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Metamath Proof Explorer

Theorem djaffvalN

Description: Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypothesis djaval.h 
|- H = ( LHyp ` K )
Assertion 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 ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 djaval.h 
 |-  H = ( LHyp ` K )
2 elex 
 |-  ( K e. V -> K e. _V )
3 fveq2 
 |-  ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
4 3 1 eqtr4di 
 |-  ( k = K -> ( LHyp ` k ) = H )
5 fveq2 
 |-  ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
6 5 fveq1d 
 |-  ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
7 6 pweqd 
 |-  ( k = K -> ~P ( ( LTrn ` k ) ` w ) = ~P ( ( LTrn ` K ) ` w ) )
8 fveq2 
 |-  ( k = K -> ( ocA ` k ) = ( ocA ` K ) )
9 8 fveq1d 
 |-  ( k = K -> ( ( ocA ` k ) ` w ) = ( ( ocA ` K ) ` w ) )
10 9 fveq1d 
 |-  ( k = K -> ( ( ( ocA ` k ) ` w ) ` x ) = ( ( ( ocA ` K ) ` w ) ` x ) )
11 9 fveq1d 
 |-  ( k = K -> ( ( ( ocA ` k ) ` w ) ` y ) = ( ( ( ocA ` K ) ` w ) ` y ) )
12 10 11 ineq12d 
 |-  ( k = K -> ( ( ( ( ocA ` k ) ` w ) ` x ) i^i ( ( ( ocA ` k ) ` w ) ` y ) ) = ( ( ( ( ocA ` K ) ` w ) ` x ) i^i ( ( ( ocA ` K ) ` w ) ` y ) ) )
13 9 12 fveq12d 
 |-  ( 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 ) ) ) )
14 7 7 13 mpoeq123dv 
 |-  ( 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 ) ) ) ) )
15 4 14 mpteq12dv 
 |-  ( 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 
 |-  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 15 16 1 mptfvmpt 
 |-  ( 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 ) ) ) ) ) )
18 2 17 syl 
 |-  ( 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 ) ) ) ) ) )