rimgim

Description: An isomorphism of rings is an isomorphism of their additive groups. (Contributed by AV, 24-Dec-2019)

		Ref	Expression
	Assertion	rimgim	$⊢ F \in (R RingIso S) \to F \in (R GrpIso S)$

Step	Hyp	Ref	Expression
1		rimrhm	$⊢ F \in (R RingIso S) \to F \in (R RingHom S)$
2		rhmghm	$⊢ F \in (R RingHom S) \to F \in (R GrpHom S)$
3	1 2	syl	$⊢ F \in (R RingIso S) \to F \in (R GrpHom S)$
4		eqid	$⊢ {Base}_{R} = {Base}_{R}$
5		eqid	$⊢ {Base}_{S} = {Base}_{S}$
6	4 5	rimf1o	$⊢ F \in (R RingIso S) \to F : {Base}_{R} \underset{1-1 onto}{⟶} {Base}_{S}$
7	4 5	isgim	$⊢ F \in (R GrpIso S) \leftrightarrow (F \in (R GrpHom S) \land F : {Base}_{R} \underset{1-1 onto}{⟶} {Base}_{S})$
8	3 6 7	sylanbrc	$⊢ F \in (R RingIso S) \to F \in (R GrpIso S)$

Metamath Proof Explorer