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Theorem thlval 16611
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 2809 . 2  |-  ( W  e.  V  ->  W  e.  _V )
2 thlval.k . . 3  |-  K  =  (toHL `  W )
3 fveq2 5541 . . . . . . . 8  |-  ( h  =  W  ->  ( CSubSp `
 h )  =  ( CSubSp `  W )
)
4 thlval.c . . . . . . . 8  |-  C  =  ( CSubSp `  W )
53, 4syl6eqr 2346 . . . . . . 7  |-  ( h  =  W  ->  ( CSubSp `
 h )  =  C )
65fveq2d 5545 . . . . . 6  |-  ( h  =  W  ->  (toInc `  ( CSubSp `  h )
)  =  (toInc `  C ) )
7 thlval.i . . . . . 6  |-  I  =  (toInc `  C )
86, 7syl6eqr 2346 . . . . 5  |-  ( h  =  W  ->  (toInc `  ( CSubSp `  h )
)  =  I )
9 fveq2 5541 . . . . . . 7  |-  ( h  =  W  ->  ( ocv `  h )  =  ( ocv `  W
) )
10 thlval.o . . . . . . 7  |-  ._|_  =  ( ocv `  W )
119, 10syl6eqr 2346 . . . . . 6  |-  ( h  =  W  ->  ( ocv `  h )  = 
._|_  )
1211opeq2d 3819 . . . . 5  |-  ( h  =  W  ->  <. ( oc `  ndx ) ,  ( ocv `  h
) >.  =  <. ( oc `  ndx ) , 
._|_  >. )
138, 12oveq12d 5892 . . . 4  |-  ( h  =  W  ->  (
(toInc `  ( CSubSp `  h ) ) sSet  <. ( oc `  ndx ) ,  ( ocv `  h
) >. )  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )
)
14 df-thl 16581 . . . 4  |- toHL  =  ( h  e.  _V  |->  ( (toInc `  ( CSubSp `  h ) ) sSet  <. ( oc `  ndx ) ,  ( ocv `  h
) >. ) )
15 ovex 5899 . . . 4  |-  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )  e.  _V
1613, 14, 15fvmpt 5618 . . 3  |-  ( W  e.  _V  ->  (toHL `  W )  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )
)
172, 16syl5eq 2340 . 2  |-  ( W  e.  _V  ->  K  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. ) )
181, 17syl 15 1  |-  ( W  e.  V  ->  K  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   ` cfv 5271  (class class class)co 5874   ndxcnx 13161   sSet csts 13162   occoc 13232  toInccipo 14270   ocvcocv 16576   CSubSpccss 16577  toHLcthl 16578
This theorem is referenced by:  thlbas  16612  thlle  16613  thloc  16615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-thl 16581
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