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 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝑁 ‘ 𝐹 ) ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		nmofval.1	⊢ 𝑁 = ( 𝑆 normOp 𝑇 )
2	1	nmof	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) → 𝑁 : ( 𝑆 GrpHom 𝑇 ) ⟶ ℝ* )
3	2	ffvelcdmda	⊢ ( ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝑁 ‘ 𝐹 ) ∈ ℝ* )
4	3	3impa	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝑁 ‘ 𝐹 ) ∈ ℝ* )