Step |
Hyp |
Ref |
Expression |
1 |
|
rnghom1.1 |
|- H = ( 2nd ` R ) |
2 |
|
rnghom1.2 |
|- U = ( GId ` H ) |
3 |
|
rnghom1.3 |
|- K = ( 2nd ` S ) |
4 |
|
rnghom1.4 |
|- V = ( GId ` K ) |
5 |
|
eqid |
|- ( 1st ` R ) = ( 1st ` R ) |
6 |
|
eqid |
|- ran ( 1st ` R ) = ran ( 1st ` R ) |
7 |
|
eqid |
|- ( 1st ` S ) = ( 1st ` S ) |
8 |
|
eqid |
|- ran ( 1st ` S ) = ran ( 1st ` S ) |
9 |
5 1 6 2 7 3 8 4
|
isrngohom |
|- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RngHom S ) <-> ( F : ran ( 1st ` R ) --> ran ( 1st ` S ) /\ ( F ` U ) = V /\ A. x e. ran ( 1st ` R ) A. y e. ran ( 1st ` R ) ( ( F ` ( x ( 1st ` R ) y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) ) ) ) |
10 |
9
|
biimpa |
|- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngHom S ) ) -> ( F : ran ( 1st ` R ) --> ran ( 1st ` S ) /\ ( F ` U ) = V /\ A. x e. ran ( 1st ` R ) A. y e. ran ( 1st ` R ) ( ( F ` ( x ( 1st ` R ) y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) ) ) |
11 |
10
|
simp2d |
|- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngHom S ) ) -> ( F ` U ) = V ) |
12 |
11
|
3impa |
|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( F ` U ) = V ) |