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Theorem djaffvalN 31932
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 2965 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5729 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 djaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2487 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5729 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5731 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76pweqd 3805 . . . . 5  |-  ( k  =  K  ->  ~P ( ( LTrn `  k
) `  w )  =  ~P ( ( LTrn `  K ) `  w
) )
8 fveq2 5729 . . . . . . 7  |-  ( k  =  K  ->  ( ocA `  k )  =  ( ocA `  K
) )
98fveq1d 5731 . . . . . 6  |-  ( k  =  K  ->  (
( ocA `  k
) `  w )  =  ( ( ocA `  K ) `  w
) )
109fveq1d 5731 . . . . . . 7  |-  ( k  =  K  ->  (
( ( ocA `  k
) `  w ) `  x )  =  ( ( ( ocA `  K
) `  w ) `  x ) )
119fveq1d 5731 . . . . . . 7  |-  ( k  =  K  ->  (
( ( ocA `  k
) `  w ) `  y )  =  ( ( ( ocA `  K
) `  w ) `  y ) )
1210, 11ineq12d 3544 . . . . . 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 5735 . . . . 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 6137 . . . 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 4288 . . 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 31931 . . 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 5743 . . . . 5  |-  ( LHyp `  K )  e.  _V
183, 17eqeltri 2507 . . . 4  |-  H  e. 
_V
1918mptex 5967 . . 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 5807 . 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 set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2957    i^i cin 3320   ~Pcpw 3800    e. cmpt 4267   ` cfv 5455    e. cmpt2 6084   LHypclh 30782   LTrncltrn 30899   ocAcocaN 31918   vAcdjaN 31930
This theorem is referenced by:  djafvalN  31933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-oprab 6086  df-mpt2 6087  df-djaN 31931
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