Theorem rnghmmgmhm

Description: A non-unital ring homomorphism is a homomorphism of multiplicative magmas. (Contributed by AV, 27-Feb-2020)

	Ref	Expression
Hypotheses	isrnghmmul.m	|- M = ( mulGrp ` R )
	isrnghmmul.n	|- N = ( mulGrp ` S )
Assertion	rnghmmgmhm	|- ( F e. ( R RngHom S ) -> F e. ( M MgmHom N ) )

Proof

Step	Hyp	Ref	Expression
1		isrnghmmul.m	|- M = ( mulGrp ` R )
2		isrnghmmul.n	|- N = ( mulGrp ` S )
3	1 2	isrnghmmul	|- ( F e. ( R RngHom S ) <-> ( ( R e. Rng /\ S e. Rng ) /\ ( F e. ( R GrpHom S ) /\ F e. ( M MgmHom N ) ) ) )
4	3	simprbi	|- ( F e. ( R RngHom S ) -> ( F e. ( R GrpHom S ) /\ F e. ( M MgmHom N ) ) )
5	4	simprd	|- ( F e. ( R RngHom S ) -> F e. ( M MgmHom N ) )