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Theorem djaffvalN 31945
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 2809 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5541 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 djaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2346 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5543 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76pweqd 3643 . . . . 5  |-  ( k  =  K  ->  ~P ( ( LTrn `  k
) `  w )  =  ~P ( ( LTrn `  K ) `  w
) )
8 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( ocA `  k )  =  ( ocA `  K
) )
98fveq1d 5543 . . . . . 6  |-  ( k  =  K  ->  (
( ocA `  k
) `  w )  =  ( ( ocA `  K ) `  w
) )
109fveq1d 5543 . . . . . . 7  |-  ( k  =  K  ->  (
( ( ocA `  k
) `  w ) `  x )  =  ( ( ( ocA `  K
) `  w ) `  x ) )
119fveq1d 5543 . . . . . . 7  |-  ( k  =  K  ->  (
( ( ocA `  k
) `  w ) `  y )  =  ( ( ( ocA `  K
) `  w ) `  y ) )
1210, 11ineq12d 3384 . . . . . 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 5547 . . . . 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 5926 . . . 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 4114 . . 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 31944 . . 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 5555 . . . . 5  |-  ( LHyp `  K )  e.  _V
183, 17eqeltri 2366 . . . 4  |-  H  e. 
_V
1918mptex 5762 . . 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 5618 . 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 15 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 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   ~Pcpw 3638    e. cmpt 4093   ` cfv 5271    e. cmpt2 5876   LHypclh 30795   LTrncltrn 30912   ocAcocaN 31931   vAcdjaN 31943
This theorem is referenced by:  djafvalN  31946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-djaN 31944
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