Theorem ishlo 28298

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 28297	. 2 ⊢ CHilOLD = (CBan ∩ CPreHilOLD)
2	1	elin2 4028	1 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))

Syntax hints:   ↔ wb 198   ∧ wa 386   ∈ wcel 2166  CPreHil_OLDccphlo 28222  CBanccbn 28273  CHil_OLDchlo 28296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-in 3805  df-hlo 28297

This theorem is referenced by:  hlobn  28299  hlph  28300  cnchl  28327  ssphlOLD  28328  hhhl  28616  hhsshlOLD  28693