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	$$⊢U\in {CHil}_{\mathrm{OLD}}\leftrightarrow \left(U\in \mathrm{CBan}\wedge U\in {CPreHil}_{\mathrm{OLD}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-hlo	$$⊢{CHil}_{\mathrm{OLD}}=\mathrm{CBan}\cap {CPreHil}_{\mathrm{OLD}}$$
2	1	elin2	$$⊢U\in {CHil}_{\mathrm{OLD}}\leftrightarrow \left(U\in \mathrm{CBan}\wedge U\in {CPreHil}_{\mathrm{OLD}}\right)$$