Theorem nmhmcl

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

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

Proof

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