Theorem nghmrcl1

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

	Ref	Expression
Assertion	nghmrcl1	⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝑆 ∈ NrmGrp )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( 𝑆 normOp 𝑇 ) = ( 𝑆 normOp 𝑇 )
2	1	isnghm	⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) ∈ ℝ ) ) )
3	2	simplbi	⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) )
4	3	simpld	⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝑆 ∈ NrmGrp )