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Theorem thlval 16914
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 2956 . 2  |-  ( W  e.  V  ->  W  e.  _V )
2 thlval.k . . 3  |-  K  =  (toHL `  W )
3 fveq2 5720 . . . . . . . 8  |-  ( h  =  W  ->  ( CSubSp `
 h )  =  ( CSubSp `  W )
)
4 thlval.c . . . . . . . 8  |-  C  =  ( CSubSp `  W )
53, 4syl6eqr 2485 . . . . . . 7  |-  ( h  =  W  ->  ( CSubSp `
 h )  =  C )
65fveq2d 5724 . . . . . 6  |-  ( h  =  W  ->  (toInc `  ( CSubSp `  h )
)  =  (toInc `  C ) )
7 thlval.i . . . . . 6  |-  I  =  (toInc `  C )
86, 7syl6eqr 2485 . . . . 5  |-  ( h  =  W  ->  (toInc `  ( CSubSp `  h )
)  =  I )
9 fveq2 5720 . . . . . . 7  |-  ( h  =  W  ->  ( ocv `  h )  =  ( ocv `  W
) )
10 thlval.o . . . . . . 7  |-  ._|_  =  ( ocv `  W )
119, 10syl6eqr 2485 . . . . . 6  |-  ( h  =  W  ->  ( ocv `  h )  = 
._|_  )
1211opeq2d 3983 . . . . 5  |-  ( h  =  W  ->  <. ( oc `  ndx ) ,  ( ocv `  h
) >.  =  <. ( oc `  ndx ) , 
._|_  >. )
138, 12oveq12d 6091 . . . 4  |-  ( h  =  W  ->  (
(toInc `  ( CSubSp `  h ) ) sSet  <. ( oc `  ndx ) ,  ( ocv `  h
) >. )  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )
)
14 df-thl 16884 . . . 4  |- toHL  =  ( h  e.  _V  |->  ( (toInc `  ( CSubSp `  h ) ) sSet  <. ( oc `  ndx ) ,  ( ocv `  h
) >. ) )
15 ovex 6098 . . . 4  |-  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )  e.  _V
1613, 14, 15fvmpt 5798 . . 3  |-  ( W  e.  _V  ->  (toHL `  W )  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )
)
172, 16syl5eq 2479 . 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 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   ` cfv 5446  (class class class)co 6073   ndxcnx 13458   sSet csts 13459   occoc 13529  toInccipo 14569   ocvcocv 16879   CSubSpccss 16880  toHLcthl 16881
This theorem is referenced by:  thlbas  16915  thlle  16916  thloc  16918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-thl 16884
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