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Metamath Proof Explorer

Theorem hlph

Description: Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007) (New usage is discouraged.)

Ref Expression
Assertion hlph ( 𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD )

Proof

Step Hyp Ref Expression
1 ishlo ⊢ ( 𝑈 ∈ CHilOLD ↔ ( 𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD ) )
2 1 simprbi ⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD )