Theorem ishlo 29150

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 29149	. 2 ⊢ CHilOLD = (CBan ∩ CPreHilOLD)
2	1	elin2 4127	1 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  CPreHil_OLDccphlo 29075  CBanccbn 29125  CHil_OLDchlo 29148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-hlo 29149

This theorem is referenced by:  hlobn  29151  hlph  29152  cnchl  29179  hhhl  29467