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Metamath Proof Explorer

Theorem nmhmcl

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

		Ref	Expression
	Hypothesis	isnmhm2.1	$⊢ N = S normOp T$
	Assertion	nmhmcl	$⊢ F \in (S NMHom T) \to N (F) \in ℝ$

Step	Hyp	Ref	Expression
1		isnmhm2.1	$⊢ N = S normOp T$
2		nmhmnghm	$⊢ F \in (S NMHom T) \to F \in (S NGHom T)$
3	1	nghmcl	$⊢ F \in (S NGHom T) \to N (F) \in ℝ$
4	2 3	syl	$⊢ F \in (S NMHom T) \to N (F) \in ℝ$