MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  thlval Unicode version

Theorem thlval 16595
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 2796 . 2  |-  ( W  e.  V  ->  W  e.  _V )
2 thlval.k . . 3  |-  K  =  (toHL `  W )
3 fveq2 5525 . . . . . . . 8  |-  ( h  =  W  ->  ( CSubSp `
 h )  =  ( CSubSp `  W )
)
4 thlval.c . . . . . . . 8  |-  C  =  ( CSubSp `  W )
53, 4syl6eqr 2333 . . . . . . 7  |-  ( h  =  W  ->  ( CSubSp `
 h )  =  C )
65fveq2d 5529 . . . . . 6  |-  ( h  =  W  ->  (toInc `  ( CSubSp `  h )
)  =  (toInc `  C ) )
7 thlval.i . . . . . 6  |-  I  =  (toInc `  C )
86, 7syl6eqr 2333 . . . . 5  |-  ( h  =  W  ->  (toInc `  ( CSubSp `  h )
)  =  I )
9 fveq2 5525 . . . . . . 7  |-  ( h  =  W  ->  ( ocv `  h )  =  ( ocv `  W
) )
10 thlval.o . . . . . . 7  |-  ._|_  =  ( ocv `  W )
119, 10syl6eqr 2333 . . . . . 6  |-  ( h  =  W  ->  ( ocv `  h )  = 
._|_  )
1211opeq2d 3803 . . . . 5  |-  ( h  =  W  ->  <. ( oc `  ndx ) ,  ( ocv `  h
) >.  =  <. ( oc `  ndx ) , 
._|_  >. )
138, 12oveq12d 5876 . . . 4  |-  ( h  =  W  ->  (
(toInc `  ( CSubSp `  h ) ) sSet  <. ( oc `  ndx ) ,  ( ocv `  h
) >. )  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )
)
14 df-thl 16565 . . . 4  |- toHL  =  ( h  e.  _V  |->  ( (toInc `  ( CSubSp `  h ) ) sSet  <. ( oc `  ndx ) ,  ( ocv `  h
) >. ) )
15 ovex 5883 . . . 4  |-  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )  e.  _V
1613, 14, 15fvmpt 5602 . . 3  |-  ( W  e.  _V  ->  (toHL `  W )  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. )
)
172, 16syl5eq 2327 . 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 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   occoc 13216  toInccipo 14254   ocvcocv 16560   CSubSpccss 16561  toHLcthl 16562
This theorem is referenced by:  thlbas  16596  thlle  16597  thloc  16599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-thl 16565
  Copyright terms: Public domain W3C validator