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Metamath Proof Explorer

Theorem djhcom

Description: Subspace join commutes. (Contributed by NM, 8-Aug-2014)

Ref Expression
Hypotheses djhcom.h 
|- H = ( LHyp ` K )
djhcom.u 
|- U = ( ( DVecH ` K ) ` W )
djhcom.v 
|- V = ( Base ` U )
djhcom.j 
|- .\/ = ( ( joinH ` K ) ` W )
djhcom.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
djhcom.x 
|- ( ph -> X C_ V )
djhcom.y 
|- ( ph -> Y C_ V )
Assertion djhcom 
|- ( ph -> ( X .\/ Y ) = ( Y .\/ X ) )

Proof

Step Hyp Ref Expression
1 djhcom.h 
 |-  H = ( LHyp ` K )
2 djhcom.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 djhcom.v 
 |-  V = ( Base ` U )
4 djhcom.j 
 |-  .\/ = ( ( joinH ` K ) ` W )
5 djhcom.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
6 djhcom.x 
 |-  ( ph -> X C_ V )
7 djhcom.y 
 |-  ( ph -> Y C_ V )
8 uncom 
 |-  ( X u. Y ) = ( Y u. X )
9 8 fveq2i 
 |-  ( ( ( ocH ` K ) ` W ) ` ( X u. Y ) ) = ( ( ( ocH ` K ) ` W ) ` ( Y u. X ) )
10 9 fveq2i 
 |-  ( ( ( ocH ` K ) ` W ) ` ( ( ( ocH ` K ) ` W ) ` ( X u. Y ) ) ) = ( ( ( ocH ` K ) ` W ) ` ( ( ( ocH ` K ) ` W ) ` ( Y u. X ) ) )
11 eqid 
 |-  ( ( ocH ` K ) ` W ) = ( ( ocH ` K ) ` W )
12 1 2 3 11 4 djhval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V /\ Y C_ V ) -> ( X .\/ Y ) = ( ( ( ocH ` K ) ` W ) ` ( ( ( ocH ` K ) ` W ) ` ( X u. Y ) ) ) )
13 5 6 7 12 syl3anc 
 |-  ( ph -> ( X .\/ Y ) = ( ( ( ocH ` K ) ` W ) ` ( ( ( ocH ` K ) ` W ) ` ( X u. Y ) ) ) )
14 1 2 3 11 4 djhval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ V ) -> ( Y .\/ X ) = ( ( ( ocH ` K ) ` W ) ` ( ( ( ocH ` K ) ` W ) ` ( Y u. X ) ) ) )
15 5 7 6 14 syl3anc 
 |-  ( ph -> ( Y .\/ X ) = ( ( ( ocH ` K ) ` W ) ` ( ( ( ocH ` K ) ` W ) ` ( Y u. X ) ) ) )
16 10 13 15 3eqtr4a 
 |-  ( ph -> ( X .\/ Y ) = ( Y .\/ X ) )