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 = ⟨ ⟨ + , · ⟩ , abs ⟩
Assertion	cnchl	⊢ 𝑈 ∈ CHilOLD

Proof

Step	Hyp	Ref	Expression
1		cnhl.6	⊢ 𝑈 = ⟨ ⟨ + , · ⟩ , abs ⟩
2	1	cnbn	⊢ 𝑈 ∈ CBan
3	1	cncph	⊢ 𝑈 ∈ CPreHilOLD
4		ishlo	⊢ ( 𝑈 ∈ CHilOLD ↔ ( 𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD ) )
5	2 3 4	mpbir2an	⊢ 𝑈 ∈ CHilOLD