Metamath Proof Explorer

Theorem rnghmrcl

Description: Reverse closure of a non-unital ring homomorphism. (Contributed by AV, 22-Feb-2020)

Ref Expression
Assertion rnghmrcl 
|- ( F e. ( R RngHom S ) -> ( R e. Rng /\ S e. Rng ) )

Proof

Step Hyp Ref Expression
1 df-rnghm 
 |-  RngHom = ( r e. Rng , s e. Rng |-> [_ ( Base ` r ) / v ]_ [_ ( Base ` s ) / w ]_ { f e. ( w ^m v ) | A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) } )
2 1 elmpocl 
 |-  ( F e. ( R RngHom S ) -> ( R e. Rng /\ S e. Rng ) )