Step |
Hyp |
Ref |
Expression |
1 |
|
dchrpt.g |
|- G = ( DChr ` N ) |
2 |
|
dchrpt.z |
|- Z = ( Z/nZ ` N ) |
3 |
|
dchrpt.d |
|- D = ( Base ` G ) |
4 |
|
dchrpt.b |
|- B = ( Base ` Z ) |
5 |
|
dchrpt.1 |
|- .1. = ( 1r ` Z ) |
6 |
|
dchrpt.n |
|- ( ph -> N e. NN ) |
7 |
|
dchrpt.n1 |
|- ( ph -> A =/= .1. ) |
8 |
|
dchrpt.u |
|- U = ( Unit ` Z ) |
9 |
|
dchrpt.h |
|- H = ( ( mulGrp ` Z ) |`s U ) |
10 |
|
dchrpt.m |
|- .x. = ( .g ` H ) |
11 |
|
dchrpt.s |
|- S = ( k e. dom W |-> ran ( n e. ZZ |-> ( n .x. ( W ` k ) ) ) ) |
12 |
|
dchrpt.au |
|- ( ph -> A e. U ) |
13 |
|
dchrpt.w |
|- ( ph -> W e. Word U ) |
14 |
|
dchrpt.2 |
|- ( ph -> H dom DProd S ) |
15 |
|
dchrpt.3 |
|- ( ph -> ( H DProd S ) = U ) |
16 |
|
dchrpt.p |
|- P = ( H dProj S ) |
17 |
|
dchrpt.o |
|- O = ( od ` H ) |
18 |
|
dchrpt.t |
|- T = ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) |
19 |
|
dchrpt.i |
|- ( ph -> I e. dom W ) |
20 |
|
dchrpt.4 |
|- ( ph -> ( ( P ` I ) ` A ) =/= .1. ) |
21 |
|
dchrpt.5 |
|- X = ( u e. U |-> ( iota h E. m e. ZZ ( ( ( P ` I ) ` u ) = ( m .x. ( W ` I ) ) /\ h = ( T ^ m ) ) ) ) |
22 |
|
fveq2 |
|- ( v = x -> ( X ` v ) = ( X ` x ) ) |
23 |
|
fveq2 |
|- ( v = y -> ( X ` v ) = ( X ` y ) ) |
24 |
|
fveq2 |
|- ( v = ( x ( .r ` Z ) y ) -> ( X ` v ) = ( X ` ( x ( .r ` Z ) y ) ) ) |
25 |
|
fveq2 |
|- ( v = ( 1r ` Z ) -> ( X ` v ) = ( X ` ( 1r ` Z ) ) ) |
26 |
|
zex |
|- ZZ e. _V |
27 |
26
|
mptex |
|- ( n e. ZZ |-> ( n .x. ( W ` k ) ) ) e. _V |
28 |
27
|
rnex |
|- ran ( n e. ZZ |-> ( n .x. ( W ` k ) ) ) e. _V |
29 |
28 11
|
dmmpti |
|- dom S = dom W |
30 |
29
|
a1i |
|- ( ph -> dom S = dom W ) |
31 |
14 30 16 19
|
dpjf |
|- ( ph -> ( P ` I ) : ( H DProd S ) --> ( S ` I ) ) |
32 |
15
|
feq2d |
|- ( ph -> ( ( P ` I ) : ( H DProd S ) --> ( S ` I ) <-> ( P ` I ) : U --> ( S ` I ) ) ) |
33 |
31 32
|
mpbid |
|- ( ph -> ( P ` I ) : U --> ( S ` I ) ) |
34 |
33
|
ffvelrnda |
|- ( ( ph /\ v e. U ) -> ( ( P ` I ) ` v ) e. ( S ` I ) ) |
35 |
19
|
adantr |
|- ( ( ph /\ v e. U ) -> I e. dom W ) |
36 |
|
oveq1 |
|- ( n = a -> ( n .x. ( W ` k ) ) = ( a .x. ( W ` k ) ) ) |
37 |
36
|
cbvmptv |
|- ( n e. ZZ |-> ( n .x. ( W ` k ) ) ) = ( a e. ZZ |-> ( a .x. ( W ` k ) ) ) |
38 |
|
fveq2 |
|- ( k = I -> ( W ` k ) = ( W ` I ) ) |
39 |
38
|
oveq2d |
|- ( k = I -> ( a .x. ( W ` k ) ) = ( a .x. ( W ` I ) ) ) |
40 |
39
|
mpteq2dv |
|- ( k = I -> ( a e. ZZ |-> ( a .x. ( W ` k ) ) ) = ( a e. ZZ |-> ( a .x. ( W ` I ) ) ) ) |
41 |
37 40
|
syl5eq |
|- ( k = I -> ( n e. ZZ |-> ( n .x. ( W ` k ) ) ) = ( a e. ZZ |-> ( a .x. ( W ` I ) ) ) ) |
42 |
41
|
rneqd |
|- ( k = I -> ran ( n e. ZZ |-> ( n .x. ( W ` k ) ) ) = ran ( a e. ZZ |-> ( a .x. ( W ` I ) ) ) ) |
43 |
42 11 28
|
fvmpt3i |
|- ( I e. dom W -> ( S ` I ) = ran ( a e. ZZ |-> ( a .x. ( W ` I ) ) ) ) |
44 |
35 43
|
syl |
|- ( ( ph /\ v e. U ) -> ( S ` I ) = ran ( a e. ZZ |-> ( a .x. ( W ` I ) ) ) ) |
45 |
34 44
|
eleqtrd |
|- ( ( ph /\ v e. U ) -> ( ( P ` I ) ` v ) e. ran ( a e. ZZ |-> ( a .x. ( W ` I ) ) ) ) |
46 |
|
eqid |
|- ( a e. ZZ |-> ( a .x. ( W ` I ) ) ) = ( a e. ZZ |-> ( a .x. ( W ` I ) ) ) |
47 |
|
ovex |
|- ( a .x. ( W ` I ) ) e. _V |
48 |
46 47
|
elrnmpti |
|- ( ( ( P ` I ) ` v ) e. ran ( a e. ZZ |-> ( a .x. ( W ` I ) ) ) <-> E. a e. ZZ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) ) |
49 |
45 48
|
sylib |
|- ( ( ph /\ v e. U ) -> E. a e. ZZ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) ) |
50 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
|
dchrptlem1 |
|- ( ( ( ph /\ v e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) ) ) -> ( X ` v ) = ( T ^ a ) ) |
51 |
|
neg1cn |
|- -u 1 e. CC |
52 |
|
2re |
|- 2 e. RR |
53 |
6
|
nnnn0d |
|- ( ph -> N e. NN0 ) |
54 |
2
|
zncrng |
|- ( N e. NN0 -> Z e. CRing ) |
55 |
|
crngring |
|- ( Z e. CRing -> Z e. Ring ) |
56 |
53 54 55
|
3syl |
|- ( ph -> Z e. Ring ) |
57 |
8 9
|
unitgrp |
|- ( Z e. Ring -> H e. Grp ) |
58 |
56 57
|
syl |
|- ( ph -> H e. Grp ) |
59 |
2 4
|
znfi |
|- ( N e. NN -> B e. Fin ) |
60 |
6 59
|
syl |
|- ( ph -> B e. Fin ) |
61 |
4 8
|
unitss |
|- U C_ B |
62 |
|
ssfi |
|- ( ( B e. Fin /\ U C_ B ) -> U e. Fin ) |
63 |
60 61 62
|
sylancl |
|- ( ph -> U e. Fin ) |
64 |
|
wrdf |
|- ( W e. Word U -> W : ( 0 ..^ ( # ` W ) ) --> U ) |
65 |
13 64
|
syl |
|- ( ph -> W : ( 0 ..^ ( # ` W ) ) --> U ) |
66 |
65
|
fdmd |
|- ( ph -> dom W = ( 0 ..^ ( # ` W ) ) ) |
67 |
19 66
|
eleqtrd |
|- ( ph -> I e. ( 0 ..^ ( # ` W ) ) ) |
68 |
65 67
|
ffvelrnd |
|- ( ph -> ( W ` I ) e. U ) |
69 |
8 9
|
unitgrpbas |
|- U = ( Base ` H ) |
70 |
69 17
|
odcl2 |
|- ( ( H e. Grp /\ U e. Fin /\ ( W ` I ) e. U ) -> ( O ` ( W ` I ) ) e. NN ) |
71 |
58 63 68 70
|
syl3anc |
|- ( ph -> ( O ` ( W ` I ) ) e. NN ) |
72 |
|
nndivre |
|- ( ( 2 e. RR /\ ( O ` ( W ` I ) ) e. NN ) -> ( 2 / ( O ` ( W ` I ) ) ) e. RR ) |
73 |
52 71 72
|
sylancr |
|- ( ph -> ( 2 / ( O ` ( W ` I ) ) ) e. RR ) |
74 |
73
|
recnd |
|- ( ph -> ( 2 / ( O ` ( W ` I ) ) ) e. CC ) |
75 |
|
cxpcl |
|- ( ( -u 1 e. CC /\ ( 2 / ( O ` ( W ` I ) ) ) e. CC ) -> ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) e. CC ) |
76 |
51 74 75
|
sylancr |
|- ( ph -> ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) e. CC ) |
77 |
18 76
|
eqeltrid |
|- ( ph -> T e. CC ) |
78 |
77
|
ad2antrr |
|- ( ( ( ph /\ v e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) ) ) -> T e. CC ) |
79 |
51
|
a1i |
|- ( ph -> -u 1 e. CC ) |
80 |
|
neg1ne0 |
|- -u 1 =/= 0 |
81 |
80
|
a1i |
|- ( ph -> -u 1 =/= 0 ) |
82 |
79 81 74
|
cxpne0d |
|- ( ph -> ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) =/= 0 ) |
83 |
18
|
neeq1i |
|- ( T =/= 0 <-> ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) =/= 0 ) |
84 |
82 83
|
sylibr |
|- ( ph -> T =/= 0 ) |
85 |
84
|
ad2antrr |
|- ( ( ( ph /\ v e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) ) ) -> T =/= 0 ) |
86 |
|
simprl |
|- ( ( ( ph /\ v e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) ) ) -> a e. ZZ ) |
87 |
78 85 86
|
expclzd |
|- ( ( ( ph /\ v e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) ) ) -> ( T ^ a ) e. CC ) |
88 |
50 87
|
eqeltrd |
|- ( ( ( ph /\ v e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) ) ) -> ( X ` v ) e. CC ) |
89 |
49 88
|
rexlimddv |
|- ( ( ph /\ v e. U ) -> ( X ` v ) e. CC ) |
90 |
|
fveqeq2 |
|- ( v = x -> ( ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) <-> ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) ) ) |
91 |
90
|
rexbidv |
|- ( v = x -> ( E. a e. ZZ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) <-> E. a e. ZZ ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) ) ) |
92 |
49
|
ralrimiva |
|- ( ph -> A. v e. U E. a e. ZZ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) ) |
93 |
92
|
adantr |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> A. v e. U E. a e. ZZ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) ) |
94 |
|
simprl |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> x e. U ) |
95 |
91 93 94
|
rspcdva |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> E. a e. ZZ ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) ) |
96 |
|
fveqeq2 |
|- ( v = y -> ( ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) <-> ( ( P ` I ) ` y ) = ( a .x. ( W ` I ) ) ) ) |
97 |
96
|
rexbidv |
|- ( v = y -> ( E. a e. ZZ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) <-> E. a e. ZZ ( ( P ` I ) ` y ) = ( a .x. ( W ` I ) ) ) ) |
98 |
|
oveq1 |
|- ( a = b -> ( a .x. ( W ` I ) ) = ( b .x. ( W ` I ) ) ) |
99 |
98
|
eqeq2d |
|- ( a = b -> ( ( ( P ` I ) ` y ) = ( a .x. ( W ` I ) ) <-> ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) |
100 |
99
|
cbvrexvw |
|- ( E. a e. ZZ ( ( P ` I ) ` y ) = ( a .x. ( W ` I ) ) <-> E. b e. ZZ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) |
101 |
97 100
|
bitrdi |
|- ( v = y -> ( E. a e. ZZ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) <-> E. b e. ZZ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) |
102 |
|
simprr |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> y e. U ) |
103 |
101 93 102
|
rspcdva |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> E. b e. ZZ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) |
104 |
|
reeanv |
|- ( E. a e. ZZ E. b e. ZZ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) <-> ( E. a e. ZZ ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ E. b e. ZZ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) |
105 |
77
|
ad2antrr |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> T e. CC ) |
106 |
84
|
ad2antrr |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> T =/= 0 ) |
107 |
|
simprll |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> a e. ZZ ) |
108 |
|
simprlr |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> b e. ZZ ) |
109 |
|
expaddz |
|- ( ( ( T e. CC /\ T =/= 0 ) /\ ( a e. ZZ /\ b e. ZZ ) ) -> ( T ^ ( a + b ) ) = ( ( T ^ a ) x. ( T ^ b ) ) ) |
110 |
105 106 107 108 109
|
syl22anc |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( T ^ ( a + b ) ) = ( ( T ^ a ) x. ( T ^ b ) ) ) |
111 |
|
simpll |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ph ) |
112 |
56
|
ad2antrr |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> Z e. Ring ) |
113 |
94
|
adantr |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> x e. U ) |
114 |
102
|
adantr |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> y e. U ) |
115 |
|
eqid |
|- ( .r ` Z ) = ( .r ` Z ) |
116 |
8 115
|
unitmulcl |
|- ( ( Z e. Ring /\ x e. U /\ y e. U ) -> ( x ( .r ` Z ) y ) e. U ) |
117 |
112 113 114 116
|
syl3anc |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( x ( .r ` Z ) y ) e. U ) |
118 |
107 108
|
zaddcld |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( a + b ) e. ZZ ) |
119 |
|
simprrl |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) ) |
120 |
|
simprrr |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) |
121 |
119 120
|
oveq12d |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( ( ( P ` I ) ` x ) ( .r ` Z ) ( ( P ` I ) ` y ) ) = ( ( a .x. ( W ` I ) ) ( .r ` Z ) ( b .x. ( W ` I ) ) ) ) |
122 |
14 30 16 19
|
dpjghm |
|- ( ph -> ( P ` I ) e. ( ( H |`s ( H DProd S ) ) GrpHom H ) ) |
123 |
15
|
oveq2d |
|- ( ph -> ( H |`s ( H DProd S ) ) = ( H |`s U ) ) |
124 |
9
|
ovexi |
|- H e. _V |
125 |
69
|
ressid |
|- ( H e. _V -> ( H |`s U ) = H ) |
126 |
124 125
|
ax-mp |
|- ( H |`s U ) = H |
127 |
123 126
|
eqtrdi |
|- ( ph -> ( H |`s ( H DProd S ) ) = H ) |
128 |
127
|
oveq1d |
|- ( ph -> ( ( H |`s ( H DProd S ) ) GrpHom H ) = ( H GrpHom H ) ) |
129 |
122 128
|
eleqtrd |
|- ( ph -> ( P ` I ) e. ( H GrpHom H ) ) |
130 |
129
|
ad2antrr |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( P ` I ) e. ( H GrpHom H ) ) |
131 |
8
|
fvexi |
|- U e. _V |
132 |
|
eqid |
|- ( mulGrp ` Z ) = ( mulGrp ` Z ) |
133 |
132 115
|
mgpplusg |
|- ( .r ` Z ) = ( +g ` ( mulGrp ` Z ) ) |
134 |
9 133
|
ressplusg |
|- ( U e. _V -> ( .r ` Z ) = ( +g ` H ) ) |
135 |
131 134
|
ax-mp |
|- ( .r ` Z ) = ( +g ` H ) |
136 |
69 135 135
|
ghmlin |
|- ( ( ( P ` I ) e. ( H GrpHom H ) /\ x e. U /\ y e. U ) -> ( ( P ` I ) ` ( x ( .r ` Z ) y ) ) = ( ( ( P ` I ) ` x ) ( .r ` Z ) ( ( P ` I ) ` y ) ) ) |
137 |
130 113 114 136
|
syl3anc |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( ( P ` I ) ` ( x ( .r ` Z ) y ) ) = ( ( ( P ` I ) ` x ) ( .r ` Z ) ( ( P ` I ) ` y ) ) ) |
138 |
58
|
ad2antrr |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> H e. Grp ) |
139 |
68
|
ad2antrr |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( W ` I ) e. U ) |
140 |
69 10 135
|
mulgdir |
|- ( ( H e. Grp /\ ( a e. ZZ /\ b e. ZZ /\ ( W ` I ) e. U ) ) -> ( ( a + b ) .x. ( W ` I ) ) = ( ( a .x. ( W ` I ) ) ( .r ` Z ) ( b .x. ( W ` I ) ) ) ) |
141 |
138 107 108 139 140
|
syl13anc |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( ( a + b ) .x. ( W ` I ) ) = ( ( a .x. ( W ` I ) ) ( .r ` Z ) ( b .x. ( W ` I ) ) ) ) |
142 |
121 137 141
|
3eqtr4d |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( ( P ` I ) ` ( x ( .r ` Z ) y ) ) = ( ( a + b ) .x. ( W ` I ) ) ) |
143 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
|
dchrptlem1 |
|- ( ( ( ph /\ ( x ( .r ` Z ) y ) e. U ) /\ ( ( a + b ) e. ZZ /\ ( ( P ` I ) ` ( x ( .r ` Z ) y ) ) = ( ( a + b ) .x. ( W ` I ) ) ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( T ^ ( a + b ) ) ) |
144 |
111 117 118 142 143
|
syl22anc |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( T ^ ( a + b ) ) ) |
145 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
|
dchrptlem1 |
|- ( ( ( ph /\ x e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) ) ) -> ( X ` x ) = ( T ^ a ) ) |
146 |
111 113 107 119 145
|
syl22anc |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( X ` x ) = ( T ^ a ) ) |
147 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
|
dchrptlem1 |
|- ( ( ( ph /\ y e. U ) /\ ( b e. ZZ /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) -> ( X ` y ) = ( T ^ b ) ) |
148 |
111 114 108 120 147
|
syl22anc |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( X ` y ) = ( T ^ b ) ) |
149 |
146 148
|
oveq12d |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( ( X ` x ) x. ( X ` y ) ) = ( ( T ^ a ) x. ( T ^ b ) ) ) |
150 |
110 144 149
|
3eqtr4d |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( ( a e. ZZ /\ b e. ZZ ) /\ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) ) |
151 |
150
|
expr |
|- ( ( ( ph /\ ( x e. U /\ y e. U ) ) /\ ( a e. ZZ /\ b e. ZZ ) ) -> ( ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) ) ) |
152 |
151
|
rexlimdvva |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( E. a e. ZZ E. b e. ZZ ( ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) ) ) |
153 |
104 152
|
syl5bir |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( ( E. a e. ZZ ( ( P ` I ) ` x ) = ( a .x. ( W ` I ) ) /\ E. b e. ZZ ( ( P ` I ) ` y ) = ( b .x. ( W ` I ) ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) ) ) |
154 |
95 103 153
|
mp2and |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) ) |
155 |
|
id |
|- ( ph -> ph ) |
156 |
|
eqid |
|- ( 1r ` Z ) = ( 1r ` Z ) |
157 |
8 156
|
1unit |
|- ( Z e. Ring -> ( 1r ` Z ) e. U ) |
158 |
56 157
|
syl |
|- ( ph -> ( 1r ` Z ) e. U ) |
159 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
160 |
|
eqid |
|- ( 0g ` H ) = ( 0g ` H ) |
161 |
160 160
|
ghmid |
|- ( ( P ` I ) e. ( H GrpHom H ) -> ( ( P ` I ) ` ( 0g ` H ) ) = ( 0g ` H ) ) |
162 |
129 161
|
syl |
|- ( ph -> ( ( P ` I ) ` ( 0g ` H ) ) = ( 0g ` H ) ) |
163 |
8 9 156
|
unitgrpid |
|- ( Z e. Ring -> ( 1r ` Z ) = ( 0g ` H ) ) |
164 |
56 163
|
syl |
|- ( ph -> ( 1r ` Z ) = ( 0g ` H ) ) |
165 |
164
|
fveq2d |
|- ( ph -> ( ( P ` I ) ` ( 1r ` Z ) ) = ( ( P ` I ) ` ( 0g ` H ) ) ) |
166 |
69 160 10
|
mulg0 |
|- ( ( W ` I ) e. U -> ( 0 .x. ( W ` I ) ) = ( 0g ` H ) ) |
167 |
68 166
|
syl |
|- ( ph -> ( 0 .x. ( W ` I ) ) = ( 0g ` H ) ) |
168 |
162 165 167
|
3eqtr4d |
|- ( ph -> ( ( P ` I ) ` ( 1r ` Z ) ) = ( 0 .x. ( W ` I ) ) ) |
169 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
|
dchrptlem1 |
|- ( ( ( ph /\ ( 1r ` Z ) e. U ) /\ ( 0 e. ZZ /\ ( ( P ` I ) ` ( 1r ` Z ) ) = ( 0 .x. ( W ` I ) ) ) ) -> ( X ` ( 1r ` Z ) ) = ( T ^ 0 ) ) |
170 |
155 158 159 168 169
|
syl22anc |
|- ( ph -> ( X ` ( 1r ` Z ) ) = ( T ^ 0 ) ) |
171 |
77
|
exp0d |
|- ( ph -> ( T ^ 0 ) = 1 ) |
172 |
170 171
|
eqtrd |
|- ( ph -> ( X ` ( 1r ` Z ) ) = 1 ) |
173 |
1 2 4 8 6 3 22 23 24 25 89 154 172
|
dchrelbasd |
|- ( ph -> ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) e. D ) |
174 |
61 12
|
sselid |
|- ( ph -> A e. B ) |
175 |
|
eleq1 |
|- ( v = A -> ( v e. U <-> A e. U ) ) |
176 |
|
fveq2 |
|- ( v = A -> ( X ` v ) = ( X ` A ) ) |
177 |
175 176
|
ifbieq1d |
|- ( v = A -> if ( v e. U , ( X ` v ) , 0 ) = if ( A e. U , ( X ` A ) , 0 ) ) |
178 |
|
eqid |
|- ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) = ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) |
179 |
|
fvex |
|- ( X ` v ) e. _V |
180 |
|
c0ex |
|- 0 e. _V |
181 |
179 180
|
ifex |
|- if ( v e. U , ( X ` v ) , 0 ) e. _V |
182 |
177 178 181
|
fvmpt3i |
|- ( A e. B -> ( ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) ` A ) = if ( A e. U , ( X ` A ) , 0 ) ) |
183 |
174 182
|
syl |
|- ( ph -> ( ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) ` A ) = if ( A e. U , ( X ` A ) , 0 ) ) |
184 |
12
|
iftrued |
|- ( ph -> if ( A e. U , ( X ` A ) , 0 ) = ( X ` A ) ) |
185 |
183 184
|
eqtrd |
|- ( ph -> ( ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) ` A ) = ( X ` A ) ) |
186 |
|
fveqeq2 |
|- ( v = A -> ( ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) <-> ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) |
187 |
186
|
rexbidv |
|- ( v = A -> ( E. a e. ZZ ( ( P ` I ) ` v ) = ( a .x. ( W ` I ) ) <-> E. a e. ZZ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) |
188 |
187 92 12
|
rspcdva |
|- ( ph -> E. a e. ZZ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) |
189 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
|
dchrptlem1 |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( X ` A ) = ( T ^ a ) ) |
190 |
18
|
oveq1i |
|- ( T ^ a ) = ( ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) ^ a ) |
191 |
189 190
|
eqtrdi |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( X ` A ) = ( ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) ^ a ) ) |
192 |
20
|
ad2antrr |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( ( P ` I ) ` A ) =/= .1. ) |
193 |
58
|
ad2antrr |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> H e. Grp ) |
194 |
68
|
ad2antrr |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( W ` I ) e. U ) |
195 |
|
simprl |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> a e. ZZ ) |
196 |
69 17 10 160
|
oddvds |
|- ( ( H e. Grp /\ ( W ` I ) e. U /\ a e. ZZ ) -> ( ( O ` ( W ` I ) ) || a <-> ( a .x. ( W ` I ) ) = ( 0g ` H ) ) ) |
197 |
193 194 195 196
|
syl3anc |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( ( O ` ( W ` I ) ) || a <-> ( a .x. ( W ` I ) ) = ( 0g ` H ) ) ) |
198 |
71
|
ad2antrr |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( O ` ( W ` I ) ) e. NN ) |
199 |
|
root1eq1 |
|- ( ( ( O ` ( W ` I ) ) e. NN /\ a e. ZZ ) -> ( ( ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) ^ a ) = 1 <-> ( O ` ( W ` I ) ) || a ) ) |
200 |
198 195 199
|
syl2anc |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( ( ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) ^ a ) = 1 <-> ( O ` ( W ` I ) ) || a ) ) |
201 |
|
simprr |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) |
202 |
5 164
|
syl5eq |
|- ( ph -> .1. = ( 0g ` H ) ) |
203 |
202
|
ad2antrr |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> .1. = ( 0g ` H ) ) |
204 |
201 203
|
eqeq12d |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( ( ( P ` I ) ` A ) = .1. <-> ( a .x. ( W ` I ) ) = ( 0g ` H ) ) ) |
205 |
197 200 204
|
3bitr4d |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( ( ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) ^ a ) = 1 <-> ( ( P ` I ) ` A ) = .1. ) ) |
206 |
205
|
necon3bid |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( ( ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) ^ a ) =/= 1 <-> ( ( P ` I ) ` A ) =/= .1. ) ) |
207 |
192 206
|
mpbird |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( ( -u 1 ^c ( 2 / ( O ` ( W ` I ) ) ) ) ^ a ) =/= 1 ) |
208 |
191 207
|
eqnetrd |
|- ( ( ( ph /\ A e. U ) /\ ( a e. ZZ /\ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) ) ) -> ( X ` A ) =/= 1 ) |
209 |
208
|
rexlimdvaa |
|- ( ( ph /\ A e. U ) -> ( E. a e. ZZ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) -> ( X ` A ) =/= 1 ) ) |
210 |
12 209
|
mpdan |
|- ( ph -> ( E. a e. ZZ ( ( P ` I ) ` A ) = ( a .x. ( W ` I ) ) -> ( X ` A ) =/= 1 ) ) |
211 |
188 210
|
mpd |
|- ( ph -> ( X ` A ) =/= 1 ) |
212 |
185 211
|
eqnetrd |
|- ( ph -> ( ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) ` A ) =/= 1 ) |
213 |
|
fveq1 |
|- ( x = ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) -> ( x ` A ) = ( ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) ` A ) ) |
214 |
213
|
neeq1d |
|- ( x = ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) -> ( ( x ` A ) =/= 1 <-> ( ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) ` A ) =/= 1 ) ) |
215 |
214
|
rspcev |
|- ( ( ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) e. D /\ ( ( v e. B |-> if ( v e. U , ( X ` v ) , 0 ) ) ` A ) =/= 1 ) -> E. x e. D ( x ` A ) =/= 1 ) |
216 |
173 212 215
|
syl2anc |
|- ( ph -> E. x e. D ( x ` A ) =/= 1 ) |