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Theorem docafvalN 31312
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 31311 . . . 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 5527 . . 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 2327 . 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 5525 . . . . . 6  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
10 docaval.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
119, 10syl6eqr 2333 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
1211pweqd 3630 . . . 4  |-  ( w  =  W  ->  ~P ( ( LTrn `  K
) `  w )  =  ~P T )
13 fveq2 5525 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  ( ( DIsoA `  K ) `  W ) )
14 docaval.i . . . . . 6  |-  I  =  ( ( DIsoA `  K
) `  W )
1513, 14syl6eqr 2333 . . . . 5  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  I )
1615cnveqd 4857 . . . . . . . . 9  |-  ( w  =  W  ->  `' ( ( DIsoA `  K
) `  w )  =  `' I )
1715rneqd 4906 . . . . . . . . . . 11  |-  ( w  =  W  ->  ran  ( ( DIsoA `  K
) `  w )  =  ran  I )
18 rabeq 2782 . . . . . . . . . . 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 15 . . . . . . . . . 10  |-  ( w  =  W  ->  { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z }  =  { z  e.  ran  I  |  x  C_  z } )
2019inteqd 3867 . . . . . . . . 9  |-  ( w  =  W  ->  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z }  =  |^| { z  e.  ran  I  |  x  C_  z } )
2116, 20fveq12d 5531 . . . . . . . 8  |-  ( w  =  W  ->  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } )  =  ( `' I `  |^| { z  e. 
ran  I  |  x 
C_  z } ) )
2221fveq2d 5529 . . . . . . 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 5525 . . . . . . 7  |-  ( w  =  W  ->  (  ._|_  `  w )  =  (  ._|_  `  W ) )
2422, 23oveq12d 5876 . . . . . 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 19 . . . . . 6  |-  ( w  =  W  ->  w  =  W )
2624, 25oveq12d 5876 . . . . 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 5531 . . . 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 4098 . . 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 2283 . . 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 5539 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
3110, 30eqeltri 2353 . . . . 5  |-  T  e. 
_V
3231pwex 4193 . . . 4  |-  ~P T  e.  _V
3332mptex 5746 . . 3  |-  ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )  e.  _V
3428, 29, 33fvmpt 5602 . 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 2335 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   |^|cint 3862    e. cmpt 4077   `'ccnv 4688   ran crn 4690   ` cfv 5255  (class class class)co 5858   occoc 13216   joincjn 14078   meetcmee 14079   LHypclh 30173   LTrncltrn 30290   DIsoAcdia 31218   ocAcocaN 31309
This theorem is referenced by:  docavalN  31313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-docaN 31310
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