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Metamath Proof Explorer

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	$⊢ F \in (S NMHom T) \to F \in (S LMHom T)$

Step	Hyp	Ref	Expression
1		isnmhm	$⊢ F \in (S NMHom T) \leftrightarrow ((S \in NrmMod \land T \in NrmMod) \land (F \in (S LMHom T) \land F \in (S NGHom T)))$
2	1	simprbi	$⊢ F \in (S NMHom T) \to (F \in (S LMHom T) \land F \in (S NGHom T))$
3	2	simpld	$⊢ F \in (S NMHom T) \to F \in (S LMHom T)$