Theorem cnchl

Description: The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cnhl.6	$$⊢U=⟨⟨+,\times ⟩,\mathrm{abs}⟩$$
Assertion	cnchl	$$⊢U\in {CHil}_{\mathrm{OLD}}$$

Proof

Step	Hyp	Ref	Expression
1		cnhl.6	$$⊢U=⟨⟨+,\times ⟩,\mathrm{abs}⟩$$
2	1	cnbn	$$⊢U\in \mathrm{CBan}$$
3	1	cncph	$$⊢U\in {CPreHil}_{\mathrm{OLD}}$$
4		ishlo	$$⊢U\in {CHil}_{\mathrm{OLD}}\leftrightarrow \left(U\in \mathrm{CBan}\wedge U\in {CPreHil}_{\mathrm{OLD}}\right)$$
5	2 3 4	mpbir2an	$$⊢U\in {CHil}_{\mathrm{OLD}}$$