MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ishlo Structured version   Visualization version   GIF version

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
StepHypRef Expression
1 df-hlo 28297 . 2 CHilOLD = (CBan ∩ CPreHilOLD)
21elin2 4028 1 (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  wcel 2166  CPreHilOLDccphlo 28222  CBanccbn 28273  CHilOLDchlo 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
  Copyright terms: Public domain W3C validator