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Metamath Proof Explorer

Theorem nvcli

Description: The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007) (New usage is discouraged.)

Ref Expression
Hypotheses nvf.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
nvf.6 ⊢ 𝑁 = ( normCV ‘ 𝑈 )
nvcli.9 ⊢ 𝑈 ∈ NrmCVec
nvcli.7 ⊢ 𝐴 ∈ 𝑋
Assertion nvcli ( 𝑁 ‘ 𝐴 ) ∈ ℝ

Proof

Step Hyp Ref Expression
1 nvf.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2 nvf.6 ⊢ 𝑁 = ( normCV ‘ 𝑈 )
3 nvcli.9 ⊢ 𝑈 ∈ NrmCVec
4 nvcli.7 ⊢ 𝐴 ∈ 𝑋
5 1 2 nvcl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) ∈ ℝ )
6 3 4 5 mp2an ⊢ ( 𝑁 ‘ 𝐴 ) ∈ ℝ