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Theorem djaffvalN 34778
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 2981 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5691 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 djaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2493 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5691 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5693 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76pweqd 3865 . . . . 5  |-  ( k  =  K  ->  ~P ( ( LTrn `  k
) `  w )  =  ~P ( ( LTrn `  K ) `  w
) )
8 fveq2 5691 . . . . . . 7  |-  ( k  =  K  ->  ( ocA `  k )  =  ( ocA `  K
) )
98fveq1d 5693 . . . . . 6  |-  ( k  =  K  ->  (
( ocA `  k
) `  w )  =  ( ( ocA `  K ) `  w
) )
109fveq1d 5693 . . . . . . 7  |-  ( k  =  K  ->  (
( ( ocA `  k
) `  w ) `  x )  =  ( ( ( ocA `  K
) `  w ) `  x ) )
119fveq1d 5693 . . . . . . 7  |-  ( k  =  K  ->  (
( ( ocA `  k
) `  w ) `  y )  =  ( ( ( ocA `  K
) `  w ) `  y ) )
1210, 11ineq12d 3553 . . . . . 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 5697 . . . . 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 6148 . . . 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 4370 . . 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 34777 . . 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 5701 . . . . 5  |-  ( LHyp `  K )  e.  _V
183, 17eqeltri 2513 . . . 4  |-  H  e. 
_V
1918mptex 5948 . . 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 5774 . 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 1369    e. wcel 1756   _Vcvv 2972    i^i cin 3327   ~Pcpw 3860    e. cmpt 4350   ` cfv 5418    e. cmpt2 6093   LHypclh 33628   LTrncltrn 33745   ocAcocaN 34764   vAcdjaN 34776
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-oprab 6095  df-mpt2 6096  df-djaN 34777
This theorem is referenced by:  djafvalN  34779
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