Metamath Proof Explorer

Theorem mapdincl

Description: Closure of dual subspace intersection for the map defined by df-mapd . (Contributed by NM, 12-Apr-2015)

Ref Expression
Hypotheses mapdincl.h 
|- H = ( LHyp ` K )
mapdincl.m 
|- M = ( ( mapd ` K ) ` W )
mapdincl.u 
|- U = ( ( DVecH ` K ) ` W )
mapdincl.c 
|- C = ( ( LCDual ` K ) ` W )
mapdincl.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdincl.x 
|- ( ph -> X e. ran M )
mapdincl.y 
|- ( ph -> Y e. ran M )
Assertion mapdincl 
|- ( ph -> ( X i^i Y ) e. ran M )

Proof

Step Hyp Ref Expression
1 mapdincl.h 
 |-  H = ( LHyp ` K )
2 mapdincl.m 
 |-  M = ( ( mapd ` K ) ` W )
3 mapdincl.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 mapdincl.c 
 |-  C = ( ( LCDual ` K ) ` W )
5 mapdincl.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
6 mapdincl.x 
 |-  ( ph -> X e. ran M )
7 mapdincl.y 
 |-  ( ph -> Y e. ran M )
8 eqid 
 |-  ( ( LCDual ` K ) ` W ) = ( ( LCDual ` K ) ` W )
9 1 8 5 lcdlmod 
 |-  ( ph -> ( ( LCDual ` K ) ` W ) e. LMod )
10 eqid 
 |-  ( LSubSp ` ( ( LCDual ` K ) ` W ) ) = ( LSubSp ` ( ( LCDual ` K ) ` W ) )
11 1 2 8 10 5 mapdrn2 
 |-  ( ph -> ran M = ( LSubSp ` ( ( LCDual ` K ) ` W ) ) )
12 6 11 eleqtrd 
 |-  ( ph -> X e. ( LSubSp ` ( ( LCDual ` K ) ` W ) ) )
13 7 11 eleqtrd 
 |-  ( ph -> Y e. ( LSubSp ` ( ( LCDual ` K ) ` W ) ) )
14 10 lssincl 
 |-  ( ( ( ( LCDual ` K ) ` W ) e. LMod /\ X e. ( LSubSp ` ( ( LCDual ` K ) ` W ) ) /\ Y e. ( LSubSp ` ( ( LCDual ` K ) ` W ) ) ) -> ( X i^i Y ) e. ( LSubSp ` ( ( LCDual ` K ) ` W ) ) )
15 9 12 13 14 syl3anc 
 |-  ( ph -> ( X i^i Y ) e. ( LSubSp ` ( ( LCDual ` K ) ` W ) ) )
16 15 11 eleqtrrd 
 |-  ( ph -> ( X i^i Y ) e. ran M )