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Theorem docaffvalN 31933
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
)
Assertion
Ref Expression
docaffvalN  |-  ( 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 )
) ) ) )
Distinct variable groups:    w, H    x, w, z, K
Allowed substitution hints:    H( x, z)    .\/ ( x, z, w)    ./\ ( x, z, w)    ._|_ ( x, z, w)    V( x, z, w)

Proof of Theorem docaffvalN
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2809 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5541 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 docaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2346 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5543 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76pweqd 3643 . . . . 5  |-  ( k  =  K  ->  ~P ( ( LTrn `  k
) `  w )  =  ~P ( ( LTrn `  K ) `  w
) )
8 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( DIsoA `  k )  =  ( DIsoA `  K )
)
98fveq1d 5543 . . . . . 6  |-  ( k  =  K  ->  (
( DIsoA `  k ) `  w )  =  ( ( DIsoA `  K ) `  w ) )
10 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
11 docaval.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
1210, 11syl6eqr 2346 . . . . . . 7  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
13 fveq2 5541 . . . . . . . . 9  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
14 docaval.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
1513, 14syl6eqr 2346 . . . . . . . 8  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
16 fveq2 5541 . . . . . . . . . 10  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
17 docaval.o . . . . . . . . . 10  |-  ._|_  =  ( oc `  K )
1816, 17syl6eqr 2346 . . . . . . . . 9  |-  ( k  =  K  ->  ( oc `  k )  = 
._|_  )
199cnveqd 4873 . . . . . . . . . 10  |-  ( k  =  K  ->  `' ( ( DIsoA `  k
) `  w )  =  `' ( ( DIsoA `  K ) `  w
) )
209rneqd 4922 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ran  ( ( DIsoA `  k
) `  w )  =  ran  ( ( DIsoA `  K ) `  w
) )
21 rabeq 2795 . . . . . . . . . . . 12  |-  ( ran  ( ( DIsoA `  k
) `  w )  =  ran  ( ( DIsoA `  K ) `  w
)  ->  { z  e.  ran  ( ( DIsoA `  k ) `  w
)  |  x  C_  z }  =  {
z  e.  ran  (
( DIsoA `  K ) `  w )  |  x 
C_  z } )
2220, 21syl 15 . . . . . . . . . . 11  |-  ( k  =  K  ->  { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z }  =  { z  e.  ran  ( ( DIsoA `  K
) `  w )  |  x  C_  z } )
2322inteqd 3883 . . . . . . . . . 10  |-  ( k  =  K  ->  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z }  =  |^| { z  e.  ran  ( ( DIsoA `  K
) `  w )  |  x  C_  z } )
2419, 23fveq12d 5547 . . . . . . . . 9  |-  ( k  =  K  ->  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } )  =  ( `' ( ( DIsoA `  K ) `  w ) `  |^| { z  e.  ran  (
( DIsoA `  K ) `  w )  |  x 
C_  z } ) )
2518, 24fveq12d 5547 . . . . . . . 8  |-  ( k  =  K  ->  (
( oc `  k
) `  ( `' ( ( DIsoA `  k
) `  w ) `  |^| { z  e. 
ran  ( ( DIsoA `  k ) `  w
)  |  x  C_  z } ) )  =  (  ._|_  `  ( `' ( ( DIsoA `  K
) `  w ) `  |^| { z  e. 
ran  ( ( DIsoA `  K ) `  w
)  |  x  C_  z } ) ) )
2618fveq1d 5543 . . . . . . . 8  |-  ( k  =  K  ->  (
( oc `  k
) `  w )  =  (  ._|_  `  w
) )
2715, 25, 26oveq123d 5895 . . . . . . 7  |-  ( k  =  K  ->  (
( ( oc `  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) )  =  ( (  ._|_  `  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } ) )  .\/  (  ._|_  `  w ) ) )
28 eqidd 2297 . . . . . . 7  |-  ( k  =  K  ->  w  =  w )
2912, 27, 28oveq123d 5895 . . . . . 6  |-  ( k  =  K  ->  (
( ( ( oc
`  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  k )
w )  =  ( ( (  ._|_  `  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } ) )  .\/  (  ._|_  `  w ) )  ./\  w ) )
309, 29fveq12d 5547 . . . . 5  |-  ( k  =  K  ->  (
( ( DIsoA `  k
) `  w ) `  ( ( ( ( oc `  k ) `
 ( `' ( ( DIsoA `  k ) `  w ) `  |^| { z  e.  ran  (
( DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  k )
w ) )  =  ( ( ( DIsoA `  K ) `  w
) `  ( (
(  ._|_  `  ( `' ( ( DIsoA `  K
) `  w ) `  |^| { z  e. 
ran  ( ( DIsoA `  K ) `  w
)  |  x  C_  z } ) )  .\/  (  ._|_  `  w )
)  ./\  w )
) )
317, 30mpteq12dv 4114 . . . 4  |-  ( k  =  K  ->  (
x  e.  ~P (
( LTrn `  k ) `  w )  |->  ( ( ( DIsoA `  k ) `  w ) `  (
( ( ( oc
`  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  k )
w ) ) )  =  ( x  e. 
~P ( ( LTrn `  K ) `  w
)  |->  ( ( (
DIsoA `  K ) `  w ) `  (
( (  ._|_  `  ( `' ( ( DIsoA `  K ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  K ) `  w )  |  x 
C_  z } ) )  .\/  (  ._|_  `  w ) )  ./\  w ) ) ) )
324, 31mpteq12dv 4114 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( x  e.  ~P ( (
LTrn `  k ) `  w )  |->  ( ( ( DIsoA `  k ) `  w ) `  (
( ( ( oc
`  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  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 ) ) ) ) )
33 df-docaN 31932 . . 3  |-  ocA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ~P (
( LTrn `  k ) `  w )  |->  ( ( ( DIsoA `  k ) `  w ) `  (
( ( ( oc
`  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  k )
w ) ) ) ) )
34 fvex 5555 . . . . 5  |-  ( LHyp `  K )  e.  _V
353, 34eqeltri 2366 . . . 4  |-  H  e. 
_V
3635mptex 5762 . . 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 ) ) ) )  e.  _V
3732, 33, 36fvmpt 5618 . 2  |-  ( 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 )
) ) ) )
381, 37syl 15 1  |-  ( 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 )
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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   LHypclh 30795   LTrncltrn 30912   DIsoAcdia 31840   ocAcocaN 31931
This theorem is referenced by:  docafvalN  31934
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-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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