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Theorem dochffval 36547
Description: Subspace orthocomplement for  DVecH vector space. (Contributed by NM, 14-Mar-2014.)
Hypotheses
Ref Expression
dochval.b  |-  B  =  ( Base `  K
)
dochval.g  |-  G  =  ( glb `  K
)
dochval.o  |-  ._|_  =  ( oc `  K )
dochval.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
dochffval  |-  ( K  e.  V  ->  ( ocH `  K )  =  ( w  e.  H  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  K ) `  w ) )  |->  ( ( ( DIsoH `  K
) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } ) ) ) ) ) )
Distinct variable groups:    y, B    w, H    x, w, y, K
Allowed substitution hints:    B( x, w)    G( x, y, w)    H( x, y)    ._|_ ( x, y, w)    V( x, y, w)

Proof of Theorem dochffval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 3127 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5872 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dochval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2526 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5872 . . . . . . . 8  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
65fveq1d 5874 . . . . . . 7  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
76fveq2d 5876 . . . . . 6  |-  ( k  =  K  ->  ( Base `  ( ( DVecH `  k ) `  w
) )  =  (
Base `  ( ( DVecH `  K ) `  w ) ) )
87pweqd 4021 . . . . 5  |-  ( k  =  K  ->  ~P ( Base `  ( ( DVecH `  k ) `  w ) )  =  ~P ( Base `  (
( DVecH `  K ) `  w ) ) )
9 fveq2 5872 . . . . . . 7  |-  ( k  =  K  ->  ( DIsoH `  k )  =  ( DIsoH `  K )
)
109fveq1d 5874 . . . . . 6  |-  ( k  =  K  ->  (
( DIsoH `  k ) `  w )  =  ( ( DIsoH `  K ) `  w ) )
11 fveq2 5872 . . . . . . . 8  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
12 dochval.o . . . . . . . 8  |-  ._|_  =  ( oc `  K )
1311, 12syl6eqr 2526 . . . . . . 7  |-  ( k  =  K  ->  ( oc `  k )  = 
._|_  )
14 fveq2 5872 . . . . . . . . 9  |-  ( k  =  K  ->  ( glb `  k )  =  ( glb `  K
) )
15 dochval.g . . . . . . . . 9  |-  G  =  ( glb `  K
)
1614, 15syl6eqr 2526 . . . . . . . 8  |-  ( k  =  K  ->  ( glb `  k )  =  G )
17 fveq2 5872 . . . . . . . . . 10  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
18 dochval.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1917, 18syl6eqr 2526 . . . . . . . . 9  |-  ( k  =  K  ->  ( Base `  k )  =  B )
2010fveq1d 5874 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( ( DIsoH `  k
) `  w ) `  y )  =  ( ( ( DIsoH `  K
) `  w ) `  y ) )
2120sseq2d 3537 . . . . . . . . 9  |-  ( k  =  K  ->  (
x  C_  ( (
( DIsoH `  k ) `  w ) `  y
)  <->  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) ) )
2219, 21rabeqbidv 3113 . . . . . . . 8  |-  ( k  =  K  ->  { y  e.  ( Base `  k
)  |  x  C_  ( ( ( DIsoH `  k ) `  w
) `  y ) }  =  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } )
2316, 22fveq12d 5878 . . . . . . 7  |-  ( k  =  K  ->  (
( glb `  k
) `  { y  e.  ( Base `  k
)  |  x  C_  ( ( ( DIsoH `  k ) `  w
) `  y ) } )  =  ( G `  { y  e.  B  |  x 
C_  ( ( (
DIsoH `  K ) `  w ) `  y
) } ) )
2413, 23fveq12d 5878 . . . . . 6  |-  ( k  =  K  ->  (
( oc `  k
) `  ( ( glb `  k ) `  { y  e.  (
Base `  k )  |  x  C_  ( ( ( DIsoH `  k ) `  w ) `  y
) } ) )  =  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } ) ) )
2510, 24fveq12d 5878 . . . . 5  |-  ( k  =  K  ->  (
( ( DIsoH `  k
) `  w ) `  ( ( oc `  k ) `  (
( glb `  k
) `  { y  e.  ( Base `  k
)  |  x  C_  ( ( ( DIsoH `  k ) `  w
) `  y ) } ) ) )  =  ( ( (
DIsoH `  K ) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) } ) ) ) )
268, 25mpteq12dv 4531 . . . 4  |-  ( k  =  K  ->  (
x  e.  ~P ( Base `  ( ( DVecH `  k ) `  w
) )  |->  ( ( ( DIsoH `  k ) `  w ) `  (
( oc `  k
) `  ( ( glb `  k ) `  { y  e.  (
Base `  k )  |  x  C_  ( ( ( DIsoH `  k ) `  w ) `  y
) } ) ) ) )  =  ( x  e.  ~P ( Base `  ( ( DVecH `  K ) `  w
) )  |->  ( ( ( DIsoH `  K ) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) } ) ) ) ) )
274, 26mpteq12dv 4531 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  k
) `  w )
)  |->  ( ( (
DIsoH `  k ) `  w ) `  (
( oc `  k
) `  ( ( glb `  k ) `  { y  e.  (
Base `  k )  |  x  C_  ( ( ( DIsoH `  k ) `  w ) `  y
) } ) ) ) ) )  =  ( w  e.  H  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  K ) `  w ) )  |->  ( ( ( DIsoH `  K
) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } ) ) ) ) ) )
28 df-doch 36546 . . 3  |-  ocH  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  k ) `  w
) )  |->  ( ( ( DIsoH `  k ) `  w ) `  (
( oc `  k
) `  ( ( glb `  k ) `  { y  e.  (
Base `  k )  |  x  C_  ( ( ( DIsoH `  k ) `  w ) `  y
) } ) ) ) ) ) )
29 fvex 5882 . . . . 5  |-  ( LHyp `  K )  e.  _V
303, 29eqeltri 2551 . . . 4  |-  H  e. 
_V
3130mptex 6142 . . 3  |-  ( w  e.  H  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  K
) `  w )
)  |->  ( ( (
DIsoH `  K ) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) } ) ) ) ) )  e. 
_V
3227, 28, 31fvmpt 5957 . 2  |-  ( K  e.  _V  ->  ( ocH `  K )  =  ( w  e.  H  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  K ) `  w ) )  |->  ( ( ( DIsoH `  K
) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } ) ) ) ) ) )
331, 32syl 16 1  |-  ( K  e.  V  ->  ( ocH `  K )  =  ( w  e.  H  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  K ) `  w ) )  |->  ( ( ( DIsoH `  K
) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    C_ wss 3481   ~Pcpw 4016    |-> cmpt 4511   ` cfv 5594   Basecbs 14507   occoc 14580   glbcglb 15447   LHypclh 35181   DVecHcdvh 36276   DIsoHcdih 36426   ocHcoch 36545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-doch 36546
This theorem is referenced by:  dochfval  36548
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