| Step |
Hyp |
Ref |
Expression |
| 1 |
|
zrhneg.1 |
|- L = ( ZRHom ` R ) |
| 2 |
|
zrhneg.2 |
|- I = ( invg ` R ) |
| 3 |
|
zrhneg.3 |
|- ( ph -> R e. Ring ) |
| 4 |
|
zrhneg.4 |
|- ( ph -> N e. ZZ ) |
| 5 |
|
zringinvg |
|- ( N e. ZZ -> -u N = ( ( invg ` ZZring ) ` N ) ) |
| 6 |
4 5
|
syl |
|- ( ph -> -u N = ( ( invg ` ZZring ) ` N ) ) |
| 7 |
6
|
fveq2d |
|- ( ph -> ( L ` -u N ) = ( L ` ( ( invg ` ZZring ) ` N ) ) ) |
| 8 |
1
|
zrhrhm |
|- ( R e. Ring -> L e. ( ZZring RingHom R ) ) |
| 9 |
|
rhmghm |
|- ( L e. ( ZZring RingHom R ) -> L e. ( ZZring GrpHom R ) ) |
| 10 |
3 8 9
|
3syl |
|- ( ph -> L e. ( ZZring GrpHom R ) ) |
| 11 |
|
zringbas |
|- ZZ = ( Base ` ZZring ) |
| 12 |
|
eqid |
|- ( invg ` ZZring ) = ( invg ` ZZring ) |
| 13 |
11 12 2
|
ghminv |
|- ( ( L e. ( ZZring GrpHom R ) /\ N e. ZZ ) -> ( L ` ( ( invg ` ZZring ) ` N ) ) = ( I ` ( L ` N ) ) ) |
| 14 |
10 4 13
|
syl2anc |
|- ( ph -> ( L ` ( ( invg ` ZZring ) ` N ) ) = ( I ` ( L ` N ) ) ) |
| 15 |
7 14
|
eqtrd |
|- ( ph -> ( L ` -u N ) = ( I ` ( L ` N ) ) ) |