Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dochffval Structured version   Unicode version

Theorem dochffval 32148
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 2965 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5729 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dochval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2487 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5729 . . . . . . . 8  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
65fveq1d 5731 . . . . . . 7  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
76fveq2d 5733 . . . . . 6  |-  ( k  =  K  ->  ( Base `  ( ( DVecH `  k ) `  w
) )  =  (
Base `  ( ( DVecH `  K ) `  w ) ) )
87pweqd 3805 . . . . 5  |-  ( k  =  K  ->  ~P ( Base `  ( ( DVecH `  k ) `  w ) )  =  ~P ( Base `  (
( DVecH `  K ) `  w ) ) )
9 fveq2 5729 . . . . . . 7  |-  ( k  =  K  ->  ( DIsoH `  k )  =  ( DIsoH `  K )
)
109fveq1d 5731 . . . . . 6  |-  ( k  =  K  ->  (
( DIsoH `  k ) `  w )  =  ( ( DIsoH `  K ) `  w ) )
11 fveq2 5729 . . . . . . . 8  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
12 dochval.o . . . . . . . 8  |-  ._|_  =  ( oc `  K )
1311, 12syl6eqr 2487 . . . . . . 7  |-  ( k  =  K  ->  ( oc `  k )  = 
._|_  )
14 fveq2 5729 . . . . . . . . 9  |-  ( k  =  K  ->  ( glb `  k )  =  ( glb `  K
) )
15 dochval.g . . . . . . . . 9  |-  G  =  ( glb `  K
)
1614, 15syl6eqr 2487 . . . . . . . 8  |-  ( k  =  K  ->  ( glb `  k )  =  G )
17 fveq2 5729 . . . . . . . . . 10  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
18 dochval.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1917, 18syl6eqr 2487 . . . . . . . . 9  |-  ( k  =  K  ->  ( Base `  k )  =  B )
2010fveq1d 5731 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( ( DIsoH `  k
) `  w ) `  y )  =  ( ( ( DIsoH `  K
) `  w ) `  y ) )
2120sseq2d 3377 . . . . . . . . 9  |-  ( k  =  K  ->  (
x  C_  ( (
( DIsoH `  k ) `  w ) `  y
)  <->  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) ) )
2219, 21rabeqbidv 2952 . . . . . . . 8  |-  ( k  =  K  ->  { y  e.  ( Base `  k
)  |  x  C_  ( ( ( DIsoH `  k ) `  w
) `  y ) }  =  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } )
2316, 22fveq12d 5735 . . . . . . 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 5735 . . . . . 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 5735 . . . . 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 4288 . . . 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 4288 . . 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 32147 . . 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 5743 . . . . 5  |-  ( LHyp `  K )  e.  _V
303, 29eqeltri 2507 . . . 4  |-  H  e. 
_V
3130mptex 5967 . . 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 5807 . 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 set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {crab 2710   _Vcvv 2957    C_ wss 3321   ~Pcpw 3800    e. cmpt 4267   ` cfv 5455   Basecbs 13470   occoc 13538   glbcglb 14401   LHypclh 30782   DVecHcdvh 31877   DIsoHcdih 32027   ocHcoch 32146
This theorem is referenced by:  dochfval  32149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-doch 32147
  Copyright terms: Public domain W3C validator