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

Theorem ishil 16618
Description: The predicate "is a Hilbert space" (over a *-division ring). A Hilbert space is a pre-Hilbert space such that all closed subspaces have a projection decomposition. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
ishil.k  |-  K  =  ( proj `  H
)
ishil.c  |-  C  =  ( CSubSp `  H )
Assertion
Ref Expression
ishil  |-  ( H  e.  Hil  <->  ( H  e.  PreHil  /\  dom  K  =  C ) )

Proof of Theorem ishil
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . 5  |-  ( h  =  H  ->  ( proj `  h )  =  ( proj `  H
) )
2 ishil.k . . . . 5  |-  K  =  ( proj `  H
)
31, 2syl6eqr 2333 . . . 4  |-  ( h  =  H  ->  ( proj `  h )  =  K )
43dmeqd 4881 . . 3  |-  ( h  =  H  ->  dom  ( proj `  h )  =  dom  K )
5 fveq2 5525 . . . 4  |-  ( h  =  H  ->  ( CSubSp `
 h )  =  ( CSubSp `  H )
)
6 ishil.c . . . 4  |-  C  =  ( CSubSp `  H )
75, 6syl6eqr 2333 . . 3  |-  ( h  =  H  ->  ( CSubSp `
 h )  =  C )
84, 7eqeq12d 2297 . 2  |-  ( h  =  H  ->  ( dom  ( proj `  h
)  =  ( CSubSp `  h )  <->  dom  K  =  C ) )
9 df-hil 16604 . 2  |-  Hil  =  { h  e.  PreHil  |  dom  ( proj `  h )  =  ( CSubSp `  h
) }
108, 9elrab2 2925 1  |-  ( H  e.  Hil  <->  ( H  e.  PreHil  /\  dom  K  =  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   dom cdm 4689   ` cfv 5255   PreHilcphl 16528   CSubSpccss 16561   projcpj 16600   Hilchs 16601
This theorem is referenced by:  ishil2  16619  hlhil  18807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-dm 4699  df-iota 5219  df-fv 5263  df-hil 16604
  Copyright terms: Public domain W3C validator