Step |
Hyp |
Ref |
Expression |
1 |
|
sylow2a.x |
|- X = ( Base ` G ) |
2 |
|
sylow2a.m |
|- ( ph -> .(+) e. ( G GrpAct Y ) ) |
3 |
|
sylow2a.p |
|- ( ph -> P pGrp G ) |
4 |
|
sylow2a.f |
|- ( ph -> X e. Fin ) |
5 |
|
sylow2a.y |
|- ( ph -> Y e. Fin ) |
6 |
|
sylow2a.z |
|- Z = { u e. Y | A. h e. X ( h .(+) u ) = u } |
7 |
|
sylow2a.r |
|- .~ = { <. x , y >. | ( { x , y } C_ Y /\ E. g e. X ( g .(+) x ) = y ) } |
8 |
|
pwfi |
|- ( Y e. Fin <-> ~P Y e. Fin ) |
9 |
5 8
|
sylib |
|- ( ph -> ~P Y e. Fin ) |
10 |
7 1
|
gaorber |
|- ( .(+) e. ( G GrpAct Y ) -> .~ Er Y ) |
11 |
2 10
|
syl |
|- ( ph -> .~ Er Y ) |
12 |
11
|
qsss |
|- ( ph -> ( Y /. .~ ) C_ ~P Y ) |
13 |
9 12
|
ssfid |
|- ( ph -> ( Y /. .~ ) e. Fin ) |
14 |
|
diffi |
|- ( ( Y /. .~ ) e. Fin -> ( ( Y /. .~ ) \ ~P Z ) e. Fin ) |
15 |
13 14
|
syl |
|- ( ph -> ( ( Y /. .~ ) \ ~P Z ) e. Fin ) |
16 |
|
gagrp |
|- ( .(+) e. ( G GrpAct Y ) -> G e. Grp ) |
17 |
2 16
|
syl |
|- ( ph -> G e. Grp ) |
18 |
1
|
pgpfi |
|- ( ( G e. Grp /\ X e. Fin ) -> ( P pGrp G <-> ( P e. Prime /\ E. n e. NN0 ( # ` X ) = ( P ^ n ) ) ) ) |
19 |
17 4 18
|
syl2anc |
|- ( ph -> ( P pGrp G <-> ( P e. Prime /\ E. n e. NN0 ( # ` X ) = ( P ^ n ) ) ) ) |
20 |
3 19
|
mpbid |
|- ( ph -> ( P e. Prime /\ E. n e. NN0 ( # ` X ) = ( P ^ n ) ) ) |
21 |
20
|
simpld |
|- ( ph -> P e. Prime ) |
22 |
|
prmz |
|- ( P e. Prime -> P e. ZZ ) |
23 |
21 22
|
syl |
|- ( ph -> P e. ZZ ) |
24 |
|
eldifi |
|- ( z e. ( ( Y /. .~ ) \ ~P Z ) -> z e. ( Y /. .~ ) ) |
25 |
5
|
adantr |
|- ( ( ph /\ z e. ( Y /. .~ ) ) -> Y e. Fin ) |
26 |
12
|
sselda |
|- ( ( ph /\ z e. ( Y /. .~ ) ) -> z e. ~P Y ) |
27 |
26
|
elpwid |
|- ( ( ph /\ z e. ( Y /. .~ ) ) -> z C_ Y ) |
28 |
25 27
|
ssfid |
|- ( ( ph /\ z e. ( Y /. .~ ) ) -> z e. Fin ) |
29 |
24 28
|
sylan2 |
|- ( ( ph /\ z e. ( ( Y /. .~ ) \ ~P Z ) ) -> z e. Fin ) |
30 |
|
hashcl |
|- ( z e. Fin -> ( # ` z ) e. NN0 ) |
31 |
29 30
|
syl |
|- ( ( ph /\ z e. ( ( Y /. .~ ) \ ~P Z ) ) -> ( # ` z ) e. NN0 ) |
32 |
31
|
nn0zd |
|- ( ( ph /\ z e. ( ( Y /. .~ ) \ ~P Z ) ) -> ( # ` z ) e. ZZ ) |
33 |
|
eldif |
|- ( z e. ( ( Y /. .~ ) \ ~P Z ) <-> ( z e. ( Y /. .~ ) /\ -. z e. ~P Z ) ) |
34 |
|
eqid |
|- ( Y /. .~ ) = ( Y /. .~ ) |
35 |
|
sseq1 |
|- ( [ w ] .~ = z -> ( [ w ] .~ C_ Z <-> z C_ Z ) ) |
36 |
|
velpw |
|- ( z e. ~P Z <-> z C_ Z ) |
37 |
35 36
|
bitr4di |
|- ( [ w ] .~ = z -> ( [ w ] .~ C_ Z <-> z e. ~P Z ) ) |
38 |
37
|
notbid |
|- ( [ w ] .~ = z -> ( -. [ w ] .~ C_ Z <-> -. z e. ~P Z ) ) |
39 |
|
fveq2 |
|- ( [ w ] .~ = z -> ( # ` [ w ] .~ ) = ( # ` z ) ) |
40 |
39
|
breq2d |
|- ( [ w ] .~ = z -> ( P || ( # ` [ w ] .~ ) <-> P || ( # ` z ) ) ) |
41 |
38 40
|
imbi12d |
|- ( [ w ] .~ = z -> ( ( -. [ w ] .~ C_ Z -> P || ( # ` [ w ] .~ ) ) <-> ( -. z e. ~P Z -> P || ( # ` z ) ) ) ) |
42 |
21
|
adantr |
|- ( ( ph /\ w e. Y ) -> P e. Prime ) |
43 |
11
|
adantr |
|- ( ( ph /\ w e. Y ) -> .~ Er Y ) |
44 |
|
simpr |
|- ( ( ph /\ w e. Y ) -> w e. Y ) |
45 |
43 44
|
erref |
|- ( ( ph /\ w e. Y ) -> w .~ w ) |
46 |
|
vex |
|- w e. _V |
47 |
46 46
|
elec |
|- ( w e. [ w ] .~ <-> w .~ w ) |
48 |
45 47
|
sylibr |
|- ( ( ph /\ w e. Y ) -> w e. [ w ] .~ ) |
49 |
48
|
ne0d |
|- ( ( ph /\ w e. Y ) -> [ w ] .~ =/= (/) ) |
50 |
11
|
ecss |
|- ( ph -> [ w ] .~ C_ Y ) |
51 |
5 50
|
ssfid |
|- ( ph -> [ w ] .~ e. Fin ) |
52 |
51
|
adantr |
|- ( ( ph /\ w e. Y ) -> [ w ] .~ e. Fin ) |
53 |
|
hashnncl |
|- ( [ w ] .~ e. Fin -> ( ( # ` [ w ] .~ ) e. NN <-> [ w ] .~ =/= (/) ) ) |
54 |
52 53
|
syl |
|- ( ( ph /\ w e. Y ) -> ( ( # ` [ w ] .~ ) e. NN <-> [ w ] .~ =/= (/) ) ) |
55 |
49 54
|
mpbird |
|- ( ( ph /\ w e. Y ) -> ( # ` [ w ] .~ ) e. NN ) |
56 |
|
pceq0 |
|- ( ( P e. Prime /\ ( # ` [ w ] .~ ) e. NN ) -> ( ( P pCnt ( # ` [ w ] .~ ) ) = 0 <-> -. P || ( # ` [ w ] .~ ) ) ) |
57 |
42 55 56
|
syl2anc |
|- ( ( ph /\ w e. Y ) -> ( ( P pCnt ( # ` [ w ] .~ ) ) = 0 <-> -. P || ( # ` [ w ] .~ ) ) ) |
58 |
|
oveq2 |
|- ( ( P pCnt ( # ` [ w ] .~ ) ) = 0 -> ( P ^ ( P pCnt ( # ` [ w ] .~ ) ) ) = ( P ^ 0 ) ) |
59 |
|
hashcl |
|- ( [ w ] .~ e. Fin -> ( # ` [ w ] .~ ) e. NN0 ) |
60 |
51 59
|
syl |
|- ( ph -> ( # ` [ w ] .~ ) e. NN0 ) |
61 |
60
|
nn0zd |
|- ( ph -> ( # ` [ w ] .~ ) e. ZZ ) |
62 |
|
ssrab2 |
|- { v e. X | ( v .(+) w ) = w } C_ X |
63 |
|
ssfi |
|- ( ( X e. Fin /\ { v e. X | ( v .(+) w ) = w } C_ X ) -> { v e. X | ( v .(+) w ) = w } e. Fin ) |
64 |
4 62 63
|
sylancl |
|- ( ph -> { v e. X | ( v .(+) w ) = w } e. Fin ) |
65 |
|
hashcl |
|- ( { v e. X | ( v .(+) w ) = w } e. Fin -> ( # ` { v e. X | ( v .(+) w ) = w } ) e. NN0 ) |
66 |
64 65
|
syl |
|- ( ph -> ( # ` { v e. X | ( v .(+) w ) = w } ) e. NN0 ) |
67 |
66
|
nn0zd |
|- ( ph -> ( # ` { v e. X | ( v .(+) w ) = w } ) e. ZZ ) |
68 |
|
dvdsmul1 |
|- ( ( ( # ` [ w ] .~ ) e. ZZ /\ ( # ` { v e. X | ( v .(+) w ) = w } ) e. ZZ ) -> ( # ` [ w ] .~ ) || ( ( # ` [ w ] .~ ) x. ( # ` { v e. X | ( v .(+) w ) = w } ) ) ) |
69 |
61 67 68
|
syl2anc |
|- ( ph -> ( # ` [ w ] .~ ) || ( ( # ` [ w ] .~ ) x. ( # ` { v e. X | ( v .(+) w ) = w } ) ) ) |
70 |
69
|
adantr |
|- ( ( ph /\ w e. Y ) -> ( # ` [ w ] .~ ) || ( ( # ` [ w ] .~ ) x. ( # ` { v e. X | ( v .(+) w ) = w } ) ) ) |
71 |
2
|
adantr |
|- ( ( ph /\ w e. Y ) -> .(+) e. ( G GrpAct Y ) ) |
72 |
4
|
adantr |
|- ( ( ph /\ w e. Y ) -> X e. Fin ) |
73 |
|
eqid |
|- { v e. X | ( v .(+) w ) = w } = { v e. X | ( v .(+) w ) = w } |
74 |
|
eqid |
|- ( G ~QG { v e. X | ( v .(+) w ) = w } ) = ( G ~QG { v e. X | ( v .(+) w ) = w } ) |
75 |
1 73 74 7
|
orbsta2 |
|- ( ( ( .(+) e. ( G GrpAct Y ) /\ w e. Y ) /\ X e. Fin ) -> ( # ` X ) = ( ( # ` [ w ] .~ ) x. ( # ` { v e. X | ( v .(+) w ) = w } ) ) ) |
76 |
71 44 72 75
|
syl21anc |
|- ( ( ph /\ w e. Y ) -> ( # ` X ) = ( ( # ` [ w ] .~ ) x. ( # ` { v e. X | ( v .(+) w ) = w } ) ) ) |
77 |
70 76
|
breqtrrd |
|- ( ( ph /\ w e. Y ) -> ( # ` [ w ] .~ ) || ( # ` X ) ) |
78 |
20
|
simprd |
|- ( ph -> E. n e. NN0 ( # ` X ) = ( P ^ n ) ) |
79 |
78
|
adantr |
|- ( ( ph /\ w e. Y ) -> E. n e. NN0 ( # ` X ) = ( P ^ n ) ) |
80 |
|
breq2 |
|- ( ( # ` X ) = ( P ^ n ) -> ( ( # ` [ w ] .~ ) || ( # ` X ) <-> ( # ` [ w ] .~ ) || ( P ^ n ) ) ) |
81 |
80
|
biimpcd |
|- ( ( # ` [ w ] .~ ) || ( # ` X ) -> ( ( # ` X ) = ( P ^ n ) -> ( # ` [ w ] .~ ) || ( P ^ n ) ) ) |
82 |
81
|
reximdv |
|- ( ( # ` [ w ] .~ ) || ( # ` X ) -> ( E. n e. NN0 ( # ` X ) = ( P ^ n ) -> E. n e. NN0 ( # ` [ w ] .~ ) || ( P ^ n ) ) ) |
83 |
77 79 82
|
sylc |
|- ( ( ph /\ w e. Y ) -> E. n e. NN0 ( # ` [ w ] .~ ) || ( P ^ n ) ) |
84 |
|
pcprmpw2 |
|- ( ( P e. Prime /\ ( # ` [ w ] .~ ) e. NN ) -> ( E. n e. NN0 ( # ` [ w ] .~ ) || ( P ^ n ) <-> ( # ` [ w ] .~ ) = ( P ^ ( P pCnt ( # ` [ w ] .~ ) ) ) ) ) |
85 |
42 55 84
|
syl2anc |
|- ( ( ph /\ w e. Y ) -> ( E. n e. NN0 ( # ` [ w ] .~ ) || ( P ^ n ) <-> ( # ` [ w ] .~ ) = ( P ^ ( P pCnt ( # ` [ w ] .~ ) ) ) ) ) |
86 |
83 85
|
mpbid |
|- ( ( ph /\ w e. Y ) -> ( # ` [ w ] .~ ) = ( P ^ ( P pCnt ( # ` [ w ] .~ ) ) ) ) |
87 |
86
|
eqcomd |
|- ( ( ph /\ w e. Y ) -> ( P ^ ( P pCnt ( # ` [ w ] .~ ) ) ) = ( # ` [ w ] .~ ) ) |
88 |
23
|
adantr |
|- ( ( ph /\ w e. Y ) -> P e. ZZ ) |
89 |
88
|
zcnd |
|- ( ( ph /\ w e. Y ) -> P e. CC ) |
90 |
89
|
exp0d |
|- ( ( ph /\ w e. Y ) -> ( P ^ 0 ) = 1 ) |
91 |
|
hash1 |
|- ( # ` 1o ) = 1 |
92 |
90 91
|
eqtr4di |
|- ( ( ph /\ w e. Y ) -> ( P ^ 0 ) = ( # ` 1o ) ) |
93 |
87 92
|
eqeq12d |
|- ( ( ph /\ w e. Y ) -> ( ( P ^ ( P pCnt ( # ` [ w ] .~ ) ) ) = ( P ^ 0 ) <-> ( # ` [ w ] .~ ) = ( # ` 1o ) ) ) |
94 |
|
df1o2 |
|- 1o = { (/) } |
95 |
|
snfi |
|- { (/) } e. Fin |
96 |
94 95
|
eqeltri |
|- 1o e. Fin |
97 |
|
hashen |
|- ( ( [ w ] .~ e. Fin /\ 1o e. Fin ) -> ( ( # ` [ w ] .~ ) = ( # ` 1o ) <-> [ w ] .~ ~~ 1o ) ) |
98 |
52 96 97
|
sylancl |
|- ( ( ph /\ w e. Y ) -> ( ( # ` [ w ] .~ ) = ( # ` 1o ) <-> [ w ] .~ ~~ 1o ) ) |
99 |
93 98
|
bitrd |
|- ( ( ph /\ w e. Y ) -> ( ( P ^ ( P pCnt ( # ` [ w ] .~ ) ) ) = ( P ^ 0 ) <-> [ w ] .~ ~~ 1o ) ) |
100 |
|
en1b |
|- ( [ w ] .~ ~~ 1o <-> [ w ] .~ = { U. [ w ] .~ } ) |
101 |
99 100
|
bitrdi |
|- ( ( ph /\ w e. Y ) -> ( ( P ^ ( P pCnt ( # ` [ w ] .~ ) ) ) = ( P ^ 0 ) <-> [ w ] .~ = { U. [ w ] .~ } ) ) |
102 |
44
|
adantr |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> w e. Y ) |
103 |
2
|
ad2antrr |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> .(+) e. ( G GrpAct Y ) ) |
104 |
1
|
gaf |
|- ( .(+) e. ( G GrpAct Y ) -> .(+) : ( X X. Y ) --> Y ) |
105 |
103 104
|
syl |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> .(+) : ( X X. Y ) --> Y ) |
106 |
|
simprl |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> h e. X ) |
107 |
105 106 102
|
fovrnd |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> ( h .(+) w ) e. Y ) |
108 |
|
eqid |
|- ( h .(+) w ) = ( h .(+) w ) |
109 |
|
oveq1 |
|- ( k = h -> ( k .(+) w ) = ( h .(+) w ) ) |
110 |
109
|
eqeq1d |
|- ( k = h -> ( ( k .(+) w ) = ( h .(+) w ) <-> ( h .(+) w ) = ( h .(+) w ) ) ) |
111 |
110
|
rspcev |
|- ( ( h e. X /\ ( h .(+) w ) = ( h .(+) w ) ) -> E. k e. X ( k .(+) w ) = ( h .(+) w ) ) |
112 |
106 108 111
|
sylancl |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> E. k e. X ( k .(+) w ) = ( h .(+) w ) ) |
113 |
7
|
gaorb |
|- ( w .~ ( h .(+) w ) <-> ( w e. Y /\ ( h .(+) w ) e. Y /\ E. k e. X ( k .(+) w ) = ( h .(+) w ) ) ) |
114 |
102 107 112 113
|
syl3anbrc |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> w .~ ( h .(+) w ) ) |
115 |
|
ovex |
|- ( h .(+) w ) e. _V |
116 |
115 46
|
elec |
|- ( ( h .(+) w ) e. [ w ] .~ <-> w .~ ( h .(+) w ) ) |
117 |
114 116
|
sylibr |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> ( h .(+) w ) e. [ w ] .~ ) |
118 |
|
simprr |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> [ w ] .~ = { U. [ w ] .~ } ) |
119 |
117 118
|
eleqtrd |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> ( h .(+) w ) e. { U. [ w ] .~ } ) |
120 |
115
|
elsn |
|- ( ( h .(+) w ) e. { U. [ w ] .~ } <-> ( h .(+) w ) = U. [ w ] .~ ) |
121 |
119 120
|
sylib |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> ( h .(+) w ) = U. [ w ] .~ ) |
122 |
48
|
adantr |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> w e. [ w ] .~ ) |
123 |
122 118
|
eleqtrd |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> w e. { U. [ w ] .~ } ) |
124 |
46
|
elsn |
|- ( w e. { U. [ w ] .~ } <-> w = U. [ w ] .~ ) |
125 |
123 124
|
sylib |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> w = U. [ w ] .~ ) |
126 |
121 125
|
eqtr4d |
|- ( ( ( ph /\ w e. Y ) /\ ( h e. X /\ [ w ] .~ = { U. [ w ] .~ } ) ) -> ( h .(+) w ) = w ) |
127 |
126
|
expr |
|- ( ( ( ph /\ w e. Y ) /\ h e. X ) -> ( [ w ] .~ = { U. [ w ] .~ } -> ( h .(+) w ) = w ) ) |
128 |
127
|
ralrimdva |
|- ( ( ph /\ w e. Y ) -> ( [ w ] .~ = { U. [ w ] .~ } -> A. h e. X ( h .(+) w ) = w ) ) |
129 |
101 128
|
sylbid |
|- ( ( ph /\ w e. Y ) -> ( ( P ^ ( P pCnt ( # ` [ w ] .~ ) ) ) = ( P ^ 0 ) -> A. h e. X ( h .(+) w ) = w ) ) |
130 |
58 129
|
syl5 |
|- ( ( ph /\ w e. Y ) -> ( ( P pCnt ( # ` [ w ] .~ ) ) = 0 -> A. h e. X ( h .(+) w ) = w ) ) |
131 |
57 130
|
sylbird |
|- ( ( ph /\ w e. Y ) -> ( -. P || ( # ` [ w ] .~ ) -> A. h e. X ( h .(+) w ) = w ) ) |
132 |
|
oveq2 |
|- ( u = w -> ( h .(+) u ) = ( h .(+) w ) ) |
133 |
|
id |
|- ( u = w -> u = w ) |
134 |
132 133
|
eqeq12d |
|- ( u = w -> ( ( h .(+) u ) = u <-> ( h .(+) w ) = w ) ) |
135 |
134
|
ralbidv |
|- ( u = w -> ( A. h e. X ( h .(+) u ) = u <-> A. h e. X ( h .(+) w ) = w ) ) |
136 |
135 6
|
elrab2 |
|- ( w e. Z <-> ( w e. Y /\ A. h e. X ( h .(+) w ) = w ) ) |
137 |
136
|
baib |
|- ( w e. Y -> ( w e. Z <-> A. h e. X ( h .(+) w ) = w ) ) |
138 |
137
|
adantl |
|- ( ( ph /\ w e. Y ) -> ( w e. Z <-> A. h e. X ( h .(+) w ) = w ) ) |
139 |
131 138
|
sylibrd |
|- ( ( ph /\ w e. Y ) -> ( -. P || ( # ` [ w ] .~ ) -> w e. Z ) ) |
140 |
1 2 3 4 5 6 7
|
sylow2alem1 |
|- ( ( ph /\ w e. Z ) -> [ w ] .~ = { w } ) |
141 |
|
simpr |
|- ( ( ph /\ w e. Z ) -> w e. Z ) |
142 |
141
|
snssd |
|- ( ( ph /\ w e. Z ) -> { w } C_ Z ) |
143 |
140 142
|
eqsstrd |
|- ( ( ph /\ w e. Z ) -> [ w ] .~ C_ Z ) |
144 |
143
|
ex |
|- ( ph -> ( w e. Z -> [ w ] .~ C_ Z ) ) |
145 |
144
|
adantr |
|- ( ( ph /\ w e. Y ) -> ( w e. Z -> [ w ] .~ C_ Z ) ) |
146 |
139 145
|
syld |
|- ( ( ph /\ w e. Y ) -> ( -. P || ( # ` [ w ] .~ ) -> [ w ] .~ C_ Z ) ) |
147 |
146
|
con1d |
|- ( ( ph /\ w e. Y ) -> ( -. [ w ] .~ C_ Z -> P || ( # ` [ w ] .~ ) ) ) |
148 |
34 41 147
|
ectocld |
|- ( ( ph /\ z e. ( Y /. .~ ) ) -> ( -. z e. ~P Z -> P || ( # ` z ) ) ) |
149 |
148
|
impr |
|- ( ( ph /\ ( z e. ( Y /. .~ ) /\ -. z e. ~P Z ) ) -> P || ( # ` z ) ) |
150 |
33 149
|
sylan2b |
|- ( ( ph /\ z e. ( ( Y /. .~ ) \ ~P Z ) ) -> P || ( # ` z ) ) |
151 |
15 23 32 150
|
fsumdvds |
|- ( ph -> P || sum_ z e. ( ( Y /. .~ ) \ ~P Z ) ( # ` z ) ) |