Metamath Proof Explorer


Theorem hhmet

Description: The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008) (New usage is discouraged.)

Ref Expression
Hypotheses hhnv.1 𝑈 = ⟨ ⟨ + , · ⟩ , norm
hhims2.2 𝐷 = ( IndMet ‘ 𝑈 )
Assertion hhmet 𝐷 ∈ ( Met ‘ ℋ )

Proof

Step Hyp Ref Expression
1 hhnv.1 𝑈 = ⟨ ⟨ + , · ⟩ , norm
2 hhims2.2 𝐷 = ( IndMet ‘ 𝑈 )
3 1 hhnv 𝑈 ∈ NrmCVec
4 1 hhba ℋ = ( BaseSet ‘ 𝑈 )
5 4 2 imsmet ( 𝑈 ∈ NrmCVec → 𝐷 ∈ ( Met ‘ ℋ ) )
6 3 5 ax-mp 𝐷 ∈ ( Met ‘ ℋ )