Theorem nmhmlmhm

Description: A normed module homomorphism is a left module homomorphism (a linear operator). (Contributed by Mario Carneiro, 18-Oct-2015)

	Ref	Expression
Assertion	nmhmlmhm	⊢ ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) )

Proof

Step	Hyp	Ref	Expression
1		isnmhm	⊢ ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ) ∧ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) )
2	1	simprbi	⊢ ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) → ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) )
3	2	simpld	⊢ ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) )