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Theorem docavalN 31935
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
docavalN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( N `  X )  =  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )
Distinct variable groups:    z, K    z, I    z, W    z, T    z, X
Allowed substitution hints:    H( z)    .\/ ( z)    ./\ ( z)    N( z)    ._|_ ( z)

Proof of Theorem docavalN
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 docaval.j . . . . 5  |-  .\/  =  ( join `  K )
2 docaval.m . . . . 5  |-  ./\  =  ( meet `  K )
3 docaval.o . . . . 5  |-  ._|_  =  ( oc `  K )
4 docaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 docaval.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
6 docaval.i . . . . 5  |-  I  =  ( ( DIsoA `  K
) `  W )
7 docaval.n . . . . 5  |-  N  =  ( ( ocA `  K
) `  W )
81, 2, 3, 4, 5, 6, 7docafvalN 31934 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  N  =  ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) ) )
98adantr 451 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  N  =  ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z } ) )  .\/  (  ._|_  `  W )
)  ./\  W )
) ) )
109fveq1d 5543 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( N `  X )  =  ( ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z } ) )  .\/  (  ._|_  `  W )
)  ./\  W )
) ) `  X
) )
11 fvex 5555 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  e.  _V
125, 11eqeltri 2366 . . . . . 6  |-  T  e. 
_V
1312elpw2 4191 . . . . 5  |-  ( X  e.  ~P T  <->  X  C_  T
)
1413biimpri 197 . . . 4  |-  ( X 
C_  T  ->  X  e.  ~P T )
1514adantl 452 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  X  e.  ~P T )
16 fvex 5555 . . 3  |-  ( I `
 ( ( ( 
._|_  `  ( `' I `  |^| { z  e. 
ran  I  |  X  C_  z } ) ) 
.\/  (  ._|_  `  W
) )  ./\  W
) )  e.  _V
17 sseq1 3212 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  C_  z  <->  X  C_  z
) )
1817rabbidv 2793 . . . . . . . . . 10  |-  ( x  =  X  ->  { z  e.  ran  I  |  x  C_  z }  =  { z  e.  ran  I  |  X  C_  z } )
1918inteqd 3883 . . . . . . . . 9  |-  ( x  =  X  ->  |^| { z  e.  ran  I  |  x  C_  z }  =  |^| { z  e. 
ran  I  |  X  C_  z } )
2019fveq2d 5545 . . . . . . . 8  |-  ( x  =  X  ->  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
)  =  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )
2120fveq2d 5545 . . . . . . 7  |-  ( x  =  X  ->  (  ._|_  `  ( `' I `  |^| { z  e. 
ran  I  |  x 
C_  z } ) )  =  (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) ) )
2221oveq1d 5889 . . . . . 6  |-  ( x  =  X  ->  (
(  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) )  =  ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) )  .\/  (  ._|_  `  W )
) )
2322oveq1d 5889 . . . . 5  |-  ( x  =  X  ->  (
( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W )  =  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) )  .\/  (  ._|_  `  W )
)  ./\  W )
)
2423fveq2d 5545 . . . 4  |-  ( x  =  X  ->  (
I `  ( (
(  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) )  =  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) )  .\/  (  ._|_  `  W )
)  ./\  W )
) )
25 eqid 2296 . . . 4  |-  ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )  =  ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )
2624, 25fvmptg 5616 . . 3  |-  ( ( X  e.  ~P T  /\  ( I `  (
( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) )  e.  _V )  -> 
( ( x  e. 
~P T  |->  ( I `
 ( ( ( 
._|_  `  ( `' I `  |^| { z  e. 
ran  I  |  x 
C_  z } ) )  .\/  (  ._|_  `  W ) )  ./\  W ) ) ) `  X )  =  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )
2715, 16, 26sylancl 643 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( (
x  e.  ~P T  |->  ( I `  (
( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) ) `  X )  =  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) )  .\/  (  ._|_  `  W )
)  ./\  W )
) )
2810, 27eqtrd 2328 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( N `  X )  =  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   |^|cint 3878    e. cmpt 4093   `'ccnv 4704   ran crn 4706   ` cfv 5271  (class class class)co 5874   occoc 13232   joincjn 14094   meetcmee 14095   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   DIsoAcdia 31840   ocAcocaN 31931
This theorem is referenced by:  docaclN  31936  diaocN  31937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-docaN 31932
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