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Theorem djafvalN 34784
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 34783 . . . 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 5698 . . 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 2487 . 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 5696 . . . . . 6  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
7 djaval.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
86, 7syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
98pweqd 3870 . . . 4  |-  ( w  =  W  ->  ~P ( ( LTrn `  K
) `  w )  =  ~P T )
10 fveq2 5696 . . . . . 6  |-  ( w  =  W  ->  (
( ocA `  K
) `  w )  =  ( ( ocA `  K ) `  W
) )
11 djaval.n . . . . . 6  |-  ._|_  =  ( ( ocA `  K
) `  W )
1210, 11syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  (
( ocA `  K
) `  w )  =  ._|_  )
1312fveq1d 5698 . . . . . 6  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  x )  =  ( 
._|_  `  x ) )
1412fveq1d 5698 . . . . . 6  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  y )  =  ( 
._|_  `  y ) )
1513, 14ineq12d 3558 . . . . 5  |-  ( w  =  W  ->  (
( ( ( ocA `  K ) `  w
) `  x )  i^i  ( ( ( ocA `  K ) `  w
) `  y )
)  =  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) )
1612, 15fveq12d 5702 . . . 4  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  ( ( ( ( ocA `  K ) `
 w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) )  =  (  ._|_  `  (
(  ._|_  `  x )  i^i  (  ._|_  `  y
) ) ) )
179, 9, 16mpt2eq123dv 6153 . . 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 2443 . . 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 5706 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
207, 19eqeltri 2513 . . . . 5  |-  T  e. 
_V
2120pwex 4480 . . . 4  |-  ~P T  e.  _V
2221, 21mpt2ex 6655 . . 3  |-  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) )  e. 
_V
2317, 18, 22fvmpt 5779 . 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 2495 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 1369    e. wcel 1756   _Vcvv 2977    i^i cin 3332   ~Pcpw 3865    e. cmpt 4355   ` cfv 5423    e. cmpt2 6098   LHypclh 33633   LTrncltrn 33750   DIsoAcdia 34678   ocAcocaN 34769   vAcdjaN 34781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-djaN 34782
This theorem is referenced by:  djavalN  34785
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