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Theorem dochfval 32222
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 32221 . . . 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 5733 . . 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 2482 . 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 5731 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
10 dochval.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
119, 10syl6eqr 2488 . . . . . . 7  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
1211fveq2d 5735 . . . . . 6  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  (
Base `  U )
)
13 dochval.v . . . . . 6  |-  V  =  ( Base `  U
)
1412, 13syl6eqr 2488 . . . . 5  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  V )
1514pweqd 3806 . . . 4  |-  ( w  =  W  ->  ~P ( Base `  ( ( DVecH `  K ) `  w ) )  =  ~P V )
16 fveq2 5731 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoH `  K ) `  w )  =  ( ( DIsoH `  K ) `  W ) )
17 dochval.i . . . . . 6  |-  I  =  ( ( DIsoH `  K
) `  W )
1816, 17syl6eqr 2488 . . . . 5  |-  ( w  =  W  ->  (
( DIsoH `  K ) `  w )  =  I )
1918fveq1d 5733 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ( DIsoH `  K
) `  w ) `  y )  =  ( I `  y ) )
2019sseq2d 3378 . . . . . . . 8  |-  ( w  =  W  ->  (
x  C_  ( (
( DIsoH `  K ) `  w ) `  y
)  <->  x  C_  ( I `
 y ) ) )
2120rabbidv 2950 . . . . . . 7  |-  ( w  =  W  ->  { y  e.  B  |  x 
C_  ( ( (
DIsoH `  K ) `  w ) `  y
) }  =  {
y  e.  B  |  x  C_  ( I `  y ) } )
2221fveq2d 5735 . . . . . 6  |-  ( w  =  W  ->  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } )  =  ( G `  { y  e.  B  |  x 
C_  ( I `  y ) } ) )
2322fveq2d 5735 . . . . 5  |-  ( w  =  W  ->  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) } ) )  =  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `  y
) } ) ) )
2418, 23fveq12d 5737 . . . 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 4290 . . 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 2438 . . 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 5745 . . . . . 6  |-  ( Base `  U )  e.  _V
2813, 27eqeltri 2508 . . . . 5  |-  V  e. 
_V
2928pwex 4385 . . . 4  |-  ~P V  e.  _V
3029mptex 5969 . . 3  |-  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  x  C_  ( I `  y ) } ) ) ) )  e. 
_V
3125, 26, 30fvmpt 5809 . 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 2490 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801    e. cmpt 4269   ` cfv 5457   Basecbs 13474   occoc 13542   glbcglb 14405   LHypclh 30855   DVecHcdvh 31950   DIsoHcdih 32100   ocHcoch 32219
This theorem is referenced by:  dochval  32223  dochfN  32228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-doch 32220
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