Step |
Hyp |
Ref |
Expression |
1 |
|
zrhpropd.1 |
|- ( ph -> B = ( Base ` K ) ) |
2 |
|
zrhpropd.2 |
|- ( ph -> B = ( Base ` L ) ) |
3 |
|
zrhpropd.3 |
|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) ) |
4 |
|
zrhpropd.4 |
|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) ) |
5 |
|
eqidd |
|- ( ph -> ( Base ` ZZring ) = ( Base ` ZZring ) ) |
6 |
|
eqidd |
|- ( ( ph /\ ( x e. ( Base ` ZZring ) /\ y e. ( Base ` ZZring ) ) ) -> ( x ( +g ` ZZring ) y ) = ( x ( +g ` ZZring ) y ) ) |
7 |
|
eqidd |
|- ( ( ph /\ ( x e. ( Base ` ZZring ) /\ y e. ( Base ` ZZring ) ) ) -> ( x ( .r ` ZZring ) y ) = ( x ( .r ` ZZring ) y ) ) |
8 |
5 1 5 2 6 3 7 4
|
rhmpropd |
|- ( ph -> ( ZZring RingHom K ) = ( ZZring RingHom L ) ) |
9 |
8
|
unieqd |
|- ( ph -> U. ( ZZring RingHom K ) = U. ( ZZring RingHom L ) ) |
10 |
|
eqid |
|- ( ZRHom ` K ) = ( ZRHom ` K ) |
11 |
10
|
zrhval |
|- ( ZRHom ` K ) = U. ( ZZring RingHom K ) |
12 |
|
eqid |
|- ( ZRHom ` L ) = ( ZRHom ` L ) |
13 |
12
|
zrhval |
|- ( ZRHom ` L ) = U. ( ZZring RingHom L ) |
14 |
9 11 13
|
3eqtr4g |
|- ( ph -> ( ZRHom ` K ) = ( ZRHom ` L ) ) |