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Metamath Proof Explorer

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