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Metamath Proof Explorer

Theorem djhjlj

Description: DVecH vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014)

Ref Expression
Hypotheses djhj.k 
|- .\/ = ( join ` K )
djhj.h 
|- H = ( LHyp ` K )
djhj.i 
|- I = ( ( DIsoH ` K ) ` W )
djhj.j 
|- J = ( ( joinH ` K ) ` W )
djhj.w 
|- ( ph -> ( K e. HL /\ W e. H ) )
djhj.x 
|- ( ph -> X e. ran I )
djhj.y 
|- ( ph -> Y e. ran I )
Assertion djhjlj 
|- ( ph -> ( X J Y ) = ( I ` ( ( `' I ` X ) .\/ ( `' I ` Y ) ) ) )

Proof

Step Hyp Ref Expression
1 djhj.k 
 |-  .\/ = ( join ` K )
2 djhj.h 
 |-  H = ( LHyp ` K )
3 djhj.i 
 |-  I = ( ( DIsoH ` K ) ` W )
4 djhj.j 
 |-  J = ( ( joinH ` K ) ` W )
5 djhj.w 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
6 djhj.x 
 |-  ( ph -> X e. ran I )
7 djhj.y 
 |-  ( ph -> Y e. ran I )
8 eqid 
 |-  ( Base ` K ) = ( Base ` K )
9 8 2 3 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` X ) e. ( Base ` K ) )
10 5 6 9 syl2anc 
 |-  ( ph -> ( `' I ` X ) e. ( Base ` K ) )
11 8 2 3 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. ran I ) -> ( `' I ` Y ) e. ( Base ` K ) )
12 5 7 11 syl2anc 
 |-  ( ph -> ( `' I ` Y ) e. ( Base ` K ) )
13 8 1 2 3 4 djhlj 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( `' I ` X ) e. ( Base ` K ) /\ ( `' I ` Y ) e. ( Base ` K ) ) ) -> ( I ` ( ( `' I ` X ) .\/ ( `' I ` Y ) ) ) = ( ( I ` ( `' I ` X ) ) J ( I ` ( `' I ` Y ) ) ) )
14 5 10 12 13 syl12anc 
 |-  ( ph -> ( I ` ( ( `' I ` X ) .\/ ( `' I ` Y ) ) ) = ( ( I ` ( `' I ` X ) ) J ( I ` ( `' I ` Y ) ) ) )
15 2 3 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )
16 5 6 15 syl2anc 
 |-  ( ph -> ( I ` ( `' I ` X ) ) = X )
17 2 3 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. ran I ) -> ( I ` ( `' I ` Y ) ) = Y )
18 5 7 17 syl2anc 
 |-  ( ph -> ( I ` ( `' I ` Y ) ) = Y )
19 16 18 oveq12d 
 |-  ( ph -> ( ( I ` ( `' I ` X ) ) J ( I ` ( `' I ` Y ) ) ) = ( X J Y ) )
20 14 19 eqtr2d 
 |-  ( ph -> ( X J Y ) = ( I ` ( ( `' I ` X ) .\/ ( `' I ` Y ) ) ) )