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Metamath Proof Explorer

Theorem djaclN

Description: Closure of subspace join for DVecA partial vector space. (Contributed by NM, 5-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses djacl.h 
|- H = ( LHyp ` K )
djacl.t 
|- T = ( ( LTrn ` K ) ` W )
djacl.i 
|- I = ( ( DIsoA ` K ) ` W )
djacl.j 
|- J = ( ( vA ` K ) ` W )
Assertion djaclN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( X C_ T /\ Y C_ T ) ) -> ( X J Y ) e. ran I )

Proof

Step Hyp Ref Expression
1 djacl.h 
 |-  H = ( LHyp ` K )
2 djacl.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 djacl.i 
 |-  I = ( ( DIsoA ` K ) ` W )
4 djacl.j 
 |-  J = ( ( vA ` K ) ` W )
5 eqid 
 |-  ( ( ocA ` K ) ` W ) = ( ( ocA ` K ) ` W )
6 1 2 3 5 4 djavalN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X C_ T /\ Y C_ T ) ) -> ( X J Y ) = ( ( ( ocA ` K ) ` W ) ` ( ( ( ( ocA ` K ) ` W ) ` X ) i^i ( ( ( ocA ` K ) ` W ) ` Y ) ) ) )
7 inss1 
 |-  ( ( ( ( ocA ` K ) ` W ) ` X ) i^i ( ( ( ocA ` K ) ` W ) ` Y ) ) C_ ( ( ( ocA ` K ) ` W ) ` X )
8 1 2 3 5 docaclN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ( ( ocA ` K ) ` W ) ` X ) e. ran I )
9 8 adantrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X C_ T /\ Y C_ T ) ) -> ( ( ( ocA ` K ) ` W ) ` X ) e. ran I )
10 1 2 3 diaelrnN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( ( ocA ` K ) ` W ) ` X ) e. ran I ) -> ( ( ( ocA ` K ) ` W ) ` X ) C_ T )
11 9 10 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X C_ T /\ Y C_ T ) ) -> ( ( ( ocA ` K ) ` W ) ` X ) C_ T )
12 7 11 sstrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X C_ T /\ Y C_ T ) ) -> ( ( ( ( ocA ` K ) ` W ) ` X ) i^i ( ( ( ocA ` K ) ` W ) ` Y ) ) C_ T )
13 1 2 3 5 docaclN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( ( ( ocA ` K ) ` W ) ` X ) i^i ( ( ( ocA ` K ) ` W ) ` Y ) ) C_ T ) -> ( ( ( ocA ` K ) ` W ) ` ( ( ( ( ocA ` K ) ` W ) ` X ) i^i ( ( ( ocA ` K ) ` W ) ` Y ) ) ) e. ran I )
14 12 13 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X C_ T /\ Y C_ T ) ) -> ( ( ( ocA ` K ) ` W ) ` ( ( ( ( ocA ` K ) ` W ) ` X ) i^i ( ( ( ocA ` K ) ` W ) ` Y ) ) ) e. ran I )
15 6 14 eqeltrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X C_ T /\ Y C_ T ) ) -> ( X J Y ) e. ran I )