Theorem hlph

Description: Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	hlph	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in {CPreHil}_{\mathrm{OLD}}$$

Proof

Step	Hyp	Ref	Expression
1		ishlo	$$⊢U\in {CHil}_{\mathrm{OLD}}\leftrightarrow \left(U\in \mathrm{CBan}\wedge U\in {CPreHil}_{\mathrm{OLD}}\right)$$
2	1	simprbi	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in {CPreHil}_{\mathrm{OLD}}$$