Theorem nghmcl

Description: A normed group homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015)

	Ref	Expression
Hypothesis	nmofval.1	⊢ 𝑁 = ( 𝑆 normOp 𝑇 )
Assertion	nghmcl	⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → ( 𝑁 ‘ 𝐹 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		nmofval.1	⊢ 𝑁 = ( 𝑆 normOp 𝑇 )
2	1	isnghm	⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) ) )
3	2	simprbi	⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) )
4	3	simprd	⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → ( 𝑁 ‘ 𝐹 ) ∈ ℝ )