Step |
Hyp |
Ref |
Expression |
1 |
|
dchrval.g |
|- G = ( DChr ` N ) |
2 |
|
dchrval.z |
|- Z = ( Z/nZ ` N ) |
3 |
|
dchrval.b |
|- B = ( Base ` Z ) |
4 |
|
dchrval.u |
|- U = ( Unit ` Z ) |
5 |
|
dchrval.n |
|- ( ph -> N e. NN ) |
6 |
|
dchrbas.b |
|- D = ( Base ` G ) |
7 |
1 2 3 4 5 6
|
dchrelbas |
|- ( ph -> ( X e. D <-> ( X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) /\ ( ( B \ U ) X. { 0 } ) C_ X ) ) ) |
8 |
|
eqid |
|- ( mulGrp ` Z ) = ( mulGrp ` Z ) |
9 |
8 3
|
mgpbas |
|- B = ( Base ` ( mulGrp ` Z ) ) |
10 |
|
eqid |
|- ( mulGrp ` CCfld ) = ( mulGrp ` CCfld ) |
11 |
|
cnfldbas |
|- CC = ( Base ` CCfld ) |
12 |
10 11
|
mgpbas |
|- CC = ( Base ` ( mulGrp ` CCfld ) ) |
13 |
9 12
|
mhmf |
|- ( X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) -> X : B --> CC ) |
14 |
13
|
adantl |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> X : B --> CC ) |
15 |
14
|
ffund |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> Fun X ) |
16 |
|
funssres |
|- ( ( Fun X /\ ( ( B \ U ) X. { 0 } ) C_ X ) -> ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) ) |
17 |
15 16
|
sylan |
|- ( ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) /\ ( ( B \ U ) X. { 0 } ) C_ X ) -> ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) ) |
18 |
|
simpr |
|- ( ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) /\ ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) ) -> ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) ) |
19 |
|
resss |
|- ( X |` dom ( ( B \ U ) X. { 0 } ) ) C_ X |
20 |
18 19
|
eqsstrrdi |
|- ( ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) /\ ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) ) -> ( ( B \ U ) X. { 0 } ) C_ X ) |
21 |
17 20
|
impbida |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> ( ( ( B \ U ) X. { 0 } ) C_ X <-> ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) ) ) |
22 |
|
0cn |
|- 0 e. CC |
23 |
|
fconst6g |
|- ( 0 e. CC -> ( ( B \ U ) X. { 0 } ) : ( B \ U ) --> CC ) |
24 |
22 23
|
mp1i |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> ( ( B \ U ) X. { 0 } ) : ( B \ U ) --> CC ) |
25 |
24
|
fdmd |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> dom ( ( B \ U ) X. { 0 } ) = ( B \ U ) ) |
26 |
25
|
reseq2d |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( X |` ( B \ U ) ) ) |
27 |
26
|
eqeq1d |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> ( ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) <-> ( X |` ( B \ U ) ) = ( ( B \ U ) X. { 0 } ) ) ) |
28 |
|
difss |
|- ( B \ U ) C_ B |
29 |
|
fssres |
|- ( ( X : B --> CC /\ ( B \ U ) C_ B ) -> ( X |` ( B \ U ) ) : ( B \ U ) --> CC ) |
30 |
14 28 29
|
sylancl |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> ( X |` ( B \ U ) ) : ( B \ U ) --> CC ) |
31 |
30
|
ffnd |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> ( X |` ( B \ U ) ) Fn ( B \ U ) ) |
32 |
24
|
ffnd |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> ( ( B \ U ) X. { 0 } ) Fn ( B \ U ) ) |
33 |
|
eqfnfv |
|- ( ( ( X |` ( B \ U ) ) Fn ( B \ U ) /\ ( ( B \ U ) X. { 0 } ) Fn ( B \ U ) ) -> ( ( X |` ( B \ U ) ) = ( ( B \ U ) X. { 0 } ) <-> A. x e. ( B \ U ) ( ( X |` ( B \ U ) ) ` x ) = ( ( ( B \ U ) X. { 0 } ) ` x ) ) ) |
34 |
31 32 33
|
syl2anc |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> ( ( X |` ( B \ U ) ) = ( ( B \ U ) X. { 0 } ) <-> A. x e. ( B \ U ) ( ( X |` ( B \ U ) ) ` x ) = ( ( ( B \ U ) X. { 0 } ) ` x ) ) ) |
35 |
|
fvres |
|- ( x e. ( B \ U ) -> ( ( X |` ( B \ U ) ) ` x ) = ( X ` x ) ) |
36 |
|
c0ex |
|- 0 e. _V |
37 |
36
|
fvconst2 |
|- ( x e. ( B \ U ) -> ( ( ( B \ U ) X. { 0 } ) ` x ) = 0 ) |
38 |
35 37
|
eqeq12d |
|- ( x e. ( B \ U ) -> ( ( ( X |` ( B \ U ) ) ` x ) = ( ( ( B \ U ) X. { 0 } ) ` x ) <-> ( X ` x ) = 0 ) ) |
39 |
38
|
ralbiia |
|- ( A. x e. ( B \ U ) ( ( X |` ( B \ U ) ) ` x ) = ( ( ( B \ U ) X. { 0 } ) ` x ) <-> A. x e. ( B \ U ) ( X ` x ) = 0 ) |
40 |
|
eldif |
|- ( x e. ( B \ U ) <-> ( x e. B /\ -. x e. U ) ) |
41 |
40
|
imbi1i |
|- ( ( x e. ( B \ U ) -> ( X ` x ) = 0 ) <-> ( ( x e. B /\ -. x e. U ) -> ( X ` x ) = 0 ) ) |
42 |
|
impexp |
|- ( ( ( x e. B /\ -. x e. U ) -> ( X ` x ) = 0 ) <-> ( x e. B -> ( -. x e. U -> ( X ` x ) = 0 ) ) ) |
43 |
|
con1b |
|- ( ( -. x e. U -> ( X ` x ) = 0 ) <-> ( -. ( X ` x ) = 0 -> x e. U ) ) |
44 |
|
df-ne |
|- ( ( X ` x ) =/= 0 <-> -. ( X ` x ) = 0 ) |
45 |
44
|
imbi1i |
|- ( ( ( X ` x ) =/= 0 -> x e. U ) <-> ( -. ( X ` x ) = 0 -> x e. U ) ) |
46 |
43 45
|
bitr4i |
|- ( ( -. x e. U -> ( X ` x ) = 0 ) <-> ( ( X ` x ) =/= 0 -> x e. U ) ) |
47 |
46
|
imbi2i |
|- ( ( x e. B -> ( -. x e. U -> ( X ` x ) = 0 ) ) <-> ( x e. B -> ( ( X ` x ) =/= 0 -> x e. U ) ) ) |
48 |
41 42 47
|
3bitri |
|- ( ( x e. ( B \ U ) -> ( X ` x ) = 0 ) <-> ( x e. B -> ( ( X ` x ) =/= 0 -> x e. U ) ) ) |
49 |
48
|
ralbii2 |
|- ( A. x e. ( B \ U ) ( X ` x ) = 0 <-> A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) ) |
50 |
39 49
|
bitri |
|- ( A. x e. ( B \ U ) ( ( X |` ( B \ U ) ) ` x ) = ( ( ( B \ U ) X. { 0 } ) ` x ) <-> A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) ) |
51 |
34 50
|
bitrdi |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> ( ( X |` ( B \ U ) ) = ( ( B \ U ) X. { 0 } ) <-> A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) ) ) |
52 |
21 27 51
|
3bitrd |
|- ( ( ph /\ X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) -> ( ( ( B \ U ) X. { 0 } ) C_ X <-> A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) ) ) |
53 |
52
|
pm5.32da |
|- ( ph -> ( ( X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) /\ ( ( B \ U ) X. { 0 } ) C_ X ) <-> ( X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) /\ A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) ) ) ) |
54 |
7 53
|
bitrd |
|- ( ph -> ( X e. D <-> ( X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) /\ A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) ) ) ) |