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Theorem djafvalN 36224
Description: Subspace join for  DVecA partial vector space. TODO: take out hypothesis .i, no longer used. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
djaval.h  |-  H  =  ( LHyp `  K
)
djaval.t  |-  T  =  ( ( LTrn `  K
) `  W )
djaval.i  |-  I  =  ( ( DIsoA `  K
) `  W )
djaval.n  |-  ._|_  =  ( ( ocA `  K
) `  W )
djaval.j  |-  J  =  ( ( vA `  K ) `  W
)
Assertion
Ref Expression
djafvalN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  J  =  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) ) )
Distinct variable groups:    x, y, K    x, T, y    x, W, y
Allowed substitution hints:    H( x, y)    I( x, y)    J( x, y)    ._|_ ( x, y)    V( x, y)

Proof of Theorem djafvalN
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 djaval.j . . 3  |-  J  =  ( ( vA `  K ) `  W
)
2 djaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
32djaffvalN 36223 . . . 4  |-  ( K  e.  V  ->  ( vA `  K )  =  ( w  e.  H  |->  ( x  e.  ~P ( ( LTrn `  K
) `  w ) ,  y  e.  ~P ( ( LTrn `  K
) `  w )  |->  ( ( ( ocA `  K ) `  w
) `  ( (
( ( ocA `  K
) `  w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) ) ) ) )
43fveq1d 5873 . . 3  |-  ( K  e.  V  ->  (
( vA `  K
) `  W )  =  ( ( w  e.  H  |->  ( x  e.  ~P ( (
LTrn `  K ) `  w ) ,  y  e.  ~P ( (
LTrn `  K ) `  w )  |->  ( ( ( ocA `  K
) `  w ) `  ( ( ( ( ocA `  K ) `
 w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) ) ) ) `  W
) )
51, 4syl5eq 2520 . 2  |-  ( K  e.  V  ->  J  =  ( ( w  e.  H  |->  ( x  e.  ~P ( (
LTrn `  K ) `  w ) ,  y  e.  ~P ( (
LTrn `  K ) `  w )  |->  ( ( ( ocA `  K
) `  w ) `  ( ( ( ( ocA `  K ) `
 w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) ) ) ) `  W
) )
6 fveq2 5871 . . . . . 6  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
7 djaval.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
86, 7syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
98pweqd 4020 . . . 4  |-  ( w  =  W  ->  ~P ( ( LTrn `  K
) `  w )  =  ~P T )
10 fveq2 5871 . . . . . 6  |-  ( w  =  W  ->  (
( ocA `  K
) `  w )  =  ( ( ocA `  K ) `  W
) )
11 djaval.n . . . . . 6  |-  ._|_  =  ( ( ocA `  K
) `  W )
1210, 11syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  (
( ocA `  K
) `  w )  =  ._|_  )
1312fveq1d 5873 . . . . . 6  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  x )  =  ( 
._|_  `  x ) )
1412fveq1d 5873 . . . . . 6  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  y )  =  ( 
._|_  `  y ) )
1513, 14ineq12d 3706 . . . . 5  |-  ( w  =  W  ->  (
( ( ( ocA `  K ) `  w
) `  x )  i^i  ( ( ( ocA `  K ) `  w
) `  y )
)  =  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) )
1612, 15fveq12d 5877 . . . 4  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  ( ( ( ( ocA `  K ) `
 w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) )  =  (  ._|_  `  (
(  ._|_  `  x )  i^i  (  ._|_  `  y
) ) ) )
179, 9, 16mpt2eq123dv 6353 . . 3  |-  ( w  =  W  ->  (
x  e.  ~P (
( LTrn `  K ) `  w ) ,  y  e.  ~P ( (
LTrn `  K ) `  w )  |->  ( ( ( ocA `  K
) `  w ) `  ( ( ( ( ocA `  K ) `
 w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) ) )  =  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) ) )
18 eqid 2467 . . 3  |-  ( w  e.  H  |->  ( x  e.  ~P ( (
LTrn `  K ) `  w ) ,  y  e.  ~P ( (
LTrn `  K ) `  w )  |->  ( ( ( ocA `  K
) `  w ) `  ( ( ( ( ocA `  K ) `
 w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) ) ) )  =  ( w  e.  H  |->  ( x  e.  ~P (
( LTrn `  K ) `  w ) ,  y  e.  ~P ( (
LTrn `  K ) `  w )  |->  ( ( ( ocA `  K
) `  w ) `  ( ( ( ( ocA `  K ) `
 w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) ) ) )
19 fvex 5881 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
207, 19eqeltri 2551 . . . . 5  |-  T  e. 
_V
2120pwex 4635 . . . 4  |-  ~P T  e.  _V
2221, 21mpt2ex 6870 . . 3  |-  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) )  e. 
_V
2317, 18, 22fvmpt 5956 . 2  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( x  e.  ~P ( ( LTrn `  K
) `  w ) ,  y  e.  ~P ( ( LTrn `  K
) `  w )  |->  ( ( ( ocA `  K ) `  w
) `  ( (
( ( ocA `  K
) `  w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) ) ) ) `  W
)  =  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) ) )
245, 23sylan9eq 2528 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  J  =  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480   ~Pcpw 4015    |-> cmpt 4510   ` cfv 5593    |-> cmpt2 6296   LHypclh 35073   LTrncltrn 35190   DIsoAcdia 36118   ocAcocaN 36209   vAcdjaN 36221
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-8 1769  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-pow 4630  ax-pr 4691  ax-un 6586
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-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-oprab 6298  df-mpt2 6299  df-1st 6794  df-2nd 6795  df-djaN 36222
This theorem is referenced by:  djavalN  36225
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