Theorem hlobn 29250

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 29249	. 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
2	1	simplbi 498	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)

Syntax hints:   → wi 4   ∈ wcel 2106  CPreHil_OLDccphlo 29174  CBanccbn 29224  CHil_OLDchlo 29247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-hlo 29248

This theorem is referenced by:  hlrel  29252  hlnv  29253  hlcmet  29256  htthlem  29279