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 ) )