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Theorem dochval 31610
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
dochval  |-  ( ( ( K  e.  Y  /\  W  e.  H
)  /\  X  C_  V
)  ->  ( N `  X )  =  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  X  C_  ( I `  y ) } ) ) ) )
Distinct variable groups:    y, B    y, K    y, W    y, X
Allowed substitution hints:    U( y)    G( y)    H( y)    I( y)    N( y)    ._|_ ( y)    V( y)    Y( y)

Proof of Theorem dochval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dochval.b . . . . 5  |-  B  =  ( Base `  K
)
2 dochval.g . . . . 5  |-  G  =  ( glb `  K
)
3 dochval.o . . . . 5  |-  ._|_  =  ( oc `  K )
4 dochval.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 dochval.i . . . . 5  |-  I  =  ( ( DIsoH `  K
) `  W )
6 dochval.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
7 dochval.v . . . . 5  |-  V  =  ( Base `  U
)
8 dochval.n . . . . 5  |-  N  =  ( ( ocH `  K
) `  W )
91, 2, 3, 4, 5, 6, 7, 8dochfval 31609 . . . 4  |-  ( ( K  e.  Y  /\  W  e.  H )  ->  N  =  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  x  C_  ( I `  y ) } ) ) ) ) )
109adantr 451 . . 3  |-  ( ( ( K  e.  Y  /\  W  e.  H
)  /\  X  C_  V
)  ->  N  =  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `
 y ) } ) ) ) ) )
1110fveq1d 5610 . 2  |-  ( ( ( K  e.  Y  /\  W  e.  H
)  /\  X  C_  V
)  ->  ( N `  X )  =  ( ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `
 y ) } ) ) ) ) `
 X ) )
12 fvex 5622 . . . . . . 7  |-  ( Base `  U )  e.  _V
137, 12eqeltri 2428 . . . . . 6  |-  V  e. 
_V
1413elpw2 4256 . . . . 5  |-  ( X  e.  ~P V  <->  X  C_  V
)
1514biimpri 197 . . . 4  |-  ( X 
C_  V  ->  X  e.  ~P V )
1615adantl 452 . . 3  |-  ( ( ( K  e.  Y  /\  W  e.  H
)  /\  X  C_  V
)  ->  X  e.  ~P V )
17 fvex 5622 . . 3  |-  ( I `
 (  ._|_  `  ( G `  { y  e.  B  |  X  C_  ( I `  y
) } ) ) )  e.  _V
18 sseq1 3275 . . . . . . . 8  |-  ( x  =  X  ->  (
x  C_  ( I `  y )  <->  X  C_  (
I `  y )
) )
1918rabbidv 2856 . . . . . . 7  |-  ( x  =  X  ->  { y  e.  B  |  x 
C_  ( I `  y ) }  =  { y  e.  B  |  X  C_  ( I `
 y ) } )
2019fveq2d 5612 . . . . . 6  |-  ( x  =  X  ->  ( G `  { y  e.  B  |  x  C_  ( I `  y
) } )  =  ( G `  {
y  e.  B  |  X  C_  ( I `  y ) } ) )
2120fveq2d 5612 . . . . 5  |-  ( x  =  X  ->  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `
 y ) } ) )  =  ( 
._|_  `  ( G `  { y  e.  B  |  X  C_  ( I `
 y ) } ) ) )
2221fveq2d 5612 . . . 4  |-  ( x  =  X  ->  (
I `  (  ._|_  `  ( G `  {
y  e.  B  |  x  C_  ( I `  y ) } ) ) )  =  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  X  C_  ( I `  y ) } ) ) ) )
23 eqid 2358 . . . 4  |-  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  x  C_  ( I `  y ) } ) ) ) )  =  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `
 y ) } ) ) ) )
2422, 23fvmptg 5683 . . 3  |-  ( ( X  e.  ~P V  /\  ( I `  (  ._|_  `  ( G `  { y  e.  B  |  X  C_  ( I `
 y ) } ) ) )  e. 
_V )  ->  (
( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `
 y ) } ) ) ) ) `
 X )  =  ( I `  (  ._|_  `  ( G `  { y  e.  B  |  X  C_  ( I `
 y ) } ) ) ) )
2516, 17, 24sylancl 643 . 2  |-  ( ( ( K  e.  Y  /\  W  e.  H
)  /\  X  C_  V
)  ->  ( (
x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G `  { y  e.  B  |  x  C_  ( I `
 y ) } ) ) ) ) `
 X )  =  ( I `  (  ._|_  `  ( G `  { y  e.  B  |  X  C_  ( I `
 y ) } ) ) ) )
2611, 25eqtrd 2390 1  |-  ( ( ( K  e.  Y  /\  W  e.  H
)  /\  X  C_  V
)  ->  ( N `  X )  =  ( I `  (  ._|_  `  ( G `  {
y  e.  B  |  X  C_  ( I `  y ) } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   {crab 2623   _Vcvv 2864    C_ wss 3228   ~Pcpw 3701    e. cmpt 4158   ` cfv 5337   Basecbs 13245   occoc 13313   glbcglb 14176   LHypclh 30242   DVecHcdvh 31337   DIsoHcdih 31487   ocHcoch 31606
This theorem is referenced by:  dochval2  31611  dochcl  31612  dochvalr  31616  dochss  31624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-doch 31607
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