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Theorem ishil 18875
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 5872 . . . . 5  |-  ( h  =  H  ->  ( proj `  h )  =  ( proj `  H
) )
2 ishil.k . . . . 5  |-  K  =  ( proj `  H
)
31, 2syl6eqr 2516 . . . 4  |-  ( h  =  H  ->  ( proj `  h )  =  K )
43dmeqd 5215 . . 3  |-  ( h  =  H  ->  dom  ( proj `  h )  =  dom  K )
5 fveq2 5872 . . . 4  |-  ( h  =  H  ->  ( CSubSp `
 h )  =  ( CSubSp `  H )
)
6 ishil.c . . . 4  |-  C  =  ( CSubSp `  H )
75, 6syl6eqr 2516 . . 3  |-  ( h  =  H  ->  ( CSubSp `
 h )  =  C )
84, 7eqeq12d 2479 . 2  |-  ( h  =  H  ->  ( dom  ( proj `  h
)  =  ( CSubSp `  h )  <->  dom  K  =  C ) )
9 df-hil 18861 . 2  |-  Hil  =  { h  e.  PreHil  |  dom  ( proj `  h )  =  ( CSubSp `  h
) }
108, 9elrab2 3259 1  |-  ( H  e.  Hil  <->  ( H  e.  PreHil  /\  dom  K  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   dom cdm 5008   ` cfv 5594   PreHilcphl 18785   CSubSpccss 18818   projcpj 18857   Hilchs 18858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-dm 5018  df-iota 5557  df-fv 5602  df-hil 18861
This theorem is referenced by:  ishil2  18876  hlhil  21983
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