Step |
Hyp |
Ref |
Expression |
1 |
|
sylow2b.x |
|- X = ( Base ` G ) |
2 |
|
sylow2b.xf |
|- ( ph -> X e. Fin ) |
3 |
|
sylow2b.h |
|- ( ph -> H e. ( SubGrp ` G ) ) |
4 |
|
sylow2b.k |
|- ( ph -> K e. ( SubGrp ` G ) ) |
5 |
|
sylow2b.a |
|- .+ = ( +g ` G ) |
6 |
|
sylow2b.r |
|- .~ = ( G ~QG K ) |
7 |
|
sylow2b.m |
|- .x. = ( x e. H , y e. ( X /. .~ ) |-> ran ( z e. y |-> ( x .+ z ) ) ) |
8 |
|
simp2 |
|- ( ( ph /\ B e. H /\ C e. X ) -> B e. H ) |
9 |
6
|
ovexi |
|- .~ e. _V |
10 |
|
simp3 |
|- ( ( ph /\ B e. H /\ C e. X ) -> C e. X ) |
11 |
|
ecelqsg |
|- ( ( .~ e. _V /\ C e. X ) -> [ C ] .~ e. ( X /. .~ ) ) |
12 |
9 10 11
|
sylancr |
|- ( ( ph /\ B e. H /\ C e. X ) -> [ C ] .~ e. ( X /. .~ ) ) |
13 |
|
simpr |
|- ( ( x = B /\ y = [ C ] .~ ) -> y = [ C ] .~ ) |
14 |
|
simpl |
|- ( ( x = B /\ y = [ C ] .~ ) -> x = B ) |
15 |
14
|
oveq1d |
|- ( ( x = B /\ y = [ C ] .~ ) -> ( x .+ z ) = ( B .+ z ) ) |
16 |
13 15
|
mpteq12dv |
|- ( ( x = B /\ y = [ C ] .~ ) -> ( z e. y |-> ( x .+ z ) ) = ( z e. [ C ] .~ |-> ( B .+ z ) ) ) |
17 |
16
|
rneqd |
|- ( ( x = B /\ y = [ C ] .~ ) -> ran ( z e. y |-> ( x .+ z ) ) = ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ) |
18 |
|
ecexg |
|- ( .~ e. _V -> [ C ] .~ e. _V ) |
19 |
9 18
|
ax-mp |
|- [ C ] .~ e. _V |
20 |
19
|
mptex |
|- ( z e. [ C ] .~ |-> ( B .+ z ) ) e. _V |
21 |
20
|
rnex |
|- ran ( z e. [ C ] .~ |-> ( B .+ z ) ) e. _V |
22 |
17 7 21
|
ovmpoa |
|- ( ( B e. H /\ [ C ] .~ e. ( X /. .~ ) ) -> ( B .x. [ C ] .~ ) = ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ) |
23 |
8 12 22
|
syl2anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( B .x. [ C ] .~ ) = ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ) |
24 |
1 6
|
eqger |
|- ( K e. ( SubGrp ` G ) -> .~ Er X ) |
25 |
4 24
|
syl |
|- ( ph -> .~ Er X ) |
26 |
25
|
ecss |
|- ( ph -> [ ( B .+ C ) ] .~ C_ X ) |
27 |
2 26
|
ssfid |
|- ( ph -> [ ( B .+ C ) ] .~ e. Fin ) |
28 |
27
|
3ad2ant1 |
|- ( ( ph /\ B e. H /\ C e. X ) -> [ ( B .+ C ) ] .~ e. Fin ) |
29 |
|
vex |
|- z e. _V |
30 |
|
elecg |
|- ( ( z e. _V /\ C e. X ) -> ( z e. [ C ] .~ <-> C .~ z ) ) |
31 |
29 10 30
|
sylancr |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( z e. [ C ] .~ <-> C .~ z ) ) |
32 |
31
|
biimpa |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ z e. [ C ] .~ ) -> C .~ z ) |
33 |
|
subgrcl |
|- ( H e. ( SubGrp ` G ) -> G e. Grp ) |
34 |
3 33
|
syl |
|- ( ph -> G e. Grp ) |
35 |
34
|
3ad2ant1 |
|- ( ( ph /\ B e. H /\ C e. X ) -> G e. Grp ) |
36 |
1
|
subgss |
|- ( H e. ( SubGrp ` G ) -> H C_ X ) |
37 |
3 36
|
syl |
|- ( ph -> H C_ X ) |
38 |
37
|
3ad2ant1 |
|- ( ( ph /\ B e. H /\ C e. X ) -> H C_ X ) |
39 |
38 8
|
sseldd |
|- ( ( ph /\ B e. H /\ C e. X ) -> B e. X ) |
40 |
1 5
|
grpcl |
|- ( ( G e. Grp /\ B e. X /\ C e. X ) -> ( B .+ C ) e. X ) |
41 |
35 39 10 40
|
syl3anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( B .+ C ) e. X ) |
42 |
41
|
adantr |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( B .+ C ) e. X ) |
43 |
35
|
adantr |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> G e. Grp ) |
44 |
39
|
adantr |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> B e. X ) |
45 |
1
|
subgss |
|- ( K e. ( SubGrp ` G ) -> K C_ X ) |
46 |
4 45
|
syl |
|- ( ph -> K C_ X ) |
47 |
|
eqid |
|- ( invg ` G ) = ( invg ` G ) |
48 |
1 47 5 6
|
eqgval |
|- ( ( G e. Grp /\ K C_ X ) -> ( C .~ z <-> ( C e. X /\ z e. X /\ ( ( ( invg ` G ) ` C ) .+ z ) e. K ) ) ) |
49 |
34 46 48
|
syl2anc |
|- ( ph -> ( C .~ z <-> ( C e. X /\ z e. X /\ ( ( ( invg ` G ) ` C ) .+ z ) e. K ) ) ) |
50 |
49
|
3ad2ant1 |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( C .~ z <-> ( C e. X /\ z e. X /\ ( ( ( invg ` G ) ` C ) .+ z ) e. K ) ) ) |
51 |
50
|
biimpa |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( C e. X /\ z e. X /\ ( ( ( invg ` G ) ` C ) .+ z ) e. K ) ) |
52 |
51
|
simp2d |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> z e. X ) |
53 |
1 5
|
grpcl |
|- ( ( G e. Grp /\ B e. X /\ z e. X ) -> ( B .+ z ) e. X ) |
54 |
43 44 52 53
|
syl3anc |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( B .+ z ) e. X ) |
55 |
1 47
|
grpinvcl |
|- ( ( G e. Grp /\ ( B .+ C ) e. X ) -> ( ( invg ` G ) ` ( B .+ C ) ) e. X ) |
56 |
35 41 55
|
syl2anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( invg ` G ) ` ( B .+ C ) ) e. X ) |
57 |
56
|
adantr |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( ( invg ` G ) ` ( B .+ C ) ) e. X ) |
58 |
1 5
|
grpass |
|- ( ( G e. Grp /\ ( ( ( invg ` G ) ` ( B .+ C ) ) e. X /\ B e. X /\ z e. X ) ) -> ( ( ( ( invg ` G ) ` ( B .+ C ) ) .+ B ) .+ z ) = ( ( ( invg ` G ) ` ( B .+ C ) ) .+ ( B .+ z ) ) ) |
59 |
43 57 44 52 58
|
syl13anc |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( ( ( ( invg ` G ) ` ( B .+ C ) ) .+ B ) .+ z ) = ( ( ( invg ` G ) ` ( B .+ C ) ) .+ ( B .+ z ) ) ) |
60 |
1 5 47
|
grpinvadd |
|- ( ( G e. Grp /\ B e. X /\ C e. X ) -> ( ( invg ` G ) ` ( B .+ C ) ) = ( ( ( invg ` G ) ` C ) .+ ( ( invg ` G ) ` B ) ) ) |
61 |
35 39 10 60
|
syl3anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( invg ` G ) ` ( B .+ C ) ) = ( ( ( invg ` G ) ` C ) .+ ( ( invg ` G ) ` B ) ) ) |
62 |
1 47
|
grpinvcl |
|- ( ( G e. Grp /\ C e. X ) -> ( ( invg ` G ) ` C ) e. X ) |
63 |
35 10 62
|
syl2anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( invg ` G ) ` C ) e. X ) |
64 |
|
eqid |
|- ( -g ` G ) = ( -g ` G ) |
65 |
1 5 47 64
|
grpsubval |
|- ( ( ( ( invg ` G ) ` C ) e. X /\ B e. X ) -> ( ( ( invg ` G ) ` C ) ( -g ` G ) B ) = ( ( ( invg ` G ) ` C ) .+ ( ( invg ` G ) ` B ) ) ) |
66 |
63 39 65
|
syl2anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( ( invg ` G ) ` C ) ( -g ` G ) B ) = ( ( ( invg ` G ) ` C ) .+ ( ( invg ` G ) ` B ) ) ) |
67 |
61 66
|
eqtr4d |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( invg ` G ) ` ( B .+ C ) ) = ( ( ( invg ` G ) ` C ) ( -g ` G ) B ) ) |
68 |
67
|
oveq1d |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( ( invg ` G ) ` ( B .+ C ) ) .+ B ) = ( ( ( ( invg ` G ) ` C ) ( -g ` G ) B ) .+ B ) ) |
69 |
1 5 64
|
grpnpcan |
|- ( ( G e. Grp /\ ( ( invg ` G ) ` C ) e. X /\ B e. X ) -> ( ( ( ( invg ` G ) ` C ) ( -g ` G ) B ) .+ B ) = ( ( invg ` G ) ` C ) ) |
70 |
35 63 39 69
|
syl3anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( ( ( invg ` G ) ` C ) ( -g ` G ) B ) .+ B ) = ( ( invg ` G ) ` C ) ) |
71 |
68 70
|
eqtrd |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( ( invg ` G ) ` ( B .+ C ) ) .+ B ) = ( ( invg ` G ) ` C ) ) |
72 |
71
|
oveq1d |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( ( ( invg ` G ) ` ( B .+ C ) ) .+ B ) .+ z ) = ( ( ( invg ` G ) ` C ) .+ z ) ) |
73 |
72
|
adantr |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( ( ( ( invg ` G ) ` ( B .+ C ) ) .+ B ) .+ z ) = ( ( ( invg ` G ) ` C ) .+ z ) ) |
74 |
59 73
|
eqtr3d |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( ( ( invg ` G ) ` ( B .+ C ) ) .+ ( B .+ z ) ) = ( ( ( invg ` G ) ` C ) .+ z ) ) |
75 |
51
|
simp3d |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( ( ( invg ` G ) ` C ) .+ z ) e. K ) |
76 |
74 75
|
eqeltrd |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( ( ( invg ` G ) ` ( B .+ C ) ) .+ ( B .+ z ) ) e. K ) |
77 |
1 47 5 6
|
eqgval |
|- ( ( G e. Grp /\ K C_ X ) -> ( ( B .+ C ) .~ ( B .+ z ) <-> ( ( B .+ C ) e. X /\ ( B .+ z ) e. X /\ ( ( ( invg ` G ) ` ( B .+ C ) ) .+ ( B .+ z ) ) e. K ) ) ) |
78 |
34 46 77
|
syl2anc |
|- ( ph -> ( ( B .+ C ) .~ ( B .+ z ) <-> ( ( B .+ C ) e. X /\ ( B .+ z ) e. X /\ ( ( ( invg ` G ) ` ( B .+ C ) ) .+ ( B .+ z ) ) e. K ) ) ) |
79 |
78
|
3ad2ant1 |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( B .+ C ) .~ ( B .+ z ) <-> ( ( B .+ C ) e. X /\ ( B .+ z ) e. X /\ ( ( ( invg ` G ) ` ( B .+ C ) ) .+ ( B .+ z ) ) e. K ) ) ) |
80 |
79
|
adantr |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( ( B .+ C ) .~ ( B .+ z ) <-> ( ( B .+ C ) e. X /\ ( B .+ z ) e. X /\ ( ( ( invg ` G ) ` ( B .+ C ) ) .+ ( B .+ z ) ) e. K ) ) ) |
81 |
42 54 76 80
|
mpbir3and |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( B .+ C ) .~ ( B .+ z ) ) |
82 |
|
ovex |
|- ( B .+ z ) e. _V |
83 |
|
ovex |
|- ( B .+ C ) e. _V |
84 |
82 83
|
elec |
|- ( ( B .+ z ) e. [ ( B .+ C ) ] .~ <-> ( B .+ C ) .~ ( B .+ z ) ) |
85 |
81 84
|
sylibr |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ C .~ z ) -> ( B .+ z ) e. [ ( B .+ C ) ] .~ ) |
86 |
32 85
|
syldan |
|- ( ( ( ph /\ B e. H /\ C e. X ) /\ z e. [ C ] .~ ) -> ( B .+ z ) e. [ ( B .+ C ) ] .~ ) |
87 |
86
|
fmpttd |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( z e. [ C ] .~ |-> ( B .+ z ) ) : [ C ] .~ --> [ ( B .+ C ) ] .~ ) |
88 |
87
|
frnd |
|- ( ( ph /\ B e. H /\ C e. X ) -> ran ( z e. [ C ] .~ |-> ( B .+ z ) ) C_ [ ( B .+ C ) ] .~ ) |
89 |
|
eqid |
|- ( z e. X |-> ( B .+ z ) ) = ( z e. X |-> ( B .+ z ) ) |
90 |
1 5 89
|
grplmulf1o |
|- ( ( G e. Grp /\ B e. X ) -> ( z e. X |-> ( B .+ z ) ) : X -1-1-onto-> X ) |
91 |
35 39 90
|
syl2anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( z e. X |-> ( B .+ z ) ) : X -1-1-onto-> X ) |
92 |
|
f1of1 |
|- ( ( z e. X |-> ( B .+ z ) ) : X -1-1-onto-> X -> ( z e. X |-> ( B .+ z ) ) : X -1-1-> X ) |
93 |
91 92
|
syl |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( z e. X |-> ( B .+ z ) ) : X -1-1-> X ) |
94 |
25
|
ecss |
|- ( ph -> [ C ] .~ C_ X ) |
95 |
94
|
3ad2ant1 |
|- ( ( ph /\ B e. H /\ C e. X ) -> [ C ] .~ C_ X ) |
96 |
|
f1ssres |
|- ( ( ( z e. X |-> ( B .+ z ) ) : X -1-1-> X /\ [ C ] .~ C_ X ) -> ( ( z e. X |-> ( B .+ z ) ) |` [ C ] .~ ) : [ C ] .~ -1-1-> X ) |
97 |
93 95 96
|
syl2anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( z e. X |-> ( B .+ z ) ) |` [ C ] .~ ) : [ C ] .~ -1-1-> X ) |
98 |
|
resmpt |
|- ( [ C ] .~ C_ X -> ( ( z e. X |-> ( B .+ z ) ) |` [ C ] .~ ) = ( z e. [ C ] .~ |-> ( B .+ z ) ) ) |
99 |
|
f1eq1 |
|- ( ( ( z e. X |-> ( B .+ z ) ) |` [ C ] .~ ) = ( z e. [ C ] .~ |-> ( B .+ z ) ) -> ( ( ( z e. X |-> ( B .+ z ) ) |` [ C ] .~ ) : [ C ] .~ -1-1-> X <-> ( z e. [ C ] .~ |-> ( B .+ z ) ) : [ C ] .~ -1-1-> X ) ) |
100 |
95 98 99
|
3syl |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( ( ( z e. X |-> ( B .+ z ) ) |` [ C ] .~ ) : [ C ] .~ -1-1-> X <-> ( z e. [ C ] .~ |-> ( B .+ z ) ) : [ C ] .~ -1-1-> X ) ) |
101 |
97 100
|
mpbid |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( z e. [ C ] .~ |-> ( B .+ z ) ) : [ C ] .~ -1-1-> X ) |
102 |
|
f1f1orn |
|- ( ( z e. [ C ] .~ |-> ( B .+ z ) ) : [ C ] .~ -1-1-> X -> ( z e. [ C ] .~ |-> ( B .+ z ) ) : [ C ] .~ -1-1-onto-> ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ) |
103 |
101 102
|
syl |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( z e. [ C ] .~ |-> ( B .+ z ) ) : [ C ] .~ -1-1-onto-> ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ) |
104 |
19
|
f1oen |
|- ( ( z e. [ C ] .~ |-> ( B .+ z ) ) : [ C ] .~ -1-1-onto-> ran ( z e. [ C ] .~ |-> ( B .+ z ) ) -> [ C ] .~ ~~ ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ) |
105 |
|
ensym |
|- ( [ C ] .~ ~~ ran ( z e. [ C ] .~ |-> ( B .+ z ) ) -> ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ~~ [ C ] .~ ) |
106 |
103 104 105
|
3syl |
|- ( ( ph /\ B e. H /\ C e. X ) -> ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ~~ [ C ] .~ ) |
107 |
4
|
3ad2ant1 |
|- ( ( ph /\ B e. H /\ C e. X ) -> K e. ( SubGrp ` G ) ) |
108 |
1 6
|
eqgen |
|- ( ( K e. ( SubGrp ` G ) /\ [ C ] .~ e. ( X /. .~ ) ) -> K ~~ [ C ] .~ ) |
109 |
107 12 108
|
syl2anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> K ~~ [ C ] .~ ) |
110 |
|
ensym |
|- ( K ~~ [ C ] .~ -> [ C ] .~ ~~ K ) |
111 |
109 110
|
syl |
|- ( ( ph /\ B e. H /\ C e. X ) -> [ C ] .~ ~~ K ) |
112 |
|
ecelqsg |
|- ( ( .~ e. _V /\ ( B .+ C ) e. X ) -> [ ( B .+ C ) ] .~ e. ( X /. .~ ) ) |
113 |
9 41 112
|
sylancr |
|- ( ( ph /\ B e. H /\ C e. X ) -> [ ( B .+ C ) ] .~ e. ( X /. .~ ) ) |
114 |
1 6
|
eqgen |
|- ( ( K e. ( SubGrp ` G ) /\ [ ( B .+ C ) ] .~ e. ( X /. .~ ) ) -> K ~~ [ ( B .+ C ) ] .~ ) |
115 |
107 113 114
|
syl2anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> K ~~ [ ( B .+ C ) ] .~ ) |
116 |
|
entr |
|- ( ( [ C ] .~ ~~ K /\ K ~~ [ ( B .+ C ) ] .~ ) -> [ C ] .~ ~~ [ ( B .+ C ) ] .~ ) |
117 |
111 115 116
|
syl2anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> [ C ] .~ ~~ [ ( B .+ C ) ] .~ ) |
118 |
|
entr |
|- ( ( ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ~~ [ C ] .~ /\ [ C ] .~ ~~ [ ( B .+ C ) ] .~ ) -> ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ~~ [ ( B .+ C ) ] .~ ) |
119 |
106 117 118
|
syl2anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ~~ [ ( B .+ C ) ] .~ ) |
120 |
|
fisseneq |
|- ( ( [ ( B .+ C ) ] .~ e. Fin /\ ran ( z e. [ C ] .~ |-> ( B .+ z ) ) C_ [ ( B .+ C ) ] .~ /\ ran ( z e. [ C ] .~ |-> ( B .+ z ) ) ~~ [ ( B .+ C ) ] .~ ) -> ran ( z e. [ C ] .~ |-> ( B .+ z ) ) = [ ( B .+ C ) ] .~ ) |
121 |
28 88 119 120
|
syl3anc |
|- ( ( ph /\ B e. H /\ C e. X ) -> ran ( z e. [ C ] .~ |-> ( B .+ z ) ) = [ ( B .+ C ) ] .~ ) |
122 |
23 121
|
eqtrd |
|- ( ( ph /\ B e. H /\ C e. X ) -> ( B .x. [ C ] .~ ) = [ ( B .+ C ) ] .~ ) |