| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dchrmhm.g |
|- G = ( DChr ` N ) |
| 2 |
|
dchrmhm.z |
|- Z = ( Z/nZ ` N ) |
| 3 |
|
dchrmhm.b |
|- D = ( Base ` G ) |
| 4 |
|
dchrelbas4.l |
|- L = ( ZRHom ` Z ) |
| 5 |
|
dchrzrh1.x |
|- ( ph -> X e. D ) |
| 6 |
1 3
|
dchrrcl |
|- ( X e. D -> N e. NN ) |
| 7 |
5 6
|
syl |
|- ( ph -> N e. NN ) |
| 8 |
7
|
nnnn0d |
|- ( ph -> N e. NN0 ) |
| 9 |
2
|
zncrng |
|- ( N e. NN0 -> Z e. CRing ) |
| 10 |
|
crngring |
|- ( Z e. CRing -> Z e. Ring ) |
| 11 |
|
eqid |
|- ( 1r ` Z ) = ( 1r ` Z ) |
| 12 |
4 11
|
zrh1 |
|- ( Z e. Ring -> ( L ` 1 ) = ( 1r ` Z ) ) |
| 13 |
8 9 10 12
|
4syl |
|- ( ph -> ( L ` 1 ) = ( 1r ` Z ) ) |
| 14 |
13
|
fveq2d |
|- ( ph -> ( X ` ( L ` 1 ) ) = ( X ` ( 1r ` Z ) ) ) |
| 15 |
1 2 3
|
dchrmhm |
|- D C_ ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) |
| 16 |
15 5
|
sselid |
|- ( ph -> X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) |
| 17 |
|
eqid |
|- ( mulGrp ` Z ) = ( mulGrp ` Z ) |
| 18 |
17 11
|
ringidval |
|- ( 1r ` Z ) = ( 0g ` ( mulGrp ` Z ) ) |
| 19 |
|
eqid |
|- ( mulGrp ` CCfld ) = ( mulGrp ` CCfld ) |
| 20 |
|
cnfld1 |
|- 1 = ( 1r ` CCfld ) |
| 21 |
19 20
|
ringidval |
|- 1 = ( 0g ` ( mulGrp ` CCfld ) ) |
| 22 |
18 21
|
mhm0 |
|- ( X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) -> ( X ` ( 1r ` Z ) ) = 1 ) |
| 23 |
16 22
|
syl |
|- ( ph -> ( X ` ( 1r ` Z ) ) = 1 ) |
| 24 |
14 23
|
eqtrd |
|- ( ph -> ( X ` ( L ` 1 ) ) = 1 ) |