z ) .- x ) ) ) |
8 |
|
sylow3lem2.k |
|- ( ph -> K e. ( P pSyl G ) ) |
9 |
|
sylow3lem2.h |
|- H = { u e. X | ( u .(+) K ) = K } |
10 |
|
sylow3lem2.n |
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. K <-> ( y .+ x ) e. K ) } |
11 |
|
pwfi |
|- ( X e. Fin <-> ~P X e. Fin ) |
12 |
3 11
|
sylib |
|- ( ph -> ~P X e. Fin ) |
13 |
|
slwsubg |
|- ( x e. ( P pSyl G ) -> x e. ( SubGrp ` G ) ) |
14 |
1
|
subgss |
|- ( x e. ( SubGrp ` G ) -> x C_ X ) |
15 |
13 14
|
syl |
|- ( x e. ( P pSyl G ) -> x C_ X ) |
16 |
13 15
|
elpwd |
|- ( x e. ( P pSyl G ) -> x e. ~P X ) |
17 |
16
|
ssriv |
|- ( P pSyl G ) C_ ~P X |
18 |
|
ssfi |
|- ( ( ~P X e. Fin /\ ( P pSyl G ) C_ ~P X ) -> ( P pSyl G ) e. Fin ) |
19 |
12 17 18
|
sylancl |
|- ( ph -> ( P pSyl G ) e. Fin ) |
20 |
|
hashcl |
|- ( ( P pSyl G ) e. Fin -> ( # ` ( P pSyl G ) ) e. NN0 ) |
21 |
19 20
|
syl |
|- ( ph -> ( # ` ( P pSyl G ) ) e. NN0 ) |
22 |
21
|
nn0cnd |
|- ( ph -> ( # ` ( P pSyl G ) ) e. CC ) |
23 |
10 1 5
|
nmzsubg |
|- ( G e. Grp -> N e. ( SubGrp ` G ) ) |
24 |
|
eqid |
|- ( G ~QG N ) = ( G ~QG N ) |
25 |
1 24
|
eqger |
|- ( N e. ( SubGrp ` G ) -> ( G ~QG N ) Er X ) |
26 |
2 23 25
|
3syl |
|- ( ph -> ( G ~QG N ) Er X ) |
27 |
26
|
qsss |
|- ( ph -> ( X /. ( G ~QG N ) ) C_ ~P X ) |
28 |
12 27
|
ssfid |
|- ( ph -> ( X /. ( G ~QG N ) ) e. Fin ) |
29 |
|
hashcl |
|- ( ( X /. ( G ~QG N ) ) e. Fin -> ( # ` ( X /. ( G ~QG N ) ) ) e. NN0 ) |
30 |
28 29
|
syl |
|- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. NN0 ) |
31 |
30
|
nn0cnd |
|- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. CC ) |
32 |
2 23
|
syl |
|- ( ph -> N e. ( SubGrp ` G ) ) |
33 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
34 |
33
|
subg0cl |
|- ( N e. ( SubGrp ` G ) -> ( 0g ` G ) e. N ) |
35 |
|
ne0i |
|- ( ( 0g ` G ) e. N -> N =/= (/) ) |
36 |
32 34 35
|
3syl |
|- ( ph -> N =/= (/) ) |
37 |
1
|
subgss |
|- ( N e. ( SubGrp ` G ) -> N C_ X ) |
38 |
2 23 37
|
3syl |
|- ( ph -> N C_ X ) |
39 |
3 38
|
ssfid |
|- ( ph -> N e. Fin ) |
40 |
|
hashnncl |
|- ( N e. Fin -> ( ( # ` N ) e. NN <-> N =/= (/) ) ) |
41 |
39 40
|
syl |
|- ( ph -> ( ( # ` N ) e. NN <-> N =/= (/) ) ) |
42 |
36 41
|
mpbird |
|- ( ph -> ( # ` N ) e. NN ) |
43 |
42
|
nncnd |
|- ( ph -> ( # ` N ) e. CC ) |
44 |
42
|
nnne0d |
|- ( ph -> ( # ` N ) =/= 0 ) |
45 |
1 2 3 4 5 6 7
|
sylow3lem1 |
|- ( ph -> .(+) e. ( G GrpAct ( P pSyl G ) ) ) |
46 |
|
eqid |
|- ( G ~QG H ) = ( G ~QG H ) |
47 |
|
eqid |
|- { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } = { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } |
48 |
1 9 46 47
|
orbsta2 |
|- ( ( ( .(+) e. ( G GrpAct ( P pSyl G ) ) /\ K e. ( P pSyl G ) ) /\ X e. Fin ) -> ( # ` X ) = ( ( # ` [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) x. ( # ` H ) ) ) |
49 |
45 8 3 48
|
syl21anc |
|- ( ph -> ( # ` X ) = ( ( # ` [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) x. ( # ` H ) ) ) |
50 |
1 24 32 3
|
lagsubg2 |
|- ( ph -> ( # ` X ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) ) |
51 |
47 1
|
gaorber |
|- ( .(+) e. ( G GrpAct ( P pSyl G ) ) -> { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } Er ( P pSyl G ) ) |
52 |
45 51
|
syl |
|- ( ph -> { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } Er ( P pSyl G ) ) |
53 |
52
|
ecss |
|- ( ph -> [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } C_ ( P pSyl G ) ) |
54 |
8
|
adantr |
|- ( ( ph /\ h e. ( P pSyl G ) ) -> K e. ( P pSyl G ) ) |
55 |
|
simpr |
|- ( ( ph /\ h e. ( P pSyl G ) ) -> h e. ( P pSyl G ) ) |
56 |
3
|
adantr |
|- ( ( ph /\ h e. ( P pSyl G ) ) -> X e. Fin ) |
57 |
1 56 55 54 5 6
|
sylow2 |
|- ( ( ph /\ h e. ( P pSyl G ) ) -> E. u e. X h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) |
58 |
|
eqcom |
|- ( ( u .(+) K ) = h <-> h = ( u .(+) K ) ) |
59 |
|
simpr |
|- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> u e. X ) |
60 |
54
|
adantr |
|- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> K e. ( P pSyl G ) ) |
61 |
|
mptexg |
|- ( K e. ( P pSyl G ) -> ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V ) |
62 |
|
rnexg |
|- ( ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V -> ran ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V ) |
63 |
60 61 62
|
3syl |
|- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ran ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V ) |
64 |
|
simpr |
|- ( ( x = u /\ y = K ) -> y = K ) |
65 |
|
simpl |
|- ( ( x = u /\ y = K ) -> x = u ) |
66 |
65
|
oveq1d |
|- ( ( x = u /\ y = K ) -> ( x .+ z ) = ( u .+ z ) ) |
67 |
66 65
|
oveq12d |
|- ( ( x = u /\ y = K ) -> ( ( x .+ z ) .- x ) = ( ( u .+ z ) .- u ) ) |
68 |
64 67
|
mpteq12dv |
|- ( ( x = u /\ y = K ) -> ( z e. y |-> ( ( x .+ z ) .- x ) ) = ( z e. K |-> ( ( u .+ z ) .- u ) ) ) |
69 |
68
|
rneqd |
|- ( ( x = u /\ y = K ) -> ran ( z e. y |-> ( ( x .+ z ) .- x ) ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) |
70 |
69 7
|
ovmpoga |
|- ( ( u e. X /\ K e. ( P pSyl G ) /\ ran ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V ) -> ( u .(+) K ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) |
71 |
59 60 63 70
|
syl3anc |
|- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( u .(+) K ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) |
72 |
71
|
eqeq2d |
|- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( h = ( u .(+) K ) <-> h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) ) |
73 |
58 72
|
syl5bb |
|- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( ( u .(+) K ) = h <-> h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) ) |
74 |
73
|
rexbidva |
|- ( ( ph /\ h e. ( P pSyl G ) ) -> ( E. u e. X ( u .(+) K ) = h <-> E. u e. X h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) ) |
75 |
57 74
|
mpbird |
|- ( ( ph /\ h e. ( P pSyl G ) ) -> E. u e. X ( u .(+) K ) = h ) |
76 |
47
|
gaorb |
|- ( K { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } h <-> ( K e. ( P pSyl G ) /\ h e. ( P pSyl G ) /\ E. u e. X ( u .(+) K ) = h ) ) |
77 |
54 55 75 76
|
syl3anbrc |
|- ( ( ph /\ h e. ( P pSyl G ) ) -> K { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } h ) |
78 |
|
elecg |
|- ( ( h e. ( P pSyl G ) /\ K e. ( P pSyl G ) ) -> ( h e. [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } <-> K { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } h ) ) |
79 |
55 54 78
|
syl2anc |
|- ( ( ph /\ h e. ( P pSyl G ) ) -> ( h e. [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } <-> K { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } h ) ) |
80 |
77 79
|
mpbird |
|- ( ( ph /\ h e. ( P pSyl G ) ) -> h e. [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) |
81 |
53 80
|
eqelssd |
|- ( ph -> [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } = ( P pSyl G ) ) |
82 |
81
|
fveq2d |
|- ( ph -> ( # ` [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) = ( # ` ( P pSyl G ) ) ) |
83 |
1 2 3 4 5 6 7 8 9 10
|
sylow3lem2 |
|- ( ph -> H = N ) |
84 |
83
|
fveq2d |
|- ( ph -> ( # ` H ) = ( # ` N ) ) |
85 |
82 84
|
oveq12d |
|- ( ph -> ( ( # ` [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) x. ( # ` H ) ) = ( ( # ` ( P pSyl G ) ) x. ( # ` N ) ) ) |
86 |
49 50 85
|
3eqtr3rd |
|- ( ph -> ( ( # ` ( P pSyl G ) ) x. ( # ` N ) ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) ) |
87 |
22 31 43 44 86
|
mulcan2ad |
|- ( ph -> ( # ` ( P pSyl G ) ) = ( # ` ( X /. ( G ~QG N ) ) ) ) |