Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  djafvalN Unicode version

Theorem djafvalN 31251
Description: Subspace join for  DVecA partial vector space. TODO: take out hypothesis .i, no longer used. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
djaval.h  |-  H  =  ( LHyp `  K
)
djaval.t  |-  T  =  ( ( LTrn `  K
) `  W )
djaval.i  |-  I  =  ( ( DIsoA `  K
) `  W )
djaval.n  |-  ._|_  =  ( ( ocA `  K
) `  W )
djaval.j  |-  J  =  ( ( vA `  K ) `  W
)
Assertion
Ref Expression
djafvalN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  J  =  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) ) )
Distinct variable groups:    x, y, K    x, T, y    x, W, y
Allowed substitution hints:    H( x, y)    I( x, y)    J( x, y)    ._|_ ( x, y)    V( x, y)

Proof of Theorem djafvalN
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 djaval.j . . 3  |-  J  =  ( ( vA `  K ) `  W
)
2 djaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
32djaffvalN 31250 . . . 4  |-  ( 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 ) ) ) ) ) )
43fveq1d 5672 . . 3  |-  ( K  e.  V  ->  (
( vA `  K
) `  W )  =  ( ( 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 ) ) ) ) ) `  W
) )
51, 4syl5eq 2433 . 2  |-  ( K  e.  V  ->  J  =  ( ( 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 ) ) ) ) ) `  W
) )
6 fveq2 5670 . . . . . 6  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
7 djaval.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
86, 7syl6eqr 2439 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
98pweqd 3749 . . . 4  |-  ( w  =  W  ->  ~P ( ( LTrn `  K
) `  w )  =  ~P T )
10 fveq2 5670 . . . . . 6  |-  ( w  =  W  ->  (
( ocA `  K
) `  w )  =  ( ( ocA `  K ) `  W
) )
11 djaval.n . . . . . 6  |-  ._|_  =  ( ( ocA `  K
) `  W )
1210, 11syl6eqr 2439 . . . . 5  |-  ( w  =  W  ->  (
( ocA `  K
) `  w )  =  ._|_  )
1312fveq1d 5672 . . . . . 6  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  x )  =  ( 
._|_  `  x ) )
1412fveq1d 5672 . . . . . 6  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  y )  =  ( 
._|_  `  y ) )
1513, 14ineq12d 3488 . . . . 5  |-  ( w  =  W  ->  (
( ( ( ocA `  K ) `  w
) `  x )  i^i  ( ( ( ocA `  K ) `  w
) `  y )
)  =  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) )
1612, 15fveq12d 5676 . . . 4  |-  ( w  =  W  ->  (
( ( ocA `  K
) `  w ) `  ( ( ( ( ocA `  K ) `
 w ) `  x )  i^i  (
( ( ocA `  K
) `  w ) `  y ) ) )  =  (  ._|_  `  (
(  ._|_  `  x )  i^i  (  ._|_  `  y
) ) ) )
179, 9, 16mpt2eq123dv 6077 . . 3  |-  ( w  =  W  ->  (
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 T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) ) )
18 eqid 2389 . . 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 ) ) ) ) )  =  ( 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 ) ) ) ) )
19 fvex 5684 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
207, 19eqeltri 2459 . . . . 5  |-  T  e. 
_V
2120pwex 4325 . . . 4  |-  ~P T  e.  _V
2221, 21mpt2ex 6366 . . 3  |-  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) )  e. 
_V
2317, 18, 22fvmpt 5747 . 2  |-  ( W  e.  H  ->  (
( 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 ) ) ) ) ) `  W
)  =  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) ) )
245, 23sylan9eq 2441 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  J  =  ( x  e.  ~P T , 
y  e.  ~P T  |->  (  ._|_  `  ( ( 
._|_  `  x )  i^i  (  ._|_  `  y ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901    i^i cin 3264   ~Pcpw 3744    e. cmpt 4209   ` cfv 5396    e. cmpt2 6024   LHypclh 30100   LTrncltrn 30217   DIsoAcdia 31145   ocAcocaN 31236   vAcdjaN 31248
This theorem is referenced by:  djavalN  31252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-djaN 31249
  Copyright terms: Public domain W3C validator