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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 )