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Theorem djaffvalN 31323
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 2796 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5525 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 djaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2333 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5527 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76pweqd 3630 . . . . 5  |-  ( k  =  K  ->  ~P ( ( LTrn `  k
) `  w )  =  ~P ( ( LTrn `  K ) `  w
) )
8 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( ocA `  k )  =  ( ocA `  K
) )
98fveq1d 5527 . . . . . 6  |-  ( k  =  K  ->  (
( ocA `  k
) `  w )  =  ( ( ocA `  K ) `  w
) )
109fveq1d 5527 . . . . . . 7  |-  ( k  =  K  ->  (
( ( ocA `  k
) `  w ) `  x )  =  ( ( ( ocA `  K
) `  w ) `  x ) )
119fveq1d 5527 . . . . . . 7  |-  ( k  =  K  ->  (
( ( ocA `  k
) `  w ) `  y )  =  ( ( ( ocA `  K
) `  w ) `  y ) )
1210, 11ineq12d 3371 . . . . . 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 5531 . . . . 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 5910 . . . 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 4098 . . 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 31322 . . 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 5539 . . . . 5  |-  ( LHyp `  K )  e.  _V
183, 17eqeltri 2353 . . . 4  |-  H  e. 
_V
1918mptex 5746 . . 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 5602 . 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 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   ~Pcpw 3625    e. cmpt 4077   ` cfv 5255    e. cmpt2 5860   LHypclh 30173   LTrncltrn 30290   ocAcocaN 31309   vAcdjaN 31321
This theorem is referenced by:  djafvalN  31324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-djaN 31322
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