Step |
Hyp |
Ref |
Expression |
1 |
|
dchrmhm.g |
|- G = ( DChr ` N ) |
2 |
|
dchrmhm.z |
|- Z = ( Z/nZ ` N ) |
3 |
|
dchrmhm.b |
|- D = ( Base ` G ) |
4 |
|
dchrn0.b |
|- B = ( Base ` Z ) |
5 |
|
dchrn0.u |
|- U = ( Unit ` Z ) |
6 |
|
dchr1cl.o |
|- .1. = ( k e. B |-> if ( k e. U , 1 , 0 ) ) |
7 |
|
dchrmulid2.t |
|- .x. = ( +g ` G ) |
8 |
|
dchrmulid2.x |
|- ( ph -> X e. D ) |
9 |
|
dchrinvcl.n |
|- K = ( k e. B |-> if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) ) |
10 |
1 3
|
dchrrcl |
|- ( X e. D -> N e. NN ) |
11 |
8 10
|
syl |
|- ( ph -> N e. NN ) |
12 |
|
fveq2 |
|- ( k = x -> ( X ` k ) = ( X ` x ) ) |
13 |
12
|
oveq2d |
|- ( k = x -> ( 1 / ( X ` k ) ) = ( 1 / ( X ` x ) ) ) |
14 |
|
fveq2 |
|- ( k = y -> ( X ` k ) = ( X ` y ) ) |
15 |
14
|
oveq2d |
|- ( k = y -> ( 1 / ( X ` k ) ) = ( 1 / ( X ` y ) ) ) |
16 |
|
fveq2 |
|- ( k = ( x ( .r ` Z ) y ) -> ( X ` k ) = ( X ` ( x ( .r ` Z ) y ) ) ) |
17 |
16
|
oveq2d |
|- ( k = ( x ( .r ` Z ) y ) -> ( 1 / ( X ` k ) ) = ( 1 / ( X ` ( x ( .r ` Z ) y ) ) ) ) |
18 |
|
fveq2 |
|- ( k = ( 1r ` Z ) -> ( X ` k ) = ( X ` ( 1r ` Z ) ) ) |
19 |
18
|
oveq2d |
|- ( k = ( 1r ` Z ) -> ( 1 / ( X ` k ) ) = ( 1 / ( X ` ( 1r ` Z ) ) ) ) |
20 |
1 2 3 4 8
|
dchrf |
|- ( ph -> X : B --> CC ) |
21 |
4 5
|
unitss |
|- U C_ B |
22 |
21
|
sseli |
|- ( k e. U -> k e. B ) |
23 |
|
ffvelrn |
|- ( ( X : B --> CC /\ k e. B ) -> ( X ` k ) e. CC ) |
24 |
20 22 23
|
syl2an |
|- ( ( ph /\ k e. U ) -> ( X ` k ) e. CC ) |
25 |
|
simpr |
|- ( ( ph /\ k e. U ) -> k e. U ) |
26 |
8
|
adantr |
|- ( ( ph /\ k e. U ) -> X e. D ) |
27 |
22
|
adantl |
|- ( ( ph /\ k e. U ) -> k e. B ) |
28 |
1 2 3 4 5 26 27
|
dchrn0 |
|- ( ( ph /\ k e. U ) -> ( ( X ` k ) =/= 0 <-> k e. U ) ) |
29 |
25 28
|
mpbird |
|- ( ( ph /\ k e. U ) -> ( X ` k ) =/= 0 ) |
30 |
24 29
|
reccld |
|- ( ( ph /\ k e. U ) -> ( 1 / ( X ` k ) ) e. CC ) |
31 |
|
1t1e1 |
|- ( 1 x. 1 ) = 1 |
32 |
31
|
eqcomi |
|- 1 = ( 1 x. 1 ) |
33 |
32
|
a1i |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> 1 = ( 1 x. 1 ) ) |
34 |
1 2 3
|
dchrmhm |
|- D C_ ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) |
35 |
8
|
adantr |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> X e. D ) |
36 |
34 35
|
sselid |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) |
37 |
|
simprl |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> x e. U ) |
38 |
21 37
|
sselid |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> x e. B ) |
39 |
|
simprr |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> y e. U ) |
40 |
21 39
|
sselid |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> y e. B ) |
41 |
|
eqid |
|- ( mulGrp ` Z ) = ( mulGrp ` Z ) |
42 |
41 4
|
mgpbas |
|- B = ( Base ` ( mulGrp ` Z ) ) |
43 |
|
eqid |
|- ( .r ` Z ) = ( .r ` Z ) |
44 |
41 43
|
mgpplusg |
|- ( .r ` Z ) = ( +g ` ( mulGrp ` Z ) ) |
45 |
|
eqid |
|- ( mulGrp ` CCfld ) = ( mulGrp ` CCfld ) |
46 |
|
cnfldmul |
|- x. = ( .r ` CCfld ) |
47 |
45 46
|
mgpplusg |
|- x. = ( +g ` ( mulGrp ` CCfld ) ) |
48 |
42 44 47
|
mhmlin |
|- ( ( X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) /\ x e. B /\ y e. B ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) ) |
49 |
36 38 40 48
|
syl3anc |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) ) |
50 |
33 49
|
oveq12d |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( 1 / ( X ` ( x ( .r ` Z ) y ) ) ) = ( ( 1 x. 1 ) / ( ( X ` x ) x. ( X ` y ) ) ) ) |
51 |
|
1cnd |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> 1 e. CC ) |
52 |
20
|
adantr |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> X : B --> CC ) |
53 |
52 38
|
ffvelrnd |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( X ` x ) e. CC ) |
54 |
52 40
|
ffvelrnd |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( X ` y ) e. CC ) |
55 |
1 2 3 4 5 35 38
|
dchrn0 |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( ( X ` x ) =/= 0 <-> x e. U ) ) |
56 |
37 55
|
mpbird |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( X ` x ) =/= 0 ) |
57 |
1 2 3 4 5 35 40
|
dchrn0 |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( ( X ` y ) =/= 0 <-> y e. U ) ) |
58 |
39 57
|
mpbird |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( X ` y ) =/= 0 ) |
59 |
51 53 51 54 56 58
|
divmuldivd |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( ( 1 / ( X ` x ) ) x. ( 1 / ( X ` y ) ) ) = ( ( 1 x. 1 ) / ( ( X ` x ) x. ( X ` y ) ) ) ) |
60 |
50 59
|
eqtr4d |
|- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( 1 / ( X ` ( x ( .r ` Z ) y ) ) ) = ( ( 1 / ( X ` x ) ) x. ( 1 / ( X ` y ) ) ) ) |
61 |
34 8
|
sselid |
|- ( ph -> X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) |
62 |
|
eqid |
|- ( 1r ` Z ) = ( 1r ` Z ) |
63 |
41 62
|
ringidval |
|- ( 1r ` Z ) = ( 0g ` ( mulGrp ` Z ) ) |
64 |
|
cnfld1 |
|- 1 = ( 1r ` CCfld ) |
65 |
45 64
|
ringidval |
|- 1 = ( 0g ` ( mulGrp ` CCfld ) ) |
66 |
63 65
|
mhm0 |
|- ( X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) -> ( X ` ( 1r ` Z ) ) = 1 ) |
67 |
61 66
|
syl |
|- ( ph -> ( X ` ( 1r ` Z ) ) = 1 ) |
68 |
67
|
oveq2d |
|- ( ph -> ( 1 / ( X ` ( 1r ` Z ) ) ) = ( 1 / 1 ) ) |
69 |
|
1div1e1 |
|- ( 1 / 1 ) = 1 |
70 |
68 69
|
eqtrdi |
|- ( ph -> ( 1 / ( X ` ( 1r ` Z ) ) ) = 1 ) |
71 |
1 2 4 5 11 3 13 15 17 19 30 60 70
|
dchrelbasd |
|- ( ph -> ( k e. B |-> if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) ) e. D ) |
72 |
9 71
|
eqeltrid |
|- ( ph -> K e. D ) |
73 |
1 2 3 7 72 8
|
dchrmul |
|- ( ph -> ( K .x. X ) = ( K oF x. X ) ) |
74 |
4
|
fvexi |
|- B e. _V |
75 |
74
|
a1i |
|- ( ph -> B e. _V ) |
76 |
|
ovex |
|- ( 1 / ( X ` k ) ) e. _V |
77 |
|
c0ex |
|- 0 e. _V |
78 |
76 77
|
ifex |
|- if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) e. _V |
79 |
78
|
a1i |
|- ( ( ph /\ k e. B ) -> if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) e. _V ) |
80 |
20
|
ffvelrnda |
|- ( ( ph /\ k e. B ) -> ( X ` k ) e. CC ) |
81 |
9
|
a1i |
|- ( ph -> K = ( k e. B |-> if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) ) ) |
82 |
20
|
feqmptd |
|- ( ph -> X = ( k e. B |-> ( X ` k ) ) ) |
83 |
75 79 80 81 82
|
offval2 |
|- ( ph -> ( K oF x. X ) = ( k e. B |-> ( if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) x. ( X ` k ) ) ) ) |
84 |
|
ovif |
|- ( if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) x. ( X ` k ) ) = if ( k e. U , ( ( 1 / ( X ` k ) ) x. ( X ` k ) ) , ( 0 x. ( X ` k ) ) ) |
85 |
80
|
adantr |
|- ( ( ( ph /\ k e. B ) /\ k e. U ) -> ( X ` k ) e. CC ) |
86 |
8
|
adantr |
|- ( ( ph /\ k e. B ) -> X e. D ) |
87 |
|
simpr |
|- ( ( ph /\ k e. B ) -> k e. B ) |
88 |
1 2 3 4 5 86 87
|
dchrn0 |
|- ( ( ph /\ k e. B ) -> ( ( X ` k ) =/= 0 <-> k e. U ) ) |
89 |
88
|
biimpar |
|- ( ( ( ph /\ k e. B ) /\ k e. U ) -> ( X ` k ) =/= 0 ) |
90 |
85 89
|
recid2d |
|- ( ( ( ph /\ k e. B ) /\ k e. U ) -> ( ( 1 / ( X ` k ) ) x. ( X ` k ) ) = 1 ) |
91 |
90
|
ifeq1da |
|- ( ( ph /\ k e. B ) -> if ( k e. U , ( ( 1 / ( X ` k ) ) x. ( X ` k ) ) , ( 0 x. ( X ` k ) ) ) = if ( k e. U , 1 , ( 0 x. ( X ` k ) ) ) ) |
92 |
80
|
mul02d |
|- ( ( ph /\ k e. B ) -> ( 0 x. ( X ` k ) ) = 0 ) |
93 |
92
|
ifeq2d |
|- ( ( ph /\ k e. B ) -> if ( k e. U , 1 , ( 0 x. ( X ` k ) ) ) = if ( k e. U , 1 , 0 ) ) |
94 |
91 93
|
eqtrd |
|- ( ( ph /\ k e. B ) -> if ( k e. U , ( ( 1 / ( X ` k ) ) x. ( X ` k ) ) , ( 0 x. ( X ` k ) ) ) = if ( k e. U , 1 , 0 ) ) |
95 |
84 94
|
syl5eq |
|- ( ( ph /\ k e. B ) -> ( if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) x. ( X ` k ) ) = if ( k e. U , 1 , 0 ) ) |
96 |
95
|
mpteq2dva |
|- ( ph -> ( k e. B |-> ( if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) x. ( X ` k ) ) ) = ( k e. B |-> if ( k e. U , 1 , 0 ) ) ) |
97 |
6 96
|
eqtr4id |
|- ( ph -> .1. = ( k e. B |-> ( if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) x. ( X ` k ) ) ) ) |
98 |
83 97
|
eqtr4d |
|- ( ph -> ( K oF x. X ) = .1. ) |
99 |
73 98
|
eqtrd |
|- ( ph -> ( K .x. X ) = .1. ) |
100 |
72 99
|
jca |
|- ( ph -> ( K e. D /\ ( K .x. X ) = .1. ) ) |