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Theorem docafvalN 34772
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 34771 . . . 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 5698 . . 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 2487 . 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 5696 . . . . . 6  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
10 docaval.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
119, 10syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
1211pweqd 3870 . . . 4  |-  ( w  =  W  ->  ~P ( ( LTrn `  K
) `  w )  =  ~P T )
13 fveq2 5696 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  ( ( DIsoA `  K ) `  W ) )
14 docaval.i . . . . . 6  |-  I  =  ( ( DIsoA `  K
) `  W )
1513, 14syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  I )
1615cnveqd 5020 . . . . . . . . 9  |-  ( w  =  W  ->  `' ( ( DIsoA `  K
) `  w )  =  `' I )
1715rneqd 5072 . . . . . . . . . . 11  |-  ( w  =  W  ->  ran  ( ( DIsoA `  K
) `  w )  =  ran  I )
18 rabeq 2971 . . . . . . . . . . 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 4138 . . . . . . . . 9  |-  ( w  =  W  ->  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z }  =  |^| { z  e.  ran  I  |  x  C_  z } )
2116, 20fveq12d 5702 . . . . . . . 8  |-  ( w  =  W  ->  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } )  =  ( `' I `  |^| { z  e. 
ran  I  |  x 
C_  z } ) )
2221fveq2d 5700 . . . . . . 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 5696 . . . . . . 7  |-  ( w  =  W  ->  (  ._|_  `  w )  =  (  ._|_  `  W ) )
2422, 23oveq12d 6114 . . . . . 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 6114 . . . . 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 5702 . . . 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 4375 . . 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 2443 . . 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 5706 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
3110, 30eqeltri 2513 . . . . 5  |-  T  e. 
_V
3231pwex 4480 . . . 4  |-  ~P T  e.  _V
3332mptex 5953 . . 3  |-  ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )  e.  _V
3428, 29, 33fvmpt 5779 . 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 2495 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 1369    e. wcel 1756   {crab 2724   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865   |^|cint 4133    e. cmpt 4355   `'ccnv 4844   ran crn 4846   ` cfv 5423  (class class class)co 6096   occoc 14251   joincjn 15119   meetcmee 15120   LHypclh 33633   LTrncltrn 33750   DIsoAcdia 34678   ocAcocaN 34769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-docaN 34770
This theorem is referenced by:  docavalN  34773
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