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Theorem djaffvalN 36331
Description: Subspace join for  DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
djaval.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
djaffvalN  |-  ( 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 ) ) ) ) ) )
Distinct variable groups:    w, H    x, w, y, K
Allowed substitution hints:    H( x, y)    V( x, y, w)

Proof of Theorem djaffvalN
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 3127 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5872 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 djaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2526 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5872 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5874 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76pweqd 4021 . . . . 5  |-  ( k  =  K  ->  ~P ( ( LTrn `  k
) `  w )  =  ~P ( ( LTrn `  K ) `  w
) )
8 fveq2 5872 . . . . . . 7  |-  ( k  =  K  ->  ( ocA `  k )  =  ( ocA `  K
) )
98fveq1d 5874 . . . . . 6  |-  ( k  =  K  ->  (
( ocA `  k
) `  w )  =  ( ( ocA `  K ) `  w
) )
109fveq1d 5874 . . . . . . 7  |-  ( k  =  K  ->  (
( ( ocA `  k
) `  w ) `  x )  =  ( ( ( ocA `  K
) `  w ) `  x ) )
119fveq1d 5874 . . . . . . 7  |-  ( k  =  K  ->  (
( ( ocA `  k
) `  w ) `  y )  =  ( ( ( ocA `  K
) `  w ) `  y ) )
1210, 11ineq12d 3706 . . . . . 6  |-  ( k  =  K  ->  (
( ( ( ocA `  k ) `  w
) `  x )  i^i  ( ( ( ocA `  k ) `  w
) `  y )
)  =  ( ( ( ( ocA `  K
) `  w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) )
139, 12fveq12d 5878 . . . . 5  |-  ( k  =  K  ->  (
( ( ocA `  k
) `  w ) `  ( ( ( ( ocA `  k ) `
 w ) `  x )  i^i  (
( ( ocA `  k
) `  w ) `  y ) ) )  =  ( ( ( ocA `  K ) `
 w ) `  ( ( ( ( ocA `  K ) `
 w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) ) )
147, 7, 13mpt2eq123dv 6354 . . . 4  |-  ( k  =  K  ->  (
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 ( (
LTrn `  K ) `  w ) ,  y  e.  ~P ( (
LTrn `  K ) `  w )  |->  ( ( ( ocA `  K
) `  w ) `  ( ( ( ( ocA `  K ) `
 w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) ) ) )
154, 14mpteq12dv 4531 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( 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 ) ) ) ) ) )
16 df-djaN 36330 . . 3  |-  vA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( 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 ) ) ) ) ) )
17 fvex 5882 . . . . 5  |-  ( LHyp `  K )  e.  _V
183, 17eqeltri 2551 . . . 4  |-  H  e. 
_V
1918mptex 6142 . . 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 ) ) ) ) )  e.  _V
2015, 16, 19fvmpt 5957 . 2  |-  ( 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 ) ) ) ) ) )
211, 20syl 16 1  |-  ( 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 ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480   ~Pcpw 4016    |-> cmpt 4511   ` cfv 5594    |-> cmpt2 6297   LHypclh 35181   LTrncltrn 35298   ocAcocaN 36317   vAcdjaN 36329
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 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
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 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-oprab 6299  df-mpt2 6300  df-djaN 36330
This theorem is referenced by:  djafvalN  36332
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