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	$$⊢W\in \mathrm{ℂHil}\leftrightarrow \left(W\in \mathrm{Ban}\wedge W\in \mathrm{CPreHil}\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-hl	$$⊢\mathrm{ℂHil}=\mathrm{Ban}\cap \mathrm{CPreHil}$$
2	1	elin2	$$⊢W\in \mathrm{ℂHil}\leftrightarrow \left(W\in \mathrm{Ban}\wedge W\in \mathrm{CPreHil}\right)$$