Theorem ishlo 30869

Description: The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
ishlo	⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))

Proof of Theorem ishlo

Step	Hyp	Ref	Expression
1		df-hlo 30868	. 2 ⊢ CHilOLD = (CBan ∩ CPreHilOLD)
2	1	elin2 4152	1 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  CPreHil_OLDccphlo 30794  CBanccbn 30844  CHil_OLDchlo 30867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-in 3905  df-hlo 30868

This theorem is referenced by:  hlobn  30870  hlph  30871  cnchl  30898  hhhl  31186