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Metamath Proof Explorer

Theorem djhval2

Description: Value of subspace join for DVecH vector space. (Contributed by NM, 6-Aug-2014)

Ref Expression
Hypotheses djhval.h 
|- H = ( LHyp ` K )
djhval.u 
|- U = ( ( DVecH ` K ) ` W )
djhval.v 
|- V = ( Base ` U )
djhval.o 
|- ._|_ = ( ( ocH ` K ) ` W )
djhval.j 
|- .\/ = ( ( joinH ` K ) ` W )
Assertion djhval2 
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ V /\ Y C_ V ) -> ( X .\/ Y ) = ( ._|_ ` ( ._|_ ` ( X u. Y ) ) ) )

Proof

Step Hyp Ref Expression
1 djhval.h 
 |-  H = ( LHyp ` K )
2 djhval.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 djhval.v 
 |-  V = ( Base ` U )
4 djhval.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
5 djhval.j 
 |-  .\/ = ( ( joinH ` K ) ` W )
6 1 2 3 4 5 djhval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X C_ V /\ Y C_ V ) ) -> ( X .\/ Y ) = ( ._|_ ` ( ( ._|_ ` X ) i^i ( ._|_ ` Y ) ) ) )
7 6 3impb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V /\ Y C_ V ) -> ( X .\/ Y ) = ( ._|_ ` ( ( ._|_ ` X ) i^i ( ._|_ ` Y ) ) ) )
8 1 2 3 4 dochdmj1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V /\ Y C_ V ) -> ( ._|_ ` ( X u. Y ) ) = ( ( ._|_ ` X ) i^i ( ._|_ ` Y ) ) )
9 8 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V /\ Y C_ V ) -> ( ._|_ ` ( ._|_ ` ( X u. Y ) ) ) = ( ._|_ ` ( ( ._|_ ` X ) i^i ( ._|_ ` Y ) ) ) )
10 7 9 eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V /\ Y C_ V ) -> ( X .\/ Y ) = ( ._|_ ` ( ._|_ ` ( X u. Y ) ) ) )