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Theorem thlval 16845
Description: Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)
Hypotheses
Ref Expression
thlval.k  |-  K  =  (toHL `  W )
thlval.c  |-  C  =  ( CSubSp `  W )
thlval.i  |-  I  =  (toInc `  C )
thlval.o  |-  ._|_  =  ( ocv `  W )
Assertion
Ref Expression
thlval  |-  ( W  e.  V  ->  K  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. ) )

Proof of Theorem thlval
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 elex 2907 . 2  |-  ( W  e.  V  ->  W  e.  _V )
2 thlval.k . . 3  |-  K  =  (toHL `  W )
3 fveq2 5668 . . . . . . . 8  |-  ( h  =  W  ->  ( CSubSp `
 h )  =  ( CSubSp `  W )
)
4 thlval.c . . . . . . . 8  |-  C  =  ( CSubSp `  W )
53, 4syl6eqr 2437 . . . . . . 7  |-  ( h  =  W  ->  ( CSubSp `
 h )  =  C )
65fveq2d 5672 . . . . . 6  |-  ( h  =  W  ->  (toInc `  ( CSubSp `  h )
)  =  (toInc `  C ) )
7 thlval.i . . . . . 6  |-  I  =  (toInc `  C )
86, 7syl6eqr 2437 . . . . 5  |-  ( h  =  W  ->  (toInc `  ( CSubSp `  h )
)  =  I )
9 fveq2 5668 . . . . . . 7  |-  ( h  =  W  ->  ( ocv `  h )  =  ( ocv `  W
) )
10 thlval.o . . . . . . 7  |-  ._|_  =  ( ocv `  W )
119, 10syl6eqr 2437 . . . . . 6  |-  ( h  =  W  ->  ( ocv `  h )  = 
._|_  )
1211opeq2d 3933 . . . . 5  |-  ( h  =  W  ->  <. ( oc `  ndx ) ,  ( ocv `  h
) >.  =  <. ( oc `  ndx ) , 
._|_  >. )
138, 12oveq12d 6038 . . . 4  |-  ( h  =  W  ->  (
(toInc `  ( CSubSp `  h ) ) sSet  <. ( oc `  ndx ) ,  ( ocv `  h
) >. )  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )
)
14 df-thl 16815 . . . 4  |- toHL  =  ( h  e.  _V  |->  ( (toInc `  ( CSubSp `  h ) ) sSet  <. ( oc `  ndx ) ,  ( ocv `  h
) >. ) )
15 ovex 6045 . . . 4  |-  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )  e.  _V
1613, 14, 15fvmpt 5745 . . 3  |-  ( W  e.  _V  ->  (toHL `  W )  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )
)
172, 16syl5eq 2431 . 2  |-  ( W  e.  _V  ->  K  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. ) )
181, 17syl 16 1  |-  ( W  e.  V  ->  K  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2899   <.cop 3760   ` cfv 5394  (class class class)co 6020   ndxcnx 13393   sSet csts 13394   occoc 13464  toInccipo 14504   ocvcocv 16810   CSubSpccss 16811  toHLcthl 16812
This theorem is referenced by:  thlbas  16846  thlle  16847  thloc  16849
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-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  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-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-thl 16815
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