Theorem isnghm2

Description: A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015)

	Ref	Expression
Hypothesis	nmofval.1	⊢ 𝑁 = ( 𝑆 normOp 𝑇 )
Assertion	isnghm2	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 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	baib	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) → ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) ) )
4	3	baibd	⊢ ( ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) )
5	4	3impa	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) )