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Theorem dochfval 31833
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 31832 . . . 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 5689 . . 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 2448 . 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 5687 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
10 dochval.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
119, 10syl6eqr 2454 . . . . . . 7  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
1211fveq2d 5691 . . . . . 6  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  (
Base `  U )
)
13 dochval.v . . . . . 6  |-  V  =  ( Base `  U
)
1412, 13syl6eqr 2454 . . . . 5  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  V )
1514pweqd 3764 . . . 4  |-  ( w  =  W  ->  ~P ( Base `  ( ( DVecH `  K ) `  w ) )  =  ~P V )
16 fveq2 5687 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoH `  K ) `  w )  =  ( ( DIsoH `  K ) `  W ) )
17 dochval.i . . . . . 6  |-  I  =  ( ( DIsoH `  K
) `  W )
1816, 17syl6eqr 2454 . . . . 5  |-  ( w  =  W  ->  (
( DIsoH `  K ) `  w )  =  I )
1918fveq1d 5689 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ( DIsoH `  K
) `  w ) `  y )  =  ( I `  y ) )
2019sseq2d 3336 . . . . . . . 8  |-  ( w  =  W  ->  (
x  C_  ( (
( DIsoH `  K ) `  w ) `  y
)  <->  x  C_  ( I `
 y ) ) )
2120rabbidv 2908 . . . . . . 7  |-  ( w  =  W  ->  { y  e.  B  |  x 
C_  ( ( (
DIsoH `  K ) `  w ) `  y
) }  =  {
y  e.  B  |  x  C_  ( I `  y ) } )
2221fveq2d 5691 . . . . . 6  |-  ( w  =  W  ->  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } )  =  ( G `  { y  e.  B  |  x 
C_  ( I `  y ) } ) )
2322fveq2d 5691 . . . . 5  |-  ( w  =  W  ->  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) } ) )  =  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `  y
) } ) ) )
2418, 23fveq12d 5693 . . . 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 4247 . . 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 2404 . . 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 5701 . . . . . 6  |-  ( Base `  U )  e.  _V
2813, 27eqeltri 2474 . . . . 5  |-  V  e. 
_V
2928pwex 4342 . . . 4  |-  ~P V  e.  _V
3029mptex 5925 . . 3  |-  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  x  C_  ( I `  y ) } ) ) ) )  e. 
_V
3125, 26, 30fvmpt 5765 . 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 2456 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 359    = wceq 1649    e. wcel 1721   {crab 2670   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759    e. cmpt 4226   ` cfv 5413   Basecbs 13424   occoc 13492   glbcglb 14355   LHypclh 30466   DVecHcdvh 31561   DIsoHcdih 31711   ocHcoch 31830
This theorem is referenced by:  dochval  31834  dochfN  31839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-doch 31831
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