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Theorem dochfval 36440
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 36439 . . . 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 5873 . . 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 2520 . 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 5871 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
10 dochval.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
119, 10syl6eqr 2526 . . . . . . 7  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
1211fveq2d 5875 . . . . . 6  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  (
Base `  U )
)
13 dochval.v . . . . . 6  |-  V  =  ( Base `  U
)
1412, 13syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  V )
1514pweqd 4020 . . . 4  |-  ( w  =  W  ->  ~P ( Base `  ( ( DVecH `  K ) `  w ) )  =  ~P V )
16 fveq2 5871 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoH `  K ) `  w )  =  ( ( DIsoH `  K ) `  W ) )
17 dochval.i . . . . . 6  |-  I  =  ( ( DIsoH `  K
) `  W )
1816, 17syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  (
( DIsoH `  K ) `  w )  =  I )
1918fveq1d 5873 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ( DIsoH `  K
) `  w ) `  y )  =  ( I `  y ) )
2019sseq2d 3537 . . . . . . . 8  |-  ( w  =  W  ->  (
x  C_  ( (
( DIsoH `  K ) `  w ) `  y
)  <->  x  C_  ( I `
 y ) ) )
2120rabbidv 3110 . . . . . . 7  |-  ( w  =  W  ->  { y  e.  B  |  x 
C_  ( ( (
DIsoH `  K ) `  w ) `  y
) }  =  {
y  e.  B  |  x  C_  ( I `  y ) } )
2221fveq2d 5875 . . . . . 6  |-  ( w  =  W  ->  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w
) `  y ) } )  =  ( G `  { y  e.  B  |  x 
C_  ( I `  y ) } ) )
2322fveq2d 5875 . . . . 5  |-  ( w  =  W  ->  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
) } ) )  =  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `  y
) } ) ) )
2418, 23fveq12d 5877 . . . 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 4530 . . 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 2467 . . 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 5881 . . . . . 6  |-  ( Base `  U )  e.  _V
2813, 27eqeltri 2551 . . . . 5  |-  V  e. 
_V
2928pwex 4635 . . . 4  |-  ~P V  e.  _V
3029mptex 6141 . . 3  |-  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  x  C_  ( I `  y ) } ) ) ) )  e. 
_V
3125, 26, 30fvmpt 5956 . 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 2528 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 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    C_ wss 3481   ~Pcpw 4015    |-> cmpt 4510   ` cfv 5593   Basecbs 14502   occoc 14575   glbcglb 15442   LHypclh 35073   DVecHcdvh 36168   DIsoHcdih 36318   ocHcoch 36437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-doch 36438
This theorem is referenced by:  dochval  36441  dochfN  36446
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