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Theorem ishil 18141
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 5689 . . . . 5  |-  ( h  =  H  ->  ( proj `  h )  =  ( proj `  H
) )
2 ishil.k . . . . 5  |-  K  =  ( proj `  H
)
31, 2syl6eqr 2491 . . . 4  |-  ( h  =  H  ->  ( proj `  h )  =  K )
43dmeqd 5040 . . 3  |-  ( h  =  H  ->  dom  ( proj `  h )  =  dom  K )
5 fveq2 5689 . . . 4  |-  ( h  =  H  ->  ( CSubSp `
 h )  =  ( CSubSp `  H )
)
6 ishil.c . . . 4  |-  C  =  ( CSubSp `  H )
75, 6syl6eqr 2491 . . 3  |-  ( h  =  H  ->  ( CSubSp `
 h )  =  C )
84, 7eqeq12d 2455 . 2  |-  ( h  =  H  ->  ( dom  ( proj `  h
)  =  ( CSubSp `  h )  <->  dom  K  =  C ) )
9 df-hil 18127 . 2  |-  Hil  =  { h  e.  PreHil  |  dom  ( proj `  h )  =  ( CSubSp `  h
) }
108, 9elrab2 3117 1  |-  ( H  e.  Hil  <->  ( H  e.  PreHil  /\  dom  K  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   dom cdm 4838   ` cfv 5416   PreHilcphl 18051   CSubSpccss 18084   projcpj 18123   Hilchs 18124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-dm 4848  df-iota 5379  df-fv 5424  df-hil 18127
This theorem is referenced by:  ishil2  18142  hlhil  20928
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