Step |
Hyp |
Ref |
Expression |
1 |
|
sumdchr.g |
|- G = ( DChr ` N ) |
2 |
|
sumdchr.d |
|- D = ( Base ` G ) |
3 |
|
sumdchr2.z |
|- Z = ( Z/nZ ` N ) |
4 |
|
sumdchr2.1 |
|- .1. = ( 1r ` Z ) |
5 |
|
sumdchr2.b |
|- B = ( Base ` Z ) |
6 |
|
sumdchr2.n |
|- ( ph -> N e. NN ) |
7 |
|
sumdchr2.x |
|- ( ph -> A e. B ) |
8 |
|
eqeq2 |
|- ( ( # ` D ) = if ( A = .1. , ( # ` D ) , 0 ) -> ( sum_ x e. D ( x ` A ) = ( # ` D ) <-> sum_ x e. D ( x ` A ) = if ( A = .1. , ( # ` D ) , 0 ) ) ) |
9 |
|
eqeq2 |
|- ( 0 = if ( A = .1. , ( # ` D ) , 0 ) -> ( sum_ x e. D ( x ` A ) = 0 <-> sum_ x e. D ( x ` A ) = if ( A = .1. , ( # ` D ) , 0 ) ) ) |
10 |
|
fveq2 |
|- ( A = .1. -> ( x ` A ) = ( x ` .1. ) ) |
11 |
1 3 2
|
dchrmhm |
|- D C_ ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) |
12 |
|
simpr |
|- ( ( ph /\ x e. D ) -> x e. D ) |
13 |
11 12
|
sselid |
|- ( ( ph /\ x e. D ) -> x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) |
14 |
|
eqid |
|- ( mulGrp ` Z ) = ( mulGrp ` Z ) |
15 |
14 4
|
ringidval |
|- .1. = ( 0g ` ( mulGrp ` Z ) ) |
16 |
|
eqid |
|- ( mulGrp ` CCfld ) = ( mulGrp ` CCfld ) |
17 |
|
cnfld1 |
|- 1 = ( 1r ` CCfld ) |
18 |
16 17
|
ringidval |
|- 1 = ( 0g ` ( mulGrp ` CCfld ) ) |
19 |
15 18
|
mhm0 |
|- ( x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) -> ( x ` .1. ) = 1 ) |
20 |
13 19
|
syl |
|- ( ( ph /\ x e. D ) -> ( x ` .1. ) = 1 ) |
21 |
10 20
|
sylan9eqr |
|- ( ( ( ph /\ x e. D ) /\ A = .1. ) -> ( x ` A ) = 1 ) |
22 |
21
|
an32s |
|- ( ( ( ph /\ A = .1. ) /\ x e. D ) -> ( x ` A ) = 1 ) |
23 |
22
|
sumeq2dv |
|- ( ( ph /\ A = .1. ) -> sum_ x e. D ( x ` A ) = sum_ x e. D 1 ) |
24 |
1 2
|
dchrfi |
|- ( N e. NN -> D e. Fin ) |
25 |
6 24
|
syl |
|- ( ph -> D e. Fin ) |
26 |
|
ax-1cn |
|- 1 e. CC |
27 |
|
fsumconst |
|- ( ( D e. Fin /\ 1 e. CC ) -> sum_ x e. D 1 = ( ( # ` D ) x. 1 ) ) |
28 |
25 26 27
|
sylancl |
|- ( ph -> sum_ x e. D 1 = ( ( # ` D ) x. 1 ) ) |
29 |
|
hashcl |
|- ( D e. Fin -> ( # ` D ) e. NN0 ) |
30 |
6 24 29
|
3syl |
|- ( ph -> ( # ` D ) e. NN0 ) |
31 |
30
|
nn0cnd |
|- ( ph -> ( # ` D ) e. CC ) |
32 |
31
|
mulid1d |
|- ( ph -> ( ( # ` D ) x. 1 ) = ( # ` D ) ) |
33 |
28 32
|
eqtrd |
|- ( ph -> sum_ x e. D 1 = ( # ` D ) ) |
34 |
33
|
adantr |
|- ( ( ph /\ A = .1. ) -> sum_ x e. D 1 = ( # ` D ) ) |
35 |
23 34
|
eqtrd |
|- ( ( ph /\ A = .1. ) -> sum_ x e. D ( x ` A ) = ( # ` D ) ) |
36 |
|
df-ne |
|- ( A =/= .1. <-> -. A = .1. ) |
37 |
6
|
adantr |
|- ( ( ph /\ A =/= .1. ) -> N e. NN ) |
38 |
|
simpr |
|- ( ( ph /\ A =/= .1. ) -> A =/= .1. ) |
39 |
7
|
adantr |
|- ( ( ph /\ A =/= .1. ) -> A e. B ) |
40 |
1 3 2 5 4 37 38 39
|
dchrpt |
|- ( ( ph /\ A =/= .1. ) -> E. y e. D ( y ` A ) =/= 1 ) |
41 |
37
|
adantr |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> N e. NN ) |
42 |
41 24
|
syl |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> D e. Fin ) |
43 |
|
simpr |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> x e. D ) |
44 |
1 3 2 5 43
|
dchrf |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> x : B --> CC ) |
45 |
39
|
adantr |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> A e. B ) |
46 |
45
|
adantr |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> A e. B ) |
47 |
44 46
|
ffvelrnd |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> ( x ` A ) e. CC ) |
48 |
42 47
|
fsumcl |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> sum_ x e. D ( x ` A ) e. CC ) |
49 |
|
0cnd |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> 0 e. CC ) |
50 |
|
simprl |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> y e. D ) |
51 |
1 3 2 5 50
|
dchrf |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> y : B --> CC ) |
52 |
51 45
|
ffvelrnd |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( y ` A ) e. CC ) |
53 |
|
subcl |
|- ( ( ( y ` A ) e. CC /\ 1 e. CC ) -> ( ( y ` A ) - 1 ) e. CC ) |
54 |
52 26 53
|
sylancl |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( y ` A ) - 1 ) e. CC ) |
55 |
|
simprr |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( y ` A ) =/= 1 ) |
56 |
|
subeq0 |
|- ( ( ( y ` A ) e. CC /\ 1 e. CC ) -> ( ( ( y ` A ) - 1 ) = 0 <-> ( y ` A ) = 1 ) ) |
57 |
52 26 56
|
sylancl |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( ( y ` A ) - 1 ) = 0 <-> ( y ` A ) = 1 ) ) |
58 |
57
|
necon3bid |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( ( y ` A ) - 1 ) =/= 0 <-> ( y ` A ) =/= 1 ) ) |
59 |
55 58
|
mpbird |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( y ` A ) - 1 ) =/= 0 ) |
60 |
|
oveq2 |
|- ( z = x -> ( y ( +g ` G ) z ) = ( y ( +g ` G ) x ) ) |
61 |
60
|
fveq1d |
|- ( z = x -> ( ( y ( +g ` G ) z ) ` A ) = ( ( y ( +g ` G ) x ) ` A ) ) |
62 |
61
|
cbvsumv |
|- sum_ z e. D ( ( y ( +g ` G ) z ) ` A ) = sum_ x e. D ( ( y ( +g ` G ) x ) ` A ) |
63 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
64 |
50
|
adantr |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> y e. D ) |
65 |
1 3 2 63 64 43
|
dchrmul |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> ( y ( +g ` G ) x ) = ( y oF x. x ) ) |
66 |
65
|
fveq1d |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> ( ( y ( +g ` G ) x ) ` A ) = ( ( y oF x. x ) ` A ) ) |
67 |
51
|
adantr |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> y : B --> CC ) |
68 |
67
|
ffnd |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> y Fn B ) |
69 |
44
|
ffnd |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> x Fn B ) |
70 |
5
|
fvexi |
|- B e. _V |
71 |
70
|
a1i |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> B e. _V ) |
72 |
|
fnfvof |
|- ( ( ( y Fn B /\ x Fn B ) /\ ( B e. _V /\ A e. B ) ) -> ( ( y oF x. x ) ` A ) = ( ( y ` A ) x. ( x ` A ) ) ) |
73 |
68 69 71 46 72
|
syl22anc |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> ( ( y oF x. x ) ` A ) = ( ( y ` A ) x. ( x ` A ) ) ) |
74 |
66 73
|
eqtrd |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ x e. D ) -> ( ( y ( +g ` G ) x ) ` A ) = ( ( y ` A ) x. ( x ` A ) ) ) |
75 |
74
|
sumeq2dv |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> sum_ x e. D ( ( y ( +g ` G ) x ) ` A ) = sum_ x e. D ( ( y ` A ) x. ( x ` A ) ) ) |
76 |
62 75
|
syl5eq |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> sum_ z e. D ( ( y ( +g ` G ) z ) ` A ) = sum_ x e. D ( ( y ` A ) x. ( x ` A ) ) ) |
77 |
|
fveq1 |
|- ( x = ( y ( +g ` G ) z ) -> ( x ` A ) = ( ( y ( +g ` G ) z ) ` A ) ) |
78 |
1
|
dchrabl |
|- ( N e. NN -> G e. Abel ) |
79 |
|
ablgrp |
|- ( G e. Abel -> G e. Grp ) |
80 |
41 78 79
|
3syl |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> G e. Grp ) |
81 |
|
eqid |
|- ( a e. D |-> ( b e. D |-> ( a ( +g ` G ) b ) ) ) = ( a e. D |-> ( b e. D |-> ( a ( +g ` G ) b ) ) ) |
82 |
81 2 63
|
grplactf1o |
|- ( ( G e. Grp /\ y e. D ) -> ( ( a e. D |-> ( b e. D |-> ( a ( +g ` G ) b ) ) ) ` y ) : D -1-1-onto-> D ) |
83 |
80 50 82
|
syl2anc |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( a e. D |-> ( b e. D |-> ( a ( +g ` G ) b ) ) ) ` y ) : D -1-1-onto-> D ) |
84 |
81 2
|
grplactval |
|- ( ( y e. D /\ z e. D ) -> ( ( ( a e. D |-> ( b e. D |-> ( a ( +g ` G ) b ) ) ) ` y ) ` z ) = ( y ( +g ` G ) z ) ) |
85 |
50 84
|
sylan |
|- ( ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) /\ z e. D ) -> ( ( ( a e. D |-> ( b e. D |-> ( a ( +g ` G ) b ) ) ) ` y ) ` z ) = ( y ( +g ` G ) z ) ) |
86 |
77 42 83 85 47
|
fsumf1o |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> sum_ x e. D ( x ` A ) = sum_ z e. D ( ( y ( +g ` G ) z ) ` A ) ) |
87 |
42 52 47
|
fsummulc2 |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( y ` A ) x. sum_ x e. D ( x ` A ) ) = sum_ x e. D ( ( y ` A ) x. ( x ` A ) ) ) |
88 |
76 86 87
|
3eqtr4rd |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( y ` A ) x. sum_ x e. D ( x ` A ) ) = sum_ x e. D ( x ` A ) ) |
89 |
48
|
mulid2d |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( 1 x. sum_ x e. D ( x ` A ) ) = sum_ x e. D ( x ` A ) ) |
90 |
88 89
|
oveq12d |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( ( y ` A ) x. sum_ x e. D ( x ` A ) ) - ( 1 x. sum_ x e. D ( x ` A ) ) ) = ( sum_ x e. D ( x ` A ) - sum_ x e. D ( x ` A ) ) ) |
91 |
48
|
subidd |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( sum_ x e. D ( x ` A ) - sum_ x e. D ( x ` A ) ) = 0 ) |
92 |
90 91
|
eqtrd |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( ( y ` A ) x. sum_ x e. D ( x ` A ) ) - ( 1 x. sum_ x e. D ( x ` A ) ) ) = 0 ) |
93 |
26
|
a1i |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> 1 e. CC ) |
94 |
52 93 48
|
subdird |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( ( y ` A ) - 1 ) x. sum_ x e. D ( x ` A ) ) = ( ( ( y ` A ) x. sum_ x e. D ( x ` A ) ) - ( 1 x. sum_ x e. D ( x ` A ) ) ) ) |
95 |
54
|
mul01d |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( ( y ` A ) - 1 ) x. 0 ) = 0 ) |
96 |
92 94 95
|
3eqtr4d |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> ( ( ( y ` A ) - 1 ) x. sum_ x e. D ( x ` A ) ) = ( ( ( y ` A ) - 1 ) x. 0 ) ) |
97 |
48 49 54 59 96
|
mulcanad |
|- ( ( ( ph /\ A =/= .1. ) /\ ( y e. D /\ ( y ` A ) =/= 1 ) ) -> sum_ x e. D ( x ` A ) = 0 ) |
98 |
40 97
|
rexlimddv |
|- ( ( ph /\ A =/= .1. ) -> sum_ x e. D ( x ` A ) = 0 ) |
99 |
36 98
|
sylan2br |
|- ( ( ph /\ -. A = .1. ) -> sum_ x e. D ( x ` A ) = 0 ) |
100 |
8 9 35 99
|
ifbothda |
|- ( ph -> sum_ x e. D ( x ` A ) = if ( A = .1. , ( # ` D ) , 0 ) ) |