Theorem hlobn 31251

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 31250	. 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
2	1	simplbi 501	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  CPreHil_OLDccphlo 31175  CBanccbn 31225  CHil_OLDchlo 31248

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-in 3911  df-hlo 31249

This theorem is used by:  hlrel  31253  hlnv  31254  hlcmet  31257  htthlem  31280