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Theorem ishil 16868
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 5668 . . . . 5  |-  ( h  =  H  ->  ( proj `  h )  =  ( proj `  H
) )
2 ishil.k . . . . 5  |-  K  =  ( proj `  H
)
31, 2syl6eqr 2437 . . . 4  |-  ( h  =  H  ->  ( proj `  h )  =  K )
43dmeqd 5012 . . 3  |-  ( h  =  H  ->  dom  ( proj `  h )  =  dom  K )
5 fveq2 5668 . . . 4  |-  ( h  =  H  ->  ( CSubSp `
 h )  =  ( CSubSp `  H )
)
6 ishil.c . . . 4  |-  C  =  ( CSubSp `  H )
75, 6syl6eqr 2437 . . 3  |-  ( h  =  H  ->  ( CSubSp `
 h )  =  C )
84, 7eqeq12d 2401 . 2  |-  ( h  =  H  ->  ( dom  ( proj `  h
)  =  ( CSubSp `  h )  <->  dom  K  =  C ) )
9 df-hil 16854 . 2  |-  Hil  =  { h  e.  PreHil  |  dom  ( proj `  h )  =  ( CSubSp `  h
) }
108, 9elrab2 3037 1  |-  ( H  e.  Hil  <->  ( H  e.  PreHil  /\  dom  K  =  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   dom cdm 4818   ` cfv 5394   PreHilcphl 16778   CSubSpccss 16811   projcpj 16850   Hilchs 16851
This theorem is referenced by:  ishil2  16869  hlhil  19211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-dm 4828  df-iota 5358  df-fv 5402  df-hil 16854
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