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Theorem dochfval 35000
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
)
dochval.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dochval.u  |-  U  =  ( ( DVecH `  K
) `  W )
dochval.v  |-  V  =  ( Base `  U
)
dochval.n  |-  N  =  ( ( ocH `  K
) `  W )
Assertion
Ref Expression
dochfval  |-  ( ( K  e.  X  /\  W  e.  H )  ->  N  =  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  x  C_  ( I `  y ) } ) ) ) ) )
Distinct variable groups:    y, B    x, y, K    x, V    x, W, y
Allowed substitution hints:    B( x)    U( x, y)    G( x, y)    H( x, y)    I( x, y)    N( x, y)    ._|_ ( x, y)    V( y)    X( x, y)

Proof of Theorem dochfval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dochval.n . . 3  |-  N  =  ( ( ocH `  K
) `  W )
2 dochval.b . . . . 5  |-  B  =  ( Base `  K
)
3 dochval.g . . . . 5  |-  G  =  ( glb `  K
)
4 dochval.o . . . . 5  |-  ._|_  =  ( oc `  K )
5 dochval.h . . . . 5  |-  H  =  ( LHyp `  K
)
62, 3, 4, 5dochffval 34999 . . . 4  |-  ( K  e.  X  ->  ( 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 ) } ) ) ) ) ) )
76fveq1d 5698 . . 3  |-  ( K  e.  X  ->  (
( ocH `  K
) `  W )  =  ( ( w  e.  H  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  K
) `  w )
)  |->  ( ( (
DIsoH `  K ) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) } ) ) ) ) ) `  W ) )
81, 7syl5eq 2487 . 2  |-  ( K  e.  X  ->  N  =  ( ( w  e.  H  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  K
) `  w )
)  |->  ( ( (
DIsoH `  K ) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) } ) ) ) ) ) `  W ) )
9 fveq2 5696 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
10 dochval.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
119, 10syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
1211fveq2d 5700 . . . . . 6  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  (
Base `  U )
)
13 dochval.v . . . . . 6  |-  V  =  ( Base `  U
)
1412, 13syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  V )
1514pweqd 3870 . . . 4  |-  ( w  =  W  ->  ~P ( Base `  ( ( DVecH `  K ) `  w ) )  =  ~P V )
16 fveq2 5696 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoH `  K ) `  w )  =  ( ( DIsoH `  K ) `  W ) )
17 dochval.i . . . . . 6  |-  I  =  ( ( DIsoH `  K
) `  W )
1816, 17syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  (
( DIsoH `  K ) `  w )  =  I )
1918fveq1d 5698 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ( DIsoH `  K
) `  w ) `  y )  =  ( I `  y ) )
2019sseq2d 3389 . . . . . . . 8  |-  ( w  =  W  ->  (
x  C_  ( (
( DIsoH `  K ) `  w ) `  y
)  <->  x  C_  ( I `
 y ) ) )
2120rabbidv 2969 . . . . . . 7  |-  ( w  =  W  ->  { y  e.  B  |  x 
C_  ( ( (
DIsoH `  K ) `  w ) `  y
) }  =  {
y  e.  B  |  x  C_  ( I `  y ) } )
2221fveq2d 5700 . . . . . 6  |-  ( w  =  W  ->  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } )  =  ( G `  { y  e.  B  |  x 
C_  ( I `  y ) } ) )
2322fveq2d 5700 . . . . 5  |-  ( w  =  W  ->  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) } ) )  =  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `  y
) } ) ) )
2418, 23fveq12d 5702 . . . 4  |-  ( w  =  W  ->  (
( ( DIsoH `  K
) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } ) ) )  =  ( I `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `
 y ) } ) ) ) )
2515, 24mpteq12dv 4375 . . 3  |-  ( w  =  W  ->  (
x  e.  ~P ( Base `  ( ( DVecH `  K ) `  w
) )  |->  ( ( ( DIsoH `  K ) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) } ) ) ) )  =  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `
 y ) } ) ) ) ) )
26 eqid 2443 . . 3  |-  ( w  e.  H  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  K
) `  w )
)  |->  ( ( (
DIsoH `  K ) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  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 ) } ) ) ) ) )
27 fvex 5706 . . . . . 6  |-  ( Base `  U )  e.  _V
2813, 27eqeltri 2513 . . . . 5  |-  V  e. 
_V
2928pwex 4480 . . . 4  |-  ~P V  e.  _V
3029mptex 5953 . . 3  |-  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  x  C_  ( I `  y ) } ) ) ) )  e. 
_V
3125, 26, 30fvmpt 5779 . 2  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  K ) `  w ) )  |->  ( ( ( DIsoH `  K
) `  w ) `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } ) ) ) ) ) `  W
)  =  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  x  C_  ( I `  y ) } ) ) ) ) )
328, 31sylan9eq 2495 1  |-  ( ( K  e.  X  /\  W  e.  H )  ->  N  =  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  x  C_  ( I `  y ) } ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2724   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865    e. cmpt 4355   ` cfv 5423   Basecbs 14179   occoc 14251   glbcglb 15118   LHypclh 33633   DVecHcdvh 34728   DIsoHcdih 34878   ocHcoch 34997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-doch 34998
This theorem is referenced by:  dochval  35001  dochfN  35006
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