Theorem lidlnsg

Description: An ideal is a normal subgroup. (Contributed by Thierry Arnoux, 14-Jan-2024)

	Ref	Expression
Assertion	lidlnsg	|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) ) -> I e. ( NrmSGrp ` R ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( LIdeal ` R ) = ( LIdeal ` R )
2	1	lidlsubg	|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) ) -> I e. ( SubGrp ` R ) )
3		ringabl	|- ( R e. Ring -> R e. Abel )
4	3	adantr	|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) ) -> R e. Abel )
5		ablnsg	|- ( R e. Abel -> ( NrmSGrp ` R ) = ( SubGrp ` R ) )
6	4 5	syl	|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) ) -> ( NrmSGrp ` R ) = ( SubGrp ` R ) )
7	2 6	eleqtrrd	|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) ) -> I e. ( NrmSGrp ` R ) )