Metamath Proof Explorer

Theorem dihmeetcl

Description: Closure of closed subspace meet for DVecH vector space. (Contributed by NM, 5-Aug-2014)

Ref Expression
Hypotheses dihmeetcl.h 
|- H = ( LHyp ` K )
dihmeetcl.i 
|- I = ( ( DIsoH ` K ) ` W )
Assertion dihmeetcl 
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( X i^i Y ) e. ran I )

Proof

Step Hyp Ref Expression
1 dihmeetcl.h 
 |-  H = ( LHyp ` K )
2 dihmeetcl.i 
 |-  I = ( ( DIsoH ` K ) ` W )
3 1 2 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )
4 3 adantrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( I ` ( `' I ` X ) ) = X )
5 1 2 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. ran I ) -> ( I ` ( `' I ` Y ) ) = Y )
6 5 adantrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( I ` ( `' I ` Y ) ) = Y )
7 4 6 ineq12d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( ( I ` ( `' I ` X ) ) i^i ( I ` ( `' I ` Y ) ) ) = ( X i^i Y ) )
8 simpl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( K e. HL /\ W e. H ) )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 9 1 2 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` X ) e. ( Base ` K ) )
11 10 adantrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( `' I ` X ) e. ( Base ` K ) )
12 9 1 2 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. ran I ) -> ( `' I ` Y ) e. ( Base ` K ) )
13 12 adantrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( `' I ` Y ) e. ( Base ` K ) )
14 eqid 
 |-  ( meet ` K ) = ( meet ` K )
15 9 14 1 2 dihmeet 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( `' I ` X ) e. ( Base ` K ) /\ ( `' I ` Y ) e. ( Base ` K ) ) -> ( I ` ( ( `' I ` X ) ( meet ` K ) ( `' I ` Y ) ) ) = ( ( I ` ( `' I ` X ) ) i^i ( I ` ( `' I ` Y ) ) ) )
16 8 11 13 15 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( I ` ( ( `' I ` X ) ( meet ` K ) ( `' I ` Y ) ) ) = ( ( I ` ( `' I ` X ) ) i^i ( I ` ( `' I ` Y ) ) ) )
17 hllat 
 |-  ( K e. HL -> K e. Lat )
18 17 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> K e. Lat )
19 9 14 latmcl 
 |-  ( ( K e. Lat /\ ( `' I ` X ) e. ( Base ` K ) /\ ( `' I ` Y ) e. ( Base ` K ) ) -> ( ( `' I ` X ) ( meet ` K ) ( `' I ` Y ) ) e. ( Base ` K ) )
20 18 11 13 19 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( ( `' I ` X ) ( meet ` K ) ( `' I ` Y ) ) e. ( Base ` K ) )
21 9 1 2 dihcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( `' I ` X ) ( meet ` K ) ( `' I ` Y ) ) e. ( Base ` K ) ) -> ( I ` ( ( `' I ` X ) ( meet ` K ) ( `' I ` Y ) ) ) e. ran I )
22 20 21 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( I ` ( ( `' I ` X ) ( meet ` K ) ( `' I ` Y ) ) ) e. ran I )
23 16 22 eqeltrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( ( I ` ( `' I ` X ) ) i^i ( I ` ( `' I ` Y ) ) ) e. ran I )
24 7 23 eqeltrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( X i^i Y ) e. ran I )