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Theorem docavalN 31848
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 31847 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  N  =  ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) ) )
98adantr 452 . . 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 5722 . 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 5734 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  e.  _V
125, 11eqeltri 2505 . . . . . 6  |-  T  e. 
_V
1312elpw2 4356 . . . . 5  |-  ( X  e.  ~P T  <->  X  C_  T
)
1413biimpri 198 . . . 4  |-  ( X 
C_  T  ->  X  e.  ~P T )
1514adantl 453 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  X  e.  ~P T )
16 fvex 5734 . . 3  |-  ( I `
 ( ( ( 
._|_  `  ( `' I `  |^| { z  e. 
ran  I  |  X  C_  z } ) ) 
.\/  (  ._|_  `  W
) )  ./\  W
) )  e.  _V
17 sseq1 3361 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  C_  z  <->  X  C_  z
) )
1817rabbidv 2940 . . . . . . . . . 10  |-  ( x  =  X  ->  { z  e.  ran  I  |  x  C_  z }  =  { z  e.  ran  I  |  X  C_  z } )
1918inteqd 4047 . . . . . . . . 9  |-  ( x  =  X  ->  |^| { z  e.  ran  I  |  x  C_  z }  =  |^| { z  e. 
ran  I  |  X  C_  z } )
2019fveq2d 5724 . . . . . . . 8  |-  ( x  =  X  ->  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
)  =  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )
2120fveq2d 5724 . . . . . . 7  |-  ( x  =  X  ->  (  ._|_  `  ( `' I `  |^| { z  e. 
ran  I  |  x 
C_  z } ) )  =  (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) ) )
2221oveq1d 6088 . . . . . 6  |-  ( x  =  X  ->  (
(  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) )  =  ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) )  .\/  (  ._|_  `  W )
) )
2322oveq1d 6088 . . . . 5  |-  ( x  =  X  ->  (
( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W )  =  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) )  .\/  (  ._|_  `  W )
)  ./\  W )
)
2423fveq2d 5724 . . . 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 2435 . . . 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 5796 . . 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 644 . 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 2467 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 359    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   |^|cint 4042    e. cmpt 4258   `'ccnv 4869   ran crn 4871   ` cfv 5446  (class class class)co 6073   occoc 13529   joincjn 14393   meetcmee 14394   HLchlt 30075   LHypclh 30708   LTrncltrn 30825   DIsoAcdia 31753   ocAcocaN 31844
This theorem is referenced by:  docaclN  31849  diaocN  31850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-docaN 31845
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