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Theorem ishil 16634
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 5541 . . . . 5  |-  ( h  =  H  ->  ( proj `  h )  =  ( proj `  H
) )
2 ishil.k . . . . 5  |-  K  =  ( proj `  H
)
31, 2syl6eqr 2346 . . . 4  |-  ( h  =  H  ->  ( proj `  h )  =  K )
43dmeqd 4897 . . 3  |-  ( h  =  H  ->  dom  ( proj `  h )  =  dom  K )
5 fveq2 5541 . . . 4  |-  ( h  =  H  ->  ( CSubSp `
 h )  =  ( CSubSp `  H )
)
6 ishil.c . . . 4  |-  C  =  ( CSubSp `  H )
75, 6syl6eqr 2346 . . 3  |-  ( h  =  H  ->  ( CSubSp `
 h )  =  C )
84, 7eqeq12d 2310 . 2  |-  ( h  =  H  ->  ( dom  ( proj `  h
)  =  ( CSubSp `  h )  <->  dom  K  =  C ) )
9 df-hil 16620 . 2  |-  Hil  =  { h  e.  PreHil  |  dom  ( proj `  h )  =  ( CSubSp `  h
) }
108, 9elrab2 2938 1  |-  ( H  e.  Hil  <->  ( H  e.  PreHil  /\  dom  K  =  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   dom cdm 4705   ` cfv 5271   PreHilcphl 16544   CSubSpccss 16577   projcpj 16616   Hilchs 16617
This theorem is referenced by:  ishil2  16635  hlhil  18823
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-3an 936  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-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-dm 4715  df-iota 5235  df-fv 5279  df-hil 16620
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