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Theorem thlval 18119
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 2980 . 2  |-  ( W  e.  V  ->  W  e.  _V )
2 thlval.k . . 3  |-  K  =  (toHL `  W )
3 fveq2 5690 . . . . . . . 8  |-  ( h  =  W  ->  ( CSubSp `
 h )  =  ( CSubSp `  W )
)
4 thlval.c . . . . . . . 8  |-  C  =  ( CSubSp `  W )
53, 4syl6eqr 2492 . . . . . . 7  |-  ( h  =  W  ->  ( CSubSp `
 h )  =  C )
65fveq2d 5694 . . . . . 6  |-  ( h  =  W  ->  (toInc `  ( CSubSp `  h )
)  =  (toInc `  C ) )
7 thlval.i . . . . . 6  |-  I  =  (toInc `  C )
86, 7syl6eqr 2492 . . . . 5  |-  ( h  =  W  ->  (toInc `  ( CSubSp `  h )
)  =  I )
9 fveq2 5690 . . . . . . 7  |-  ( h  =  W  ->  ( ocv `  h )  =  ( ocv `  W
) )
10 thlval.o . . . . . . 7  |-  ._|_  =  ( ocv `  W )
119, 10syl6eqr 2492 . . . . . 6  |-  ( h  =  W  ->  ( ocv `  h )  = 
._|_  )
1211opeq2d 4065 . . . . 5  |-  ( h  =  W  ->  <. ( oc `  ndx ) ,  ( ocv `  h
) >.  =  <. ( oc `  ndx ) , 
._|_  >. )
138, 12oveq12d 6108 . . . 4  |-  ( h  =  W  ->  (
(toInc `  ( CSubSp `  h ) ) sSet  <. ( oc `  ndx ) ,  ( ocv `  h
) >. )  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )
)
14 df-thl 18089 . . . 4  |- toHL  =  ( h  e.  _V  |->  ( (toInc `  ( CSubSp `  h ) ) sSet  <. ( oc `  ndx ) ,  ( ocv `  h
) >. ) )
15 ovex 6115 . . . 4  |-  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )  e.  _V
1613, 14, 15fvmpt 5773 . . 3  |-  ( W  e.  _V  ->  (toHL `  W )  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )
)
172, 16syl5eq 2486 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2971   <.cop 3882   ` cfv 5417  (class class class)co 6090   ndxcnx 14170   sSet csts 14171   occoc 14245  toInccipo 15320   ocvcocv 18084   CSubSpccss 18085  toHLcthl 18086
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
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-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-ov 6093  df-thl 18089
This theorem is referenced by:  thlbas  18120  thlle  18121  thloc  18123
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