Theorem nmcl

Description: The norm of a normed group is closed in the reals. (Contributed by Mario Carneiro, 4-Oct-2015)

	Ref	Expression
Hypotheses	nmf.x	⊢ 𝑋 = ( Base ‘ 𝐺 )
	nmf.n	⊢ 𝑁 = ( norm ‘ 𝐺 )
Assertion	nmcl	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		nmf.x	⊢ 𝑋 = ( Base ‘ 𝐺 )
2		nmf.n	⊢ 𝑁 = ( norm ‘ 𝐺 )
3	1 2	nmf	⊢ ( 𝐺 ∈ NrmGrp → 𝑁 : 𝑋 ⟶ ℝ )
4	3	ffvelcdmda	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) ∈ ℝ )