Theorem nmocl

Description: The operator norm of an operator is an extended real. (Contributed by Mario Carneiro, 18-Oct-2015)

	Ref	Expression
Hypothesis	nmofval.1	|- N = ( S normOp T )
Assertion	nmocl	|- ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) -> ( N ` F ) e. RR* )

Proof

Step	Hyp	Ref	Expression
1		nmofval.1	|- N = ( S normOp T )
2	1	nmof	|- ( ( S e. NrmGrp /\ T e. NrmGrp ) -> N : ( S GrpHom T ) --> RR* )
3	2	ffvelcdmda	|- ( ( ( S e. NrmGrp /\ T e. NrmGrp ) /\ F e. ( S GrpHom T ) ) -> ( N ` F ) e. RR* )
4	3	3impa	|- ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) -> ( N ` F ) e. RR* )