This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dihlss

Description: The value of isomorphism H is a subspace. (Contributed by NM, 6-Mar-2014)

Ref Expression
Hypotheses dihlss.b 
|- B = ( Base ` K )
dihlss.h 
|- H = ( LHyp ` K )
dihlss.i 
|- I = ( ( DIsoH ` K ) ` W )
dihlss.u 
|- U = ( ( DVecH ` K ) ` W )
dihlss.s 
|- S = ( LSubSp ` U )
Assertion dihlss 
|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( I ` X ) e. S )

Proof

Step Hyp Ref Expression
1 dihlss.b 
 |-  B = ( Base ` K )
2 dihlss.h 
 |-  H = ( LHyp ` K )
3 dihlss.i 
 |-  I = ( ( DIsoH ` K ) ` W )
4 dihlss.u 
 |-  U = ( ( DVecH ` K ) ` W )
5 dihlss.s 
 |-  S = ( LSubSp ` U )
6 eqid 
 |-  ( le ` K ) = ( le ` K )
7 eqid 
 |-  ( ( DIsoB ` K ) ` W ) = ( ( DIsoB ` K ) ` W )
8 1 6 2 3 7 dihvalb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X ( le ` K ) W ) ) -> ( I ` X ) = ( ( ( DIsoB ` K ) ` W ) ` X ) )
9 1 6 2 4 7 5 diblss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X ( le ` K ) W ) ) -> ( ( ( DIsoB ` K ) ` W ) ` X ) e. S )
10 8 9 eqeltrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X ( le ` K ) W ) ) -> ( I ` X ) e. S )
11 10 anassrs 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> ( I ` X ) e. S )
12 eqid 
 |-  ( join ` K ) = ( join ` K )
13 eqid 
 |-  ( meet ` K ) = ( meet ` K )
14 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
15 eqid 
 |-  ( ( DIsoC ` K ) ` W ) = ( ( DIsoC ` K ) ` W )
16 eqid 
 |-  ( LSSum ` U ) = ( LSSum ` U )
17 1 6 12 13 14 2 3 7 15 4 5 16 dihlsscpre 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X ( le ` K ) W ) ) -> ( I ` X ) e. S )
18 17 anassrs 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ -. X ( le ` K ) W ) -> ( I ` X ) e. S )
19 11 18 pm2.61dan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( I ` X ) e. S )