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Theorem docafvalN 36325
Description: Subspace orthocomplement for  DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
docaval.j  |-  .\/  =  ( join `  K )
docaval.m  |-  ./\  =  ( meet `  K )
docaval.o  |-  ._|_  =  ( oc `  K )
docaval.h  |-  H  =  ( LHyp `  K
)
docaval.t  |-  T  =  ( ( LTrn `  K
) `  W )
docaval.i  |-  I  =  ( ( DIsoA `  K
) `  W )
docaval.n  |-  N  =  ( ( ocA `  K
) `  W )
Assertion
Ref Expression
docafvalN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  N  =  ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) ) )
Distinct variable groups:    x, z, K    x, I, z    x, T    x, W, z
Allowed substitution hints:    T( z)    H( x, z)    .\/ ( x, z)    ./\ (
x, z)    N( x, z)   
._|_ ( x, z)    V( x, z)

Proof of Theorem docafvalN
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 docaval.n . . 3  |-  N  =  ( ( ocA `  K
) `  W )
2 docaval.j . . . . 5  |-  .\/  =  ( join `  K )
3 docaval.m . . . . 5  |-  ./\  =  ( meet `  K )
4 docaval.o . . . . 5  |-  ._|_  =  ( oc `  K )
5 docaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
62, 3, 4, 5docaffvalN 36324 . . . 4  |-  ( K  e.  V  ->  ( ocA `  K )  =  ( w  e.  H  |->  ( x  e.  ~P ( ( LTrn `  K
) `  w )  |->  ( ( ( DIsoA `  K ) `  w
) `  ( (
(  ._|_  `  ( `' ( ( DIsoA `  K
) `  w ) `  |^| { z  e. 
ran  ( ( DIsoA `  K ) `  w
)  |  x  C_  z } ) )  .\/  (  ._|_  `  w )
)  ./\  w )
) ) ) )
76fveq1d 5874 . . 3  |-  ( K  e.  V  ->  (
( ocA `  K
) `  W )  =  ( ( w  e.  H  |->  ( x  e.  ~P ( (
LTrn `  K ) `  w )  |->  ( ( ( DIsoA `  K ) `  w ) `  (
( (  ._|_  `  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } ) )  .\/  (  ._|_  `  w ) )  ./\  w ) ) ) ) `  W ) )
81, 7syl5eq 2520 . 2  |-  ( K  e.  V  ->  N  =  ( ( w  e.  H  |->  ( x  e.  ~P ( (
LTrn `  K ) `  w )  |->  ( ( ( DIsoA `  K ) `  w ) `  (
( (  ._|_  `  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } ) )  .\/  (  ._|_  `  w ) )  ./\  w ) ) ) ) `  W ) )
9 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
10 docaval.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
119, 10syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
1211pweqd 4021 . . . 4  |-  ( w  =  W  ->  ~P ( ( LTrn `  K
) `  w )  =  ~P T )
13 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  ( ( DIsoA `  K ) `  W ) )
14 docaval.i . . . . . 6  |-  I  =  ( ( DIsoA `  K
) `  W )
1513, 14syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  I )
1615cnveqd 5184 . . . . . . . . 9  |-  ( w  =  W  ->  `' ( ( DIsoA `  K
) `  w )  =  `' I )
1715rneqd 5236 . . . . . . . . . . 11  |-  ( w  =  W  ->  ran  ( ( DIsoA `  K
) `  w )  =  ran  I )
18 rabeq 3112 . . . . . . . . . . 11  |-  ( ran  ( ( DIsoA `  K
) `  w )  =  ran  I  ->  { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z }  =  { z  e.  ran  I  |  x  C_  z } )
1917, 18syl 16 . . . . . . . . . 10  |-  ( w  =  W  ->  { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z }  =  { z  e.  ran  I  |  x  C_  z } )
2019inteqd 4293 . . . . . . . . 9  |-  ( w  =  W  ->  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z }  =  |^| { z  e.  ran  I  |  x  C_  z } )
2116, 20fveq12d 5878 . . . . . . . 8  |-  ( w  =  W  ->  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } )  =  ( `' I `  |^| { z  e. 
ran  I  |  x 
C_  z } ) )
2221fveq2d 5876 . . . . . . 7  |-  ( w  =  W  ->  (  ._|_  `  ( `' ( ( DIsoA `  K ) `  w ) `  |^| { z  e.  ran  (
( DIsoA `  K ) `  w )  |  x 
C_  z } ) )  =  (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z } ) ) )
23 fveq2 5872 . . . . . . 7  |-  ( w  =  W  ->  (  ._|_  `  w )  =  (  ._|_  `  W ) )
2422, 23oveq12d 6313 . . . . . 6  |-  ( w  =  W  ->  (
(  ._|_  `  ( `' ( ( DIsoA `  K
) `  w ) `  |^| { z  e. 
ran  ( ( DIsoA `  K ) `  w
)  |  x  C_  z } ) )  .\/  (  ._|_  `  w )
)  =  ( ( 
._|_  `  ( `' I `  |^| { z  e. 
ran  I  |  x 
C_  z } ) )  .\/  (  ._|_  `  W ) ) )
25 id 22 . . . . . 6  |-  ( w  =  W  ->  w  =  W )
2624, 25oveq12d 6313 . . . . 5  |-  ( w  =  W  ->  (
( (  ._|_  `  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } ) )  .\/  (  ._|_  `  w ) )  ./\  w )  =  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) )
2715, 26fveq12d 5878 . . . 4  |-  ( w  =  W  ->  (
( ( DIsoA `  K
) `  w ) `  ( ( (  ._|_  `  ( `' ( (
DIsoA `  K ) `  w ) `  |^| { z  e.  ran  (
( DIsoA `  K ) `  w )  |  x 
C_  z } ) )  .\/  (  ._|_  `  w ) )  ./\  w ) )  =  ( I `  (
( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )
2812, 27mpteq12dv 4531 . . 3  |-  ( w  =  W  ->  (
x  e.  ~P (
( LTrn `  K ) `  w )  |->  ( ( ( DIsoA `  K ) `  w ) `  (
( (  ._|_  `  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } ) )  .\/  (  ._|_  `  w ) )  ./\  w ) ) )  =  ( x  e. 
~P T  |->  ( I `
 ( ( ( 
._|_  `  ( `' I `  |^| { z  e. 
ran  I  |  x 
C_  z } ) )  .\/  (  ._|_  `  W ) )  ./\  W ) ) ) )
29 eqid 2467 . . 3  |-  ( w  e.  H  |->  ( x  e.  ~P ( (
LTrn `  K ) `  w )  |->  ( ( ( DIsoA `  K ) `  w ) `  (
( (  ._|_  `  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } ) )  .\/  (  ._|_  `  w ) )  ./\  w ) ) ) )  =  ( w  e.  H  |->  ( x  e.  ~P ( (
LTrn `  K ) `  w )  |->  ( ( ( DIsoA `  K ) `  w ) `  (
( (  ._|_  `  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } ) )  .\/  (  ._|_  `  w ) )  ./\  w ) ) ) )
30 fvex 5882 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
3110, 30eqeltri 2551 . . . . 5  |-  T  e. 
_V
3231pwex 4636 . . . 4  |-  ~P T  e.  _V
3332mptex 6142 . . 3  |-  ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )  e.  _V
3428, 29, 33fvmpt 5957 . 2  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( x  e.  ~P ( ( LTrn `  K
) `  w )  |->  ( ( ( DIsoA `  K ) `  w
) `  ( (
(  ._|_  `  ( `' ( ( DIsoA `  K
) `  w ) `  |^| { z  e. 
ran  ( ( DIsoA `  K ) `  w
)  |  x  C_  z } ) )  .\/  (  ._|_  `  w )
)  ./\  w )
) ) ) `  W )  =  ( x  e.  ~P T  |->  ( I `  (
( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) ) )
358, 34sylan9eq 2528 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  N  =  ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) ) )
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 4016   |^|cint 4288    |-> cmpt 4511   `'ccnv 5004   ran crn 5006   ` cfv 5594  (class class class)co 6295   occoc 14579   joincjn 15447   meetcmee 15448   LHypclh 35186   LTrncltrn 35303   DIsoAcdia 36231   ocAcocaN 36322
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
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 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-docaN 36323
This theorem is referenced by:  docavalN  36326
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