Theorem hlobn 31030

Description: Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
hlobn	⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)

Proof of Theorem hlobn

Step	Hyp	Ref	Expression
1		ishlo 31029	. 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
2	1	simplbi 499	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)

Syntax hints:   → wi 4   ∈ wcel 2136  CPreHil_OLDccphlo 30954  CBanccbn 31004  CHil_OLDchlo 31027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1557  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-v 3450  df-in 3906  df-hlo 31028

This theorem is referenced by:  hlrel  31032  hlnv  31033  hlcmet  31036  htthlem  31059