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Theorem dochffval 32161
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 2809 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5541 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dochval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2346 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
65fveq1d 5543 . . . . . . 7  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
76fveq2d 5545 . . . . . 6  |-  ( k  =  K  ->  ( Base `  ( ( DVecH `  k ) `  w
) )  =  (
Base `  ( ( DVecH `  K ) `  w ) ) )
87pweqd 3643 . . . . 5  |-  ( k  =  K  ->  ~P ( Base `  ( ( DVecH `  k ) `  w ) )  =  ~P ( Base `  (
( DVecH `  K ) `  w ) ) )
9 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( DIsoH `  k )  =  ( DIsoH `  K )
)
109fveq1d 5543 . . . . . 6  |-  ( k  =  K  ->  (
( DIsoH `  k ) `  w )  =  ( ( DIsoH `  K ) `  w ) )
11 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
12 dochval.o . . . . . . . 8  |-  ._|_  =  ( oc `  K )
1311, 12syl6eqr 2346 . . . . . . 7  |-  ( k  =  K  ->  ( oc `  k )  = 
._|_  )
14 fveq2 5541 . . . . . . . . 9  |-  ( k  =  K  ->  ( glb `  k )  =  ( glb `  K
) )
15 dochval.g . . . . . . . . 9  |-  G  =  ( glb `  K
)
1614, 15syl6eqr 2346 . . . . . . . 8  |-  ( k  =  K  ->  ( glb `  k )  =  G )
17 fveq2 5541 . . . . . . . . . 10  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
18 dochval.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1917, 18syl6eqr 2346 . . . . . . . . 9  |-  ( k  =  K  ->  ( Base `  k )  =  B )
2010fveq1d 5543 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( ( DIsoH `  k
) `  w ) `  y )  =  ( ( ( DIsoH `  K
) `  w ) `  y ) )
2120sseq2d 3219 . . . . . . . . 9  |-  ( k  =  K  ->  (
x  C_  ( (
( DIsoH `  k ) `  w ) `  y
)  <->  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) ) )
2219, 21rabeqbidv 2796 . . . . . . . 8  |-  ( k  =  K  ->  { y  e.  ( Base `  k
)  |  x  C_  ( ( ( DIsoH `  k ) `  w
) `  y ) }  =  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } )
2316, 22fveq12d 5547 . . . . . . 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 5547 . . . . . 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 5547 . . . . 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 4114 . . . 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 4114 . . 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 32160 . . 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 5555 . . . . 5  |-  ( LHyp `  K )  e.  _V
303, 29eqeltri 2366 . . . 4  |-  H  e. 
_V
3130mptex 5762 . . 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 5618 . 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 15 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 set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638    e. cmpt 4093   ` cfv 5271   Basecbs 13164   occoc 13232   glbcglb 14093   LHypclh 30795   DVecHcdvh 31890   DIsoHcdih 32040   ocHcoch 32159
This theorem is referenced by:  dochfval  32162
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-rep 4147  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  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-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-doch 32160
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