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

Theorem docaffvalN 36211
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 3127 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5871 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 docaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2526 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5871 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5873 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76pweqd 4020 . . . . 5  |-  ( k  =  K  ->  ~P ( ( LTrn `  k
) `  w )  =  ~P ( ( LTrn `  K ) `  w
) )
8 fveq2 5871 . . . . . . 7  |-  ( k  =  K  ->  ( DIsoA `  k )  =  ( DIsoA `  K )
)
98fveq1d 5873 . . . . . 6  |-  ( k  =  K  ->  (
( DIsoA `  k ) `  w )  =  ( ( DIsoA `  K ) `  w ) )
10 fveq2 5871 . . . . . . . 8  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
11 docaval.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
1210, 11syl6eqr 2526 . . . . . . 7  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
13 fveq2 5871 . . . . . . . . 9  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
14 docaval.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
1513, 14syl6eqr 2526 . . . . . . . 8  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
16 fveq2 5871 . . . . . . . . . 10  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
17 docaval.o . . . . . . . . . 10  |-  ._|_  =  ( oc `  K )
1816, 17syl6eqr 2526 . . . . . . . . 9  |-  ( k  =  K  ->  ( oc `  k )  = 
._|_  )
199cnveqd 5183 . . . . . . . . . 10  |-  ( k  =  K  ->  `' ( ( DIsoA `  k
) `  w )  =  `' ( ( DIsoA `  K ) `  w
) )
209rneqd 5235 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ran  ( ( DIsoA `  k
) `  w )  =  ran  ( ( DIsoA `  K ) `  w
) )
21 rabeq 3112 . . . . . . . . . . . 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 16 . . . . . . . . . . 11  |-  ( k  =  K  ->  { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z }  =  { z  e.  ran  ( ( DIsoA `  K
) `  w )  |  x  C_  z } )
2322inteqd 4292 . . . . . . . . . 10  |-  ( k  =  K  ->  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z }  =  |^| { z  e.  ran  ( ( DIsoA `  K
) `  w )  |  x  C_  z } )
2419, 23fveq12d 5877 . . . . . . . . 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 5877 . . . . . . . 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 5873 . . . . . . . 8  |-  ( k  =  K  ->  (
( oc `  k
) `  w )  =  (  ._|_  `  w
) )
2715, 25, 26oveq123d 6315 . . . . . . 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 2468 . . . . . . 7  |-  ( k  =  K  ->  w  =  w )
2912, 27, 28oveq123d 6315 . . . . . 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 5877 . . . . 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 4530 . . . 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 4530 . . 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 36210 . . 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 5881 . . . . 5  |-  ( LHyp `  K )  e.  _V
353, 34eqeltri 2551 . . . 4  |-  H  e. 
_V
3635mptex 6141 . . 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 5956 . 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 16 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 setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    C_ wss 3481   ~Pcpw 4015   |^|cint 4287    |-> cmpt 4510   `'ccnv 5003   ran crn 5005   ` cfv 5593  (class class class)co 6294   occoc 14575   joincjn 15443   meetcmee 15444   LHypclh 35073   LTrncltrn 35190   DIsoAcdia 36118   ocAcocaN 36209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-docaN 36210
This theorem is referenced by:  docafvalN  36212
  Copyright terms: Public domain W3C validator