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Metamath Proof Explorer

Theorem ishlo

Description: The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007) (New usage is discouraged.)

Ref Expression
Assertion ishlo ( 𝑈 ∈ CHilOLD ↔ ( 𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD ) )

Proof

Step Hyp Ref Expression
1 df-hlo ⊢ CHilOLD = ( CBan ∩ CPreHilOLD )
2 1 elin2 ⊢ ( 𝑈 ∈ CHilOLD ↔ ( 𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD ) )