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Theorem djafvalN 31869
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 31868 . . . 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 5722 . . 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 2479 . 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 5720 . . . . . 6  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
7 djaval.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
86, 7syl6eqr 2485 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
98pweqd 3796 . . . 4  |-  ( w  =  W  ->  ~P ( ( LTrn `  K
) `  w )  =  ~P T )
10 fveq2 5720 . . . . . 6  |-  ( w  =  W  ->  (
( ocA `  K
) `  w )  =  ( ( ocA `  K ) `  W
) )
11 djaval.n . . . . . 6  |-  ._|_  =  ( ( ocA `  K
) `  W )
1210, 11syl6eqr 2485 . . . . 5  |-  ( w  =  W  ->  (
( ocA `  K
) `  w )  =  ._|_  )
1312fveq1d 5722 . . . . . 6  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  x )  =  ( 
._|_  `  x ) )
1412fveq1d 5722 . . . . . 6  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  y )  =  ( 
._|_  `  y ) )
1513, 14ineq12d 3535 . . . . 5  |-  ( w  =  W  ->  (
( ( ( ocA `  K ) `  w
) `  x )  i^i  ( ( ( ocA `  K ) `  w
) `  y )
)  =  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) )
1612, 15fveq12d 5726 . . . 4  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  ( ( ( ( ocA `  K ) `
 w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) )  =  (  ._|_  `  (
(  ._|_  `  x )  i^i  (  ._|_  `  y
) ) ) )
179, 9, 16mpt2eq123dv 6128 . . 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 2435 . . 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 5734 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
207, 19eqeltri 2505 . . . . 5  |-  T  e. 
_V
2120pwex 4374 . . . 4  |-  ~P T  e.  _V
2221, 21mpt2ex 6417 . . 3  |-  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) )  e. 
_V
2317, 18, 22fvmpt 5798 . 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 2487 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  J  =  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311   ~Pcpw 3791    e. cmpt 4258   ` cfv 5446    e. cmpt2 6075   LHypclh 30718   LTrncltrn 30835   DIsoAcdia 31763   ocAcocaN 31854   vAcdjaN 31866
This theorem is referenced by:  djavalN  31870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-djaN 31867
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