| 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 |
|
dchrzrh1.a |
|- ( ph -> A e. ZZ ) |
| 7 |
|
dchrzrh1.c |
|- ( ph -> C e. ZZ ) |
| 8 |
1 3
|
dchrrcl |
|- ( X e. D -> N e. NN ) |
| 9 |
5 8
|
syl |
|- ( ph -> N e. NN ) |
| 10 |
9
|
nnnn0d |
|- ( ph -> N e. NN0 ) |
| 11 |
2
|
zncrng |
|- ( N e. NN0 -> Z e. CRing ) |
| 12 |
10 11
|
syl |
|- ( ph -> Z e. CRing ) |
| 13 |
|
crngring |
|- ( Z e. CRing -> Z e. Ring ) |
| 14 |
12 13
|
syl |
|- ( ph -> Z e. Ring ) |
| 15 |
4
|
zrhrhm |
|- ( Z e. Ring -> L e. ( ZZring RingHom Z ) ) |
| 16 |
14 15
|
syl |
|- ( ph -> L e. ( ZZring RingHom Z ) ) |
| 17 |
|
zringbas |
|- ZZ = ( Base ` ZZring ) |
| 18 |
|
zringmulr |
|- x. = ( .r ` ZZring ) |
| 19 |
|
eqid |
|- ( .r ` Z ) = ( .r ` Z ) |
| 20 |
17 18 19
|
rhmmul |
|- ( ( L e. ( ZZring RingHom Z ) /\ A e. ZZ /\ C e. ZZ ) -> ( L ` ( A x. C ) ) = ( ( L ` A ) ( .r ` Z ) ( L ` C ) ) ) |
| 21 |
16 6 7 20
|
syl3anc |
|- ( ph -> ( L ` ( A x. C ) ) = ( ( L ` A ) ( .r ` Z ) ( L ` C ) ) ) |
| 22 |
21
|
fveq2d |
|- ( ph -> ( X ` ( L ` ( A x. C ) ) ) = ( X ` ( ( L ` A ) ( .r ` Z ) ( L ` C ) ) ) ) |
| 23 |
1 2 3
|
dchrmhm |
|- D C_ ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) |
| 24 |
23 5
|
sselid |
|- ( ph -> X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) |
| 25 |
|
eqid |
|- ( Base ` Z ) = ( Base ` Z ) |
| 26 |
17 25
|
rhmf |
|- ( L e. ( ZZring RingHom Z ) -> L : ZZ --> ( Base ` Z ) ) |
| 27 |
16 26
|
syl |
|- ( ph -> L : ZZ --> ( Base ` Z ) ) |
| 28 |
27 6
|
ffvelcdmd |
|- ( ph -> ( L ` A ) e. ( Base ` Z ) ) |
| 29 |
27 7
|
ffvelcdmd |
|- ( ph -> ( L ` C ) e. ( Base ` Z ) ) |
| 30 |
|
eqid |
|- ( mulGrp ` Z ) = ( mulGrp ` Z ) |
| 31 |
30 25
|
mgpbas |
|- ( Base ` Z ) = ( Base ` ( mulGrp ` Z ) ) |
| 32 |
30 19
|
mgpplusg |
|- ( .r ` Z ) = ( +g ` ( mulGrp ` Z ) ) |
| 33 |
|
eqid |
|- ( mulGrp ` CCfld ) = ( mulGrp ` CCfld ) |
| 34 |
|
cnfldmul |
|- x. = ( .r ` CCfld ) |
| 35 |
33 34
|
mgpplusg |
|- x. = ( +g ` ( mulGrp ` CCfld ) ) |
| 36 |
31 32 35
|
mhmlin |
|- ( ( X e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) /\ ( L ` A ) e. ( Base ` Z ) /\ ( L ` C ) e. ( Base ` Z ) ) -> ( X ` ( ( L ` A ) ( .r ` Z ) ( L ` C ) ) ) = ( ( X ` ( L ` A ) ) x. ( X ` ( L ` C ) ) ) ) |
| 37 |
24 28 29 36
|
syl3anc |
|- ( ph -> ( X ` ( ( L ` A ) ( .r ` Z ) ( L ` C ) ) ) = ( ( X ` ( L ` A ) ) x. ( X ` ( L ` C ) ) ) ) |
| 38 |
22 37
|
eqtrd |
|- ( ph -> ( X ` ( L ` ( A x. C ) ) ) = ( ( X ` ( L ` A ) ) x. ( X ` ( L ` C ) ) ) ) |