Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  docavalN Unicode version

Theorem docavalN 31240
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 31239 . . . 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 5672 . 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 5684 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  e.  _V
125, 11eqeltri 2459 . . . . . 6  |-  T  e. 
_V
1312elpw2 4307 . . . . 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 5684 . . 3  |-  ( I `
 ( ( ( 
._|_  `  ( `' I `  |^| { z  e. 
ran  I  |  X  C_  z } ) ) 
.\/  (  ._|_  `  W
) )  ./\  W
) )  e.  _V
17 sseq1 3314 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  C_  z  <->  X  C_  z
) )
1817rabbidv 2893 . . . . . . . . . 10  |-  ( x  =  X  ->  { z  e.  ran  I  |  x  C_  z }  =  { z  e.  ran  I  |  X  C_  z } )
1918inteqd 3999 . . . . . . . . 9  |-  ( x  =  X  ->  |^| { z  e.  ran  I  |  x  C_  z }  =  |^| { z  e. 
ran  I  |  X  C_  z } )
2019fveq2d 5674 . . . . . . . 8  |-  ( x  =  X  ->  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
)  =  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )
2120fveq2d 5674 . . . . . . 7  |-  ( x  =  X  ->  (  ._|_  `  ( `' I `  |^| { z  e. 
ran  I  |  x 
C_  z } ) )  =  (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) ) )
2221oveq1d 6037 . . . . . 6  |-  ( x  =  X  ->  (
(  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) )  =  ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) )  .\/  (  ._|_  `  W )
) )
2322oveq1d 6037 . . . . 5  |-  ( x  =  X  ->  (
( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  x  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W )  =  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) )  .\/  (  ._|_  `  W )
)  ./\  W )
)
2423fveq2d 5674 . . . 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 2389 . . . 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 5745 . . 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 2421 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 1649    e. wcel 1717   {crab 2655   _Vcvv 2901    C_ wss 3265   ~Pcpw 3744   |^|cint 3994    e. cmpt 4209   `'ccnv 4819   ran crn 4821   ` cfv 5396  (class class class)co 6022   occoc 13466   joincjn 14330   meetcmee 14331   HLchlt 29467   LHypclh 30100   LTrncltrn 30217   DIsoAcdia 31145   ocAcocaN 31236
This theorem is referenced by:  docaclN  31241  diaocN  31242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-docaN 31237
  Copyright terms: Public domain W3C validator