Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
2 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
3 |
|
eqid |
|- ( .r ` S ) = ( .r ` S ) |
4 |
1 2 3
|
isrnghm |
|- ( F e. ( R RngHomo S ) <-> ( ( R e. Rng /\ S e. Rng ) /\ ( F e. ( R GrpHom S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) ) ) ) |
5 |
|
simprl |
|- ( ( ( R e. Rng /\ S e. Rng ) /\ ( F e. ( R GrpHom S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) ( .r ` S ) ( F ` y ) ) ) ) -> F e. ( R GrpHom S ) ) |
6 |
4 5
|
sylbi |
|- ( F e. ( R RngHomo S ) -> F e. ( R GrpHom S ) ) |