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Metamath Proof Explorer

Theorem ishl

Description: The predicate "is a subcomplex 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) (Revised by Mario Carneiro, 15-Oct-2015)

		Ref	Expression
	Assertion	ishl	⊢ ( 𝑊 ∈ ℂHil ↔ ( 𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil ) )

Proof

Step	Hyp	Ref	Expression
1		df-hl	⊢ ℂHil = ( Ban ∩ ℂPreHil )
2	1	elin2	⊢ ( 𝑊 ∈ ℂHil ↔ ( 𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil ) )