MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ishl Unicode version

Theorem ishl 18795
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.) (Revised by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
ishl  |-  ( W  e.  CHil  <->  ( W  e. Ban  /\  W  e.  CPreHil ) )

Proof of Theorem ishl
StepHypRef Expression
1 df-hl 18775 . 2  |-  CHil  =  (Ban  i^i  CPreHil )
21elin2 3372 1  |-  ( W  e.  CHil  <->  ( W  e. Ban  /\  W  e.  CPreHil ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1696   CPreHilccph 18618  Bancbn 18771   CHilchl 18772
This theorem is referenced by:  hlbn  18796  hlcph  18797  ishl2  18803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-hl 18775
  Copyright terms: Public domain W3C validator