Theorem nghmrcl1

Description: Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015)

	Ref	Expression
Assertion	nghmrcl1	|- ( F e. ( S NGHom T ) -> S e. NrmGrp )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( S normOp T ) = ( S normOp T )
2	1	isnghm	|- ( F e. ( S NGHom T ) <-> ( ( S e. NrmGrp /\ T e. NrmGrp ) /\ ( F e. ( S GrpHom T ) /\ ( ( S normOp T ) ` F ) e. RR ) ) )
3	2	simplbi	|- ( F e. ( S NGHom T ) -> ( S e. NrmGrp /\ T e. NrmGrp ) )
4	3	simpld	|- ( F e. ( S NGHom T ) -> S e. NrmGrp )